Forest dynamics at regional scales: predictive models constrained with inventory data Emily Rebecca Lines Downing College University of Cambridge Dissertation submitted to the University of Cambridge for the Degree of Doctor of Philosophy This dissertation is my own work and contains nothing which is the outcome of work done in collaboration with others, except as specified in the text. For Denis and Margaret Oates, who inspire me. Acknowledgements I would like to thank my two supervisors, David Coomes and Drew Purves, for their unending and invaluable help and support during my PhD, and without whom this research would not exist. David, you taught me pretty much everything I know about trees (which you may or may not see as a compliment), and without your advice I would still be floundering in the world of kerplunk modelling. Drew, you taught me that enjoying maths and computing, and producing valuable, applicable ecological research are not mutually exclusive, and that the environment needs mathematicians! My sincere thanks also go to my collaborators in Spain: Miguel Zavala, whose insights and perspectives have enriched my knowledge of Mediterranean ecology, and Paloma Ruiz-Benito, whose patience and good humour enabled me to understand and use the Spanish forest inventory. Thank you to Microsoft Research for funding and supporting this project. The many members and visitors of the Forest Ecology Group who have passed through during my time here have all helped me out and broadened my understanding of ecology, so thank you to you all: David Wright, Raffaele Lafortezza, Olivier Flores, Robbie Holdaway, Andrew Tanentzap, Laura Spence, Mindy Syfert, Irantzu Primicia, Andres Plaza-Aguilar, Pieter De Frenne, Rebecca Spriggs, Tommaso Jucker and Beatrice Wedeux. Special thanks to Will Simonson and Paco Rodriguez-Sanchez for helping me out with my Spanish ecology, and to Beccy Wilebore and Jessica Royles (though not strictly a group member), frankly for putting up with me. If I've forgotten anyone, I'm sorry, and thank you too. Thanks to my family: my Mum, Dad and sister Kate, for your understanding and support. Hannah Slater, sharing this experience with you has made it so much easier! Finally, the list of things I have to thank Christopher Bosworth for is endless, so thank you for them all. Summary Forest ecosystems store more carbon than the atmosphere and harbour the majority of the world's biodiversity, yet their response to changing climate is uncertain. Forest simulation models make landscape-level predictions of forest dynamics by scaling from key tree-level processes, but models typically have no climate dependency. In this thesis I demonstrate how large-scale national inventories combined with improvements in computational methods mean that models that incorporate the climate dependency of demographic processes may be parameterised at regional scales. In Chapter One I outline historical approaches to modelling forest dynamics and present a discussion of competing methods of parameterisation and model selection. In Chapter Two I present a model of individual tree mortality in the eastern United States which incorporates species, climatic and competitive effects parameterised using Markov Chain Monte Carlo methods. The remainder of the thesis concentrates on modelling Spanish forest dynamics, so in Chapter Three I present a brief introduction to Spanish forest ecology. In Chapter Four I examine how aboveground allometry - the scaling of tree height and crown shape - varies with climate and competition in Spain for 26 species. Hierarchical modelling suggests that scaling theories based on wood properties do not explain differences between species, but climatic factors, and in particular hydraulic limitations, do. In Chapter Five I parameterise a model of recruitment in Spanish forests using Approximate Bayesian Computation, a novel computational method which allows parameterisation of individual-based models without individual-based data, and demonstrate that it produces ecologically reasonable results. Chapter Six presents a forest dynamics model parameterised for the major native species in Spain and tests whether it is able to reproduce observed species-climate distributions. Finally, in Chapter Seven I discuss the main findings of the thesis and avenues for extending this research. To facilitate publication of the thesis, Chapters Two and Four-Six are written as manuscripts for peer- reviewed journals. Since multiple authors contributed data and supervisory support, I use the pronoun "we" rather than "I", and the contributions of each author is described at the end of each of these chapters. At the time of submission, Chapter Two has been published and Chapter Four is in press: Lines, E.R., Coomes, D.A., Purves, D.W., 2010. Influences of forest structure, climate and species composition on tree mortality across the Eastern US. PLoS ONE 5, e13212. Lines, E.R., Zavala, M.Á., Purves, D.W., Coomes, D.A., in press. Predictable changes in aboveground allometry of trees along gradients of temperature, aridity and competition. Global Ecology and Biogeography. i Contents List of tables.......................................................................................................................................... vi List of figures ....................................................................................................................................... vii 1 Introduction ................................................................................................................................... 1 1.1 Motivation for modelling forest dynamics .............................................................................. 2 1.1.1 Forests in models of global vegetation................................................................................ 2 1.1.2 Models of the effects of human-induced and natural disturbance ...................................... 3 1.2 Approaches to modelling forest dynamics .............................................................................. 4 1.2.1 Patch, gap and individual-based spatially explicit models .................................................. 4 1.2.2 The Perfect Plasticity Approximation: reducing model complexity ................................... 6 1.2.3 Physiology-based gap models ............................................................................................. 7 1.2.4 Metabolic scaling theory: scaling from individual metabolism and biomechanics to whole-forest dynamics ..................................................................................................................... 7 1.3 Specifying models of forest processes .................................................................................... 9 1.3.1 Growth ................................................................................................................................ 9 1.3.2 Allometry .......................................................................................................................... 10 1.3.3 Mortality ........................................................................................................................... 10 1.3.4 Recruitment ....................................................................................................................... 11 1.4 Parameterisation and model selection ................................................................................... 12 1.4.1 Frequentist approaches to parameterisation ...................................................................... 12 1.4.2 Philosophical problems of frequentist hypothesis testing ................................................. 13 1.4.3 Frequentist likelihood-based approaches to parameterisation .......................................... 14 1.4.4 Bayesian inference ............................................................................................................ 14 1.4.5 Parameterisation without sufficient data ........................................................................... 17 1.4.6 Models selection: methods for testing ecological hypotheses .......................................... 18 1.4.7 Bayesian methods for forest dynamics modelling ............................................................ 20 ii 2 Influences of forest structure, climate and species composition on tree mortality across the Eastern US ........................................................................................................................................... 23 2.1 Introduction ........................................................................................................................... 23 2.2 Materials and methods .......................................................................................................... 25 2.2.1 Forest Inventory data ........................................................................................................ 25 2.2.2 Environmental data ........................................................................................................... 26 2.2.3 Model description ............................................................................................................. 27 2.2.4 Parameter estimation ......................................................................................................... 28 2.2.5 Model selection ................................................................................................................. 29 2.2.6 Parameter significance ...................................................................................................... 31 2.2.7 Interpretation of the results ............................................................................................... 31 2.3 Results ................................................................................................................................... 33 2.3.1 Model selection ................................................................................................................. 33 2.3.2 Species-mortality relationships ......................................................................................... 34 2.3.3 Size-mortality relationship ................................................................................................ 34 2.3.4 Environment-mortality relationships ................................................................................ 36 2.3.5 Geographical variation in mortality .................................................................................. 37 2.4 Discussion ............................................................................................................................. 38 2.4.1 Size and stand structure ..................................................................................................... 38 2.4.2 Within vs across species variation in mortality along climatic gradients ......................... 40 2.4.3 Species-mortality responses to changing climate ............................................................. 42 2.4.4 Limitations ........................................................................................................................ 42 2.4.5 Conclusion ........................................................................................................................ 43 3 A brief introduction to Spanish forests ...................................................................................... 45 3.1 Introduction ........................................................................................................................... 45 3.2 Climate, topography and tree species distributions ............................................................... 47 3.3 Adaptation to drought ........................................................................................................... 49 3.4 Adaptation to forests fires ..................................................................................................... 50 3.5 Land use change, threatened species and climate change ..................................................... 51 iii 3.6 The Spanish Forest Inventory ............................................................................................... 52 4 Predictable changes in aboveground allometry of trees along gradients of temperature, aridity and competition ...................................................................................................................... 53 4.1 Introduction ........................................................................................................................... 54 4.2 Materials and Methods .......................................................................................................... 56 4.2.1 Inventory data and climate variables................................................................................. 56 4.2.2 Species-level wood traits and critical buckling heights .................................................... 57 4.2.3 Statistical models .............................................................................................................. 57 4.2.4 Interpreting the results ...................................................................................................... 59 4.3 Results ................................................................................................................................... 59 4.3.1 Height and crown allometries varied among species and along environmental gradients 59 4.3.2 (H1) Tree heights were greater and crown diameters narrower in competitive neighbourhoods ............................................................................................................................. 61 4.3.3 (H2) The elastic similarity model provided an upper limit on height at a given DBH and trees in dense forests will grow to their critical height .................................................................. 61 4.3.4 (H3) Tree height and crown diameter for a given DBH was lower in areas of longer drought 62 4.3.5 (H4) Within-species height increased with temperature, but there was no trend in crown diameter ......................................................................................................................................... 64 4.4 Discussion ............................................................................................................................. 65 4.4.1 Allometric scaling varies substantially with species ......................................................... 65 4.4.2 Asymmetric competition for light was a strong determinant of DBH-H allometry .......... 67 4.4.3 Species were shorter in drought areas but individuals did not respond consistently ........ 67 4.4.4 Within-species height increased with temperature, but across species the trend was the opposite .......................................................................................................................................... 67 4.4.5 Within-species variation in allometry along climatic gradients is smaller than across- species differences ......................................................................................................................... 68 4.4.6 Sources of unexplained variation ...................................................................................... 69 4.4.7 Conclusions ....................................................................................................................... 69 iv 5 Inferences from aggregated count data using Approximate Bayesian Computation: deriving juvenile recruitment rates from inventory data ............................................................................... 71 5.1 Introduction ........................................................................................................................... 72 5.2 ABC Algorithms ................................................................................................................... 74 5.3 Materials and Methods .......................................................................................................... 76 5.3.1 Data ................................................................................................................................... 76 5.3.2 Prior information for growth and mortality rates .............................................................. 78 5.3.3 Simulation model .............................................................................................................. 78 5.3.4 Assessing the algorithm on test data: method ................................................................... 79 5.3.5 Performance of the algorithm on test data and choice of metric ....................................... 80 5.3.6 Implementation of the ABC-SMC algorithm on the inventory data ................................. 80 5.4 Results from inventory data .................................................................................................. 82 5.4.1 Convergence of the ABC-SMC algorithm ........................................................................ 82 5.4.2 Average recruitment rates and the effect of canopy cover ................................................ 82 5.4.3 Relationships between recruitment, growth and mortality................................................ 86 5.5 Discussion ............................................................................................................................. 87 5.5.1 Parameterisation and predicted recruitment rates ............................................................. 87 5.5.2 Other factors affecting Mediterranean juvenile recruitment ............................................. 87 5.5.3 Implementation of ABC .................................................................................................... 88 5.5.4 Applications of ABC in Ecology ...................................................................................... 89 6 Regional successional dynamics simulated from tree-level processes ..................................... 91 6.1 Introduction ........................................................................................................................... 91 6.2 Simulation model .................................................................................................................. 93 6.2.1 Canopy area of taller trees, CAIh, as a measure of competition ........................................ 93 6.2.2 Demographic rates and allometric equations .................................................................... 94 6.2.3 Predictions of successional dominance ............................................................................. 95 6.2.4 Simulation steps ................................................................................................................ 96 6.2.5 Observed species dominance ............................................................................................ 97 6.3 Results ................................................................................................................................... 98 v 6.3.1 Models of growth and mortality ........................................................................................ 98 6.3.2 Observed and predicted pine dominance .......................................................................... 99 6.3.3 Observed and predicted angiosperm dominance ............................................................. 104 6.4 Discussion ........................................................................................................................... 106 6.4.1 Growth and mortality rates varied with size, climate and competition ........................... 106 6.4.2 Early successional dominance was determined by both growth and mortality ............... 107 6.4.3 Mismatches in pine dominance are likely due to lack of fire dynamics ......................... 107 6.4.4 Late successional dominance dynamics .......................................................................... 108 6.4.5 Conclusions: climate-driven simulation models at a regional scale................................ 108 7 Discussion ................................................................................................................................... 113 7.1 Tree size and competition as primary determinants of individual-level rates ..................... 113 7.2 Climate dependency of tree-level processes and inventory data......................................... 115 7.3 Limitations of the use of forest inventory data for simulation models ............................... 116 7.4 Concluding remarks ............................................................................................................ 118 Appendix A: Eastern US mortality ................................................................................................. 121 Appendix B: Aboveground allometry ............................................................................................. 131 Appendix C: ABC recruitment model ............................................................................................ 141 Appendix D: Simulation model ........................................................................................................ 157 References .......................................................................................................................................... 171 vi List of tables Table 2.1 Species' predicted annual mortality rates. 34 Table 4.1 Wood, modulus of elasticity and number of samples used for allometric models for each species. 62 Table 5.1 Number of plots, shade tolerance and recruitment, growth and mortality rates 81 in a fixed environment for each species. Table 6.1 The 14 species modelled, their maximum observed DBH and the upper limit of drought length the species was simulated in. 97 Table A.1 Comparison of 36 models tested using AIC and BIC . 121 Table A.2 Fitted parameter values for each of the 21 common species. 122 Table B.1 Parameter values for species-level covariates of height-DBH relationships 135 fitted using the hierarchical Gibbs sampler. Table B.2 Parameter values for species’ height-DBH scaling relationships. 136 Table B.3 Across-species correlations between predicted heights and crown widths and species’ mean environment. 137 Table B.4 Priors and initial values for height-DBH and crown diameter-DBH model parameters used in MCMC model fitting. 137 Table C.1 Amount of data for each species used to parameterise the crown diameter model. 144 Table C.2 Comparison of crown diameter models tested with AIC scores for each model. 145 Table C.3 List of species in the dataset, number of plot each is found in and the crown diameter allometric equations used to calculate crown area for each species. 146 Table C.4 Comparison of stem growth rate models tested with AIC scores for each model. 149 Table C.5 Comparison of mortality rate models tested with AIC scores for each model. 149 Table C.6 Best-fit growth rate model parameter values and credible intervals. 150 Table C.7 Best-fit mortality rate model parameter values and credible intervals. 151 Table D.1 Details of the data used for each model for the 14 species included in the analysis. 161 Table D.2 Growth model comparison with functional forms and AIC values. 162 Table D.3 Mortality model comparison with functional forms and AIC values. 163 Table D.4 Parameter values for the best-fit growth model for each species. 164 Table D.5 Parameter values for the best-fit mortality model for each species. 165 Table D.6 Parameter values for the height-diameter allometric model for each species. 166 vii List of figures Figure 1.1 The horizontal structure of a JABOWA-style patch model, a gap model, and a SORTIE-style individual-based model. 6 Figure 2.1 Observed and predicted mortality against stem size and environmental gradients. 36 Figure 2.2 Predicted changes in species' mortality rates with increases in temperature and precipitation. 39 Figure 2.3 Regional variation in mortality due to variation in each component of the model. 41 Figure 3.1 Maps of average annual temperature (°C) and annual precipitation rate (mm) in Spain in the period 1940-1996. 45 Figure 3.2 Distribution of the approximately 90,000 forest plots in the Second Spanish Forest Inventory (IFN2) representing the distribution of Spanish forests. 46 Figure 3.3 Forest species dominance across Spain. 48 Figure 4.1 Fitted parameters for each species’ height-DBH scaling relationship. 60 Figure 4.2 Height-DBH data and fitted models for two common species (Pinus halepensis and Quercus robur). 63 Figure 4.3 Predicted changes in height along gradients of climate and competition. 64 Figure 4.4 Predicted changes in crown diameter along gradients of climate and competition. 66 Figure 5.1 Observed large juvenile count densities for each species. 77 Figure 5.2 Predicted growth and mortality rates used as priors for simulation model. 83 Figure 5.3 Estimated and true parameter values using simulated test data. 83 Figure 5.4 Fitted values and credible intervals for recruitment, growth and mortality parameters. 84 Figure 5.5 Predicted recruitment rates for each species against total plot canopy area. 85 Figure 5.6 Boxplots of differences between conifer and angiosperm recruitment rates and their responses to changes in plot canopy area. 86 Figure 6.1 Observed 95% climatic distribution of species in the inventory. 98 Figure 6.2 Predicted growth and mortality rates for each species along stem size and competitive gradients. 99 Figure 6.3 Observed climatic range of dominance of the six pine species. 101 viii Figure 6.4 Model predictions for climatic range of dominance of the six pine species. 102 Figure 6.5 Predicted dominance, growth, mortality and crown diameter for pine species along a temperature gradient. 103 Figure 6.6 Observed climatic range of dominance of the eight angiosperm species. 105 Figure 6.7 Model predictions for changes in climatic range of dominance of the eight angiosperm species simulated over 500 years. 109 Figure A.1 Observed and predicted mortality rates against maximum wind speed. 126 Figure A.2 Observed and predicted mortality rates against soil type. 126 Figure A.3 Observed and predicted mortality rates against plot basal area. 127 Figure A.4 Observed versus predicted plot-averaged mortality rates. 127 Figure A.5 Patterns of mortality due to regional variation in stand structure and environmental conditions alone. 128 Figure A.6 Regional patterns of differences between observed and predicted mortality rates. 129 Figure B.1 Height-diameter data, fitted allometric relationship and critical buckling height. 138 Figure B.2 Predicted changes in height along gradients of climate and competition with species names. 139 Figure B.3 Predicted changes in crown diameter along gradients of climate and competition with species names. 140 Figure C.1 Observed and predicted crown diameters for each species. 152 Figure C.2 Observed and predicted diameters for each species at the second time step. 154 Figure C.3 Observed and predicted mortality rates for each species. 155 Figure D.1 Predicted and observed tree growth rates for each species along size, competition and climatic gradients. 161 Figure D.2 Predicted and observed tree mortality rates for each species along size, competition and climatic gradients. 162 Figure D.3 Predicted and observed tree height for each species along size, competition and climatic gradients. 163 Figure D.4 Climatic distribution of predicted tallest pine species after 30 years of simulation. 164 Chapter 1 Introduction 1 1 Introduction Simulation models form a vital part of ecological research, allowing for the exploration of systems at larger geographic scales and over longer time frames than is possible through field measurements or experimentation alone. This thesis aims to develop methods for parameterising a forest simulation model from inventory data collected on a regional scale. No previous forest simulations have explicitly modelled multiple processes for multiple species as a function of environment, but without these factors long-term predictions of forest dynamics in the face of climate change and disturbance cannot be made. Many forest simulation models have been developed from data collected in small, intensively studied areas of forest comprising of a handful of species. This is predominantly for two reasons: firstly, many forest models have large numbers of parameters, for which many measurements must be taken (e.g. the detailed light environment measurements in SORTIE, Pacala et al. 1996)- data which are not available for large numbers of species and sites (Lichstein et al., 2010). Secondly, computational limitations have prevented detailed models being simulated at regional or global scales (Moorcroft et al., 2001; Purves and Pacala, 2008). Long-term forest inventory data, which are becoming increasingly available in many parts of the world, provide valuable information that may be used to parameterise forest models for many species and across wide environmental gradients (Didion et al., 2009). Moreover, careful examination of the detail required to reproduce observed forest dynamics (e.g. Deutschman et al. 1999), and new simplified forest models, e.g. PPA (Purves et al., 2008), mean that simulation models may now be developed on regional scales and rigorously parameterised for large numbers of species, and can incorporate climatic and environmental dependency in their demographic functions. Combined with increased computational power and more sophisticated algorithms, these advances mean that more detailed models than ever before can be parameterised, and simulations run in greater detail and over wider areas. In this chapter I explain the motivations for modelling forest processes at large scales in terms of the importance of better understanding the effects of climate change and human and natural disturbance on biodiversity, species composition, succession, productivity and carbon storage. I also describe some approaches and methods used to construct and parameterise models, and in particular I discuss the appropriateness and effectiveness of different statistical approaches in terms of ecological models. Chapter 1 Introduction 2 1.1 Motivation for modelling forest dynamics Simulation models are particularly important for making predictions in forest ecology research, since processes of interest often happen at substantially longer time scales than the average research project (or even career) and predictions are often made over decades or centuries. As I discuss in this section, questions about the effect of management practices, long term disturbance events and climate change on forest structure and productivity, biodiversity, carbon storage, species composition and succession must therefore be interrogated by carefully building models that capture the observed behaviour of a forest system, and by running simulation experiments. 1.1.1 Forests in models of global vegetation Forests store more carbon than is held in the atmosphere (Millenium Ecosystems Assessmemt, 2005), but ecologically realistic models of their dynamics are not currently included in global vegetation models (Purves and Pacala, 2008). The response of forests to changing climate and atmospheric CO2 levels, as well as their role in mitigating the effects of human-induced greenhouse gases through feedbacks to the atmosphere, is a large source of uncertainty in current vegetation models and therefore the global climate models (GCMs) coupled to them (Purves and Pacala, 2008; Sitch et al., 2008; Lapola et al., 2009; Fisher et al., 2010; Galbraith et al., 2010). Globally, forests are currently thought to act as a significant sink for carbon dioxide (Pan et al., 2011), but it is not known for how long they will continue to act in this way (Lewis, 2006; Hyvönen et al., 2007). Empirical studies in many parts of the world have shown forests accumulating carbon (per unit area). In temperate regions this is attributed to regeneration following land abandonment (Caspersen et al., 2000) and the fertilizing effects of nitrogen deposition from human activities (Thomas et al., 2010), whilst in old- growth tropical forests it may be a result of increased productivity under increased atmospheric CO2 levels (Lewis, 2006; Phillips et al., 2008). However, some studies of coupled vegetation-atmosphere models predict large scale dieback of forests due to climate change, such as increased drought in Amazonia (e.g. Cox et al. 2004). Moreover, some forest-climate feedbacks have been shown to amplify the impact of human-induced greenhouse gas emissions on global temperatures: in boreal regions increased forest productivity and cover in coniferous forests is thought to accelerate global warming as the associated reduction in winter surface albedo (compared with snow on unforested land) causes temperature to increase (Bonan, 2008). Dynamic global vegetation models (DGVMs) include many simplifying assumptions about forest dynamics which strongly influence the predictions they make (Sitch et al., 2008; Fisher et al., 2010). Species are grouped together and assigned parameter values according to plant functional types (PFTs) (Prentice et al., 2007). For trees, these groupings may be extremely simplistic (e.g. broadleaf and needleleaf in TRIFFID, Cox 2001) or may differentiate both between tree types (e.g. deciduous or Chapter 1 Introduction 3 evergreen, broadleaf or needleleaf) and climatic region (tropical, temperate, boreal) (e.g. ORCHIDEE, Krinner et al., 2005). However, within DGVMs there is little consensus on which PFT classification is sufficient and the approach has been criticised for a number of reasons, such as ignoring important species' differences that drive vegetation composition (Quillet et al., 2010). Species within PFTs are modelled as having the same effect on ecosystem functioning and showing the same response to environmental factors such as climate stressors (McMahon et al., 2011), but in many instances this is unlikely to be true (Lavorel et al., 2007). A mismatch in the level of detail modelled for different forest processes in many DGVMs makes them unsatisfactory for making long-term predictions. Forest growth, represented as carbon assimilation, is described in great detail using photosynthesis, stomatal conductance and respiration models which, in the most detailed DGVMs, are applied in half-hour steps. Other processes are not represented in anything like such detail: seeding establishment is modelled uniformly in each grid cell for climatically appropriate PFTs, and tree mortality is described as a function of carbon balance and simple disturbance events (Sitch et al., 2008), despite the fact that these processes are key components of the terrestrial carbon cycle and have been shown to be sensitive to climate change (e.g. Camarero and Gutiérrez, 2007; van Mantgem and Stephenson, 2007). Moreover, DGVMs assume no limit on rates of migration of species or PFTs in response to climate (Prentice et al., 2007), yet timescales of species migration are likely to be highly variable within PFTs (Neilson et al., 2005) and for many lowland areas species migration is unlikely to be fast enough to keep up with changing climate (Loarie et al., 2009). 1.1.2 Models of the effects of human-induced and natural disturbance Simulation models are vital tools for addressing questions on the impact that management and disturbance have on forests. Stand models have long been used in the forestry sector to calculate growth rates and yields (Changhui, 2000a), and are usually parameterised with site-specific data to make predictions for commercial timber production over large areas (Liu and Ashton, 1995). Changes in forest management practices in many parts of the world, from monospecific even-age plantations to multi-age multi-species stands, have increased the role of detailed ecologically-structured forest models that simulate the effects of management strategies on a wider set of forest goods and management objectives (Wolfslehner and Seidl, 2010). For example, simulation models have been used to assess the effects of forest management strategies on ecosystem structure and function (Changhui, 2000b), biodiversity and carbon stocks (Hynynen et al., 2005; Taylor et al., 2008; Swanson, 2009), nutrient reserves (Blanco et al., 2005) and forest structure (Newton et al., 2011), and to assess the need for different management practices under climate change scenarios (Linder, 2000; Garcia-Gonzalo et al., 2008). Simulations may also be used to predict the long-term impact of human activity on forest structure, for example the impact of land use on species diversity in a tropical forest Chapter 1 Introduction 4 (Teixeira et al., 2009; Uriarte et al., 2009), or the regrowth of temperate forests after land abandonment (Evans and Kelley, 2008). Simulation models are particularly valuable when predicting the effects of natural disturbance events at landscape scales, such as hurricanes, storms, forest fires and diseases, because such events are typically highly infrequent but have extreme impacts. Models of the relationship between wind and fire events and vegetation have revealed impacts on species composition and stand biomass (Hickler et al., 2004), size structure (Liedloff and Cook, 2007; Uriarte et al., 2009) and net primary productivity (Keane et al., 1996). Simulations also allow predictions to be made about the effects of changes in disturbance occurrence rates, for example the effect of large-scale disturbance on forest carbon flux (Coomes et al., 2012), the effect of climate change-induced changes to fire frequency on carbon storage (Thornley and Cannell, 2004), pest and disease outbreaks (Bergot et al., 2004; Kurz et al., 2008), and interactions between different disturbances and management practices (He and Mladenoff, 1999; Mitchell et al., 2009; James et al., 2010). 1.2 Approaches to modelling forest dynamics The many different structures of forest simulation models reflect both their diverse purposes and the data available with which to parameterise them. Models may have as their smallest component an individual tree or a small patch of forest, and may be spatially explicit (recording the exact location of each tree in relation to all others) or implicit (assuming a uniform horizontal structure within each component of the model). Models which omit the individual variation of each tree have been found to have significant limitations in terms of predictive ability (Grimm and Railsback, 2005). Spatial structure is fundamental to understanding community structure (Crawley, 1997), so whole stand models require many simplifying and unrealistic assumptions that are unnecessary in individual-based models and can only approximate a few aspects of forest dynamics (Huston et al., 1988; Vanclay, 1995). In this section I summarise the different types of forest dynamics models according to their structure, smallest unit of interest, assumptions and applications. 1.2.1 Patch, gap and individual-based spatially explicit models Patch models simulate forest dynamics as a mosaic or composite of small patches of forest, with a patch (the smallest spatially referenced unit) usually corresponding to the size of either one or a small number of mature individual trees (Shugart and Smith, 1996; Bugmann, 2001). The aim of patch models is often to simulate and therefore better understand changes in species composition and succession in relation to environmental conditions (Grimm and Railsback, 2005). Chapter 1 Introduction 5 Patch models may be subdivided into two categories (Gratzer et al., 2004): those which model the transitions of vegetation types between patches (cellular automata/state-transition models) but omit within-patch dynamics, and those which model the dynamics of individual trees within a simulated patch (e.g. JABOWA, Botkin et al., 1972, FORET, Shugart & West 1977). Cellular automata/state-transition models are structured on regular lattices or arrays and the state of each cell is followed over time, with the transition of a given cell between states determined by a transition rule and some local interaction with a defined neighbourhood of other cells (Karafyllidis and Thanailakis, 1997; Balzter et al., 1998). Such state transition models have been used to describe succession (Kessell and Potter, 1980) and dispersal dynamics (Green, 1989), the impact of disturbances such as fire (Karafyllidis and Thanailakis, 1997), and land-use change (Soares-Filho et al., 2002). Patch (gap) models based on individual trees follow birth, growth and death processes as functions of abiotic and biotic environmental variables such as available light, crowding, temperature and evapotranspiration, but only consider vertical spatial structure and assume horizontal homogeneity within a patch. Patch size is set to correspond with the size of a very large tree (100m 2 in JABOWA), and leaves are located at the top of each tree in a thin layer. This means that the light environment of each patch is controlled by the largest tree and ‘gaps’ are opened up in a patch when the large trees die, leading to successional dynamics governed by interspecific competition for light. Growth (usually calculated annually) is strongly determined by the light available, and mortality is often negatively dependent on growth. Patches within a forest are modelled as having different successional statuses, but early forms of gap models assumed, for computational reasons, no interaction between patches, and therefore ignored important processes such as dispersal and shading between patches (Bugmann et al., 1996, Fig 1.1a). Later gap models relaxed the assumption of no interactions between patches and model the forest landscape as a mosaic of interconnected patches (e.g. ZELIG, Urban, 1990, Fig. 1.1b), meaning that trees in a given cell may be shaded by those in neighbouring cells as well as within their own (Urban et al., 1991). Patch interactions also allow the inclusion of long-range dispersal, thereby improving the representation of understory dynamics, and allow studies of the impact of large herbivores which move between patches (e.g. FORGRA, Jorritsma et al., 1999). The realism of gap models has also been improved by removing the assumption of ‘flat top’ leaf arrangement, which resulted in strong asymmetry of competition for light and the largest trees having a very large advantage over all smaller ones (Leemans, 1991), and instead modelling vertical canopy structure through individual tree crown depths, which allows shading within crowns (e.g. FORSKA, Leemans and Prentice, 1989). Individual-based spatially explicit models also simulate the recruitment, growth, allometry and mortality of individuals in a plot, but record the specific location of each tree (Fig. 1.1c). These models typically involve detailed calculation of the effect of competition on the light available to each stem using three-dimensional calculations of crown geometry (e.g. SORTIE, Pacala et al. 1993), and Chapter 1 Introduction 6 therefore require detailed field measurements for parameterisation. Since they model each stem individually, these models allow gaps of any size to be created (rather than fixing patch size), and are therefore better able to reproduce successional dynamics (Pacala and Deutschman, 1995). However, such a high level of detail may not be required to accurately reproduce forest dynamics in all cases (Busing and Mailly, 2004; Strigul et al., 2008), and modelling small patches on the scale of a few trees instead of individual stems may be sufficient for predicting forest dynamics (Deutschman et al., 1999). Figure 1.1 The horizontal structure of a) a simple JABOWA-style patch/gap model with non- interacting patches of forest, b) a patch/gap model with interactions between patches, for example through shading or seed dispersal, and c) a SORTIE-style individual-based spatially-explicit model. 1.2.2 The Perfect Plasticity Approximation: reducing model complexity The high number of parameters and correspondingly detailed field data needed to parameterise spatially explicit individual-based models such as SORTIE reduces their practicality for simulating forest dynamics at large scales or in novel locations. The Perfect Plasticity Approximation (PPA) model (Purves et al., 2007, 2008) is a cohort-based canopy competition model that models individual tree processes but treats all stems within a cohort as the same. By considering height-structured competition for light as the most important process governing forest dynamics, and assuming that the canopy of a forest is space filling and plastic in its response to gaps, the need for the light environment of individual trees to be calculated is removed. This assumption not only dramatically reduces the computational effort required to run simulations (compared to spatially-explicit models such as SORTIE), but also reduces the number of parameters required, making the model able to be parameterised from standard inventory data collected over thousands of sites. For example, two parameters for each species are required for modelling each of growth and mortality rates – one describing the rates of understory trees and one for those in the canopy. A more recent adaptation of the PPA framework has relaxed the ‘in or out’ of canopy assumption by predicting growth and mortality as a function of individual tree size and the projected area of crowns taller than the tree Chapter 1 Introduction 7 (Bohlman and Pacala, 2011; Caspersen et al., 2011; Coomes et al., 2012). This gives a more continuous definition of asymmetric competition for light than simple canopy/understory status classification, and has been used to gain insight into how individual growth and mortality rates determine stand productivity. However, this framework requires the continuous competition- dependency of demographic processes to be specified. 1.2.3 Physiology-based gap models Physiology-based gap models seek to link ecosystem structure with physiological processes at the tree level such as photosynthesis, respiration and water and nutrient use, which are modelled on timescales of a day or even less (e.g. HYBRID, Friend et al., 1993). These models respond to changes in atmospheric CO2 and climate, and therefore contain the detail necessary to predict the response of both individual tree processes such as growth, seed production and phenology, and whole-forest characteristics such as succession and carbon storage (e.g. van der Meer et al., 2002). Physiology- based models have also been used to examine the effects of fires on ecosystem characteristics such as net primary productivity and nutrient cycling (Keane et al., 1996), and to predict productivity in managed stands (e.g. PROMOD, Sands et al., 2000). Because of the level of detail included, physiology-based models typically require very large numbers of parameters whose values are poorly known and often constrained only for special cases (Mohren and Burkhart, 1994; Mäkelä et al., 2000), and despite the inclusion of physiological processes which respond to the environment, models formulated for one region may not perform well in other climates (e.g. seasonal drought in the Mediterranean, Morales et al., 2005). 1.2.4 Metabolic scaling theory: scaling from individual metabolism and biomechanics to whole- forest dynamics Metabolic scaling theory (West et al., 1997, 1999, 2009; Enquist et al., 1999, 2009; Enquist and Niklas, 2002) makes a variety of predictions about individual tree and whole-forest properties based on assumptions about individual metabolic rates. The theory assumes that resources are used efficiently and are distributed through the tree (or organism in general) through an optimal, self- similar, hierarchical, volume-filling vascular branching network, through which hydraulic resistance is minimised. These assumptions lead to the fundamental scaling relationship that leaf mass, and hence whole-plant photosynthetic (metabolic) rate, scales as a 3/4 power of body mass. This, along with the specified relationship between mass (M) and stem diameter (D), M  D8/3, leads to a set of predictions about individual tree allometry and function. Metabolic scaling theory predicts that tree height (H) scales with D as H  D2/3, that diameter growth rate, (dD/dt) scales with D as dD/dt  D1/3, and that individual mortality is solely derived from competitive thinning and scales with D -2/3 (Enquist et al., 2009). The theory predicts that the exponents in these power-law relationships are invariant Chapter 1 Introduction 8 with body size and that differences between species may be accounted for with differences in the coefficients of the power function (Enquist et al., 1999). The appeal of such a model as a framework for individual-based models is therefore strong, because only the coefficient need be determined for each of these three processes and each species, thereby reducing the amount of data required to parameterise the models. The theory is scaled up to whole-forest dynamics by assuming no recruitment limitation, that resource use and supply are equal, birth and death rates are equal and that stems' allometry and demographic rates are as defined above, allowing predictions such as stand size distribution, canopy structure and resource use (West et al., 2009). The effects of disturbance on succession and temporal dynamics are also predicted, but there are discrepancies between model predictions and data. In particular mismatches are caused by the simplifying assumptions that there is no recruitment limitation and that mortality is solely size dependent and caused by competitive thinning (Enquist et al., 2009). Metabolic scaling theory has proved contentious for several reasons. The theory has been found to be effective in explaining body size from metabolism across very large scale differences in body size, but within a ten-fold difference in size, the theory explains only 20 per cent of variation in body size, and even less in more similar-sized organisms (Tilman et al., 2004). Whilst the theory may therefore be useful in examining entire ecosystems, for example in terms of resource use, it has much more limited applicability within communities with similar-sized members, such as forests. The theory ignores the effect on growth of asymmetric competition between plants of different sizes and the effects of gap-creating disturbances within a forest (Coomes et al., 2003), for example increases in crown area in trees next to a gap to take advantage of extra light available. The use of invariant fixed scaling laws for describing forest structure and dynamics has therefore been questioned by many using comparison with data (e.g. Muller-Landau et al. 2006a, 2006b, Russo et al. 2007, Návar 2009). The very fact that the predictions of the metabolic theory are independent of species, competition (in particular asymmetric competition for light), forest location and tree age conflicts specifically with studies which have found relationships between these and allometry (Canham et al., 1994; Niklas, 1995; Mencuccini, 2002; Chen and Li, 2003), growth (Li et al., 2005; Reich et al., 2006; Coomes and Allen, 2009) and mortality (Muller-Landau, Condit, Chave, et al., 2006). In this way the wisdom of ignoring the variability in experimental data and forcing species to share certain invariant traits has been questioned (Agutter and Wheatley, 2004). Chapter 1 Introduction 9 1.3 Specifying models of forest processes Forest simulation models such as SORTIE and PPA typically contain a set of submodels that describe individual growth and mortality, aboveground allometry and recruitment or regeneration, and a representation of the effects of competition on these processes. Functional forms for these processes and methods to parameterise them from data vary, and choice depends on considerations such as the forest type of interest and the purpose of the model. The choice of functional form for a model is highly dependent on the purpose of the model or the hypotheses to be tested. For example, if species diversity and succession are of interest then species-specific parameters to describe species’ interactions are required, but in tropical forest communities this is likely to be impossible due to the very large number of species, so species may be grouped by functional type (e.g. Vanclay, 1989). The earliest forest dynamics models used simple functional forms with no climatic dependency of processes, and may be appropriate for simple models, for example of plantation productivity, whereas large-scale long-term models which address questions of climate change require submodels to be responsive to climatic changes. The choice of predictors and functional forms is therefore critical to a model being sufficient for purpose and will impact the model's transferability to other ecosystems and climatic conditions. Climatic, competitive, resource and disturbance dependencies will be highly dependent on the forest system of interest, so data exploration should always be an important part of model building. Although competition for light is often considered the most important limitation in closed-canopy forests, other factors such as drought, nutrient limitations and belowground competition (e.g. root competition for nutrients, Platt et al., 2004) may be more important in other ecosystems, which must be taken into consideration when re-parameterising any model for a region for which it was not built. In this section I discuss some approaches to determining appropriate models for four major forest processes: growth, canopy development, mortality and recruitment. 1.3.1 Growth Models of tree growth abound in ecological and forestry literature, and are frequently parameterised using diameter increment data collected over two or more time steps from permanent plots. Differences in species’ growth rates are important for succession (Huston and Smith, 1987), and may be related to species’ traits such as wood density (Poorter et al., 2010) and xylem hydraulic traits (Russo et al., 2010). Models frequently include the effect of competition for light, a key limitation to tree growth (e.g. Canham et al., 1994, 2004; Pacala et al., 1996; King et al., 2005; Wyckoff and Clark, 2005). Competition for, or limitations of, other resources such as water (Karin, 1996) and soil nutrients (Coomes and Grubb, 1998), has also been found to be a key driver of growth in many regions. The climate dependency of growth, with effects of temperature and altitude (Coomes and Chapter 1 Introduction 10 Allen, 2007a), evapotranspiration (Wickramasinghe, 1988) and drought (Martínez-Vilalta et al., 2011), has also been well documented. These effects are often included as scalars to an hypothesised ‘maximum’ growth rate, although a more mechanistic, physiology-based approach has been taken for the effects of some factors on photosynthesis and respiration (e.g. temperature, Bonan and Sirois, 1992). 1.3.2 Allometry Models of aboveground allometry define relationships between stem diameter and tree height, crown width and crown depth. The shape a tree takes within its environment determines the amount of light its leaves are able to intercept, which is an important driver of individual tree growth, mortality and fecundity, and therefore of whole forest structure (Kohyama, 1991; Pacala et al., 1996; Purves et al., 2007). Whole canopy structure determines understory light conditions, knowledge of which allows prediction of which species' saplings will grow and survive to the reach the canopy and form the next generation of adult trees (Horn, 1971). Accurate prediction of tree and crown shape through allometric relationships from diameter is therefore vital for accurate simulation models, which have been shown to be sensitive to allometric scaling (e.g. ALLOCATE, Tilman 1988; SORTIE, Pacala et al. 1996; PPA, Purves et al. 2008). Aboveground allometry has been found to be affected by competition, because trees grow taller to capture more light and restrict their crown width in densely pack stands (Henry and Aarssen, 1999; Bragg, 2001; Muth and Bazzaz, 2003; Weiner, 2004) , and by temperature, because long and severe frosts can cause embolism and branch loss (Lemoine et al., 1999; Wang et al., 2006). Most field-parameterised allometric relationships have been defined for small samples in single locations, and surprisingly little is known about how scaling relationships vary within and among species along environmental gradients at regional scales (Wang et al., 2006; Méndez-Alonzo et al., 2008), although my work in Chapter Four has shown that such variation can be very pronounced. 1.3.3 Mortality Models of individual tree mortality are less developed than those for growth and allometry. Approaches to simulating stem loss commonly stress the role of suppression of smaller stems beneath the forest canopy by linking a higher probability of mortality with lower growth (e.g. FORMIX, Bossel and Krieger 1991; SORTIE, Pacala et al. 1993) and the effects of large-scale natural disturbances such as fire and windthrow (e.g. FORSKA, Prentice et al. 1993). One reason why mortality models are not as common as those for growth and allometry is that mortality is a rare, discrete event that requires long-term, large-scale monitoring in order to pick up both quantifiable continuous patterns in mortality along environmental gradients and the effects of rare large-scale disturbance events (e.g. hurricanes and earthquakes). Chapter 1 Introduction 11 Mortality has been found to vary along gradients of precipitation and temperature (Voelker et al., 2008; van Mantgem et al., 2009), and my work in Chapter Two shows that both average rates and variation along gradients may be highly species-specific. U-shaped, size-dependent mortality appears to be a common and quantifiable feature of forests (Buchman et al., 1983; Monserud and Sterba, 1999; Umeki, 2002; Coomes and Allen, 2007b; Chao et al., 2008) and one that I have found from examining mortality in the eastern United States (Chapter Two). This pattern is likely to be a result of high mortality in small suppressed trees, low mortality in canopy trees, and an increase in mortality in larger and older trees due to senescence and/or increased exposure to disturbances, although the relative importance of these processes may depend on stand age (Coomes and Allen, 2007b). Small trees do not, however, always suffer from the presence of adults; in Mediterranean conditions where water is limited and desiccation risk from direct sunlight is high, and in areas with high wind speed, the survival of small trees may be higher in the presence of adults (Taylor and MacLean, 2007; Gómez-Aparicio et al., 2008). 1.3.4 Recruitment Since juvenile recruitment and dynamics are crucial determinants of forest community dynamics (Shibata and Nakashizuka, 1995; Kobe, 1996), accurate representation of dispersal and recruitment limitation is vital to produce realistic simulations of succession and spatial dynamics (Ribbens et al., 1994). Models of recruitment vary in their treatment of seedling dispersal and recruitment, with the simplest ones assuming that seed is always available for regeneration (Clark et al., 1998) and many combining dispersal and establishment together in one process (Busing and Mailly, 2004). On the other hand, spatially-explicit individual-based forest simulation models typically model recruitment using a seed dispersal kernel, where seedlings establish in a location with probability related to the distance to conspecific adults, and the number of seedlings established per adult varies with species, adult size and shading from adult trees (e.g. Busing, 1991; SORTIE, Pacala et al. 1996; TROLL, Chave, 1999). The spatially implicit PPA simulates recruitment as a function of adult canopy tree density (Purves et al., 2008) with species-specific functions. Long-distance dispersal mechanisms are generally ignored in models of forest dynamics, but are likely to be important for determining shifts in species' distributions with rapid climate change (Hampe, 2011). Permanent-plot datasets such as national inventories rarely contain detailed information on recruitment rates, but approximate juvenile densities are sometimes recorded and in Chapter Five I parameterise a model of recruitment rates using such data. The presence and density of conspecific adults is a well-recognised determinant of seedling recruitment, but many other factors may affect seedling dynamics. For example, small scale spatial heterogeneity and microsite quality are important for seedling establishment (Nathan and Muller- Landau, 2000), although such effects are difficult to measure and quantify. Canopy gaps and competition from understory shrubs (Beckage et al., 2000), soil moisture and drought (Lloret et al., Chapter 1 Introduction 12 2004; Urbieta et al., 2008; Mendoza et al., 2009), facilitation through protection from water stress and direct sunlight by 'nurse' plants (Lookingbill and Zavala, 2000; Smit et al., 2008; Plieninger et al., 2010), and fire frequency (Lloret et al., 2003) have all been found to affect seedling recruitment, yet there are few models describing continuous changes in recruitment rates along climatic or competitive gradients. 1.4 Parameterisation and model selection In this section I discuss the different statistical methods commonly used to select and parameterise ecological models from data. I discuss classical frequentist methods (e.g. null-hypothesis least-squares regression) and frequentist likelihood methods (e.g. simulated annealing), and compare them with Bayesian inference to demonstrate why Bayesian methods may often be a better approach for forest models. Frequentist and Bayesian methods differ fundamentally in their treatment of models and data. Frequentist methods assume parameter values (the underlying model) to be fixed and consider all datasets which those parameters could generate, whereas Bayesian methods assume the data to be fixed and consider all parameter values that could have led to them. I also compare frequentist and Bayesian methods of model selection, and present the reasons why I consider Bayesian methods to be the more natural approach for comparing ecological hypotheses posed using competing models, as they can calculate the probability of each hypothesis being true when given the available data. 1.4.1 Frequentist approaches to parameterisation The most commonly used method for parameterising ecological models of continuous processes is undoubtedly frequentist least-squares linear regression, typically using an ordinary least squares (OLS) optimisation. Its appeal lies in the simplicity with which OLS predicts a response Y from a single variable X and states whether a relationship is significant. However, underlying this approach are assumptions that are inappropriate for many ecological situations. For example, fitting a linear model between only two variables assumes a fixed linear relationship that, as the predictor variable increases, the dependent variable will also increase proportionally. Extrapolation of such relationships beyond the range of the data is likely to give unrealistic predictions in many ecological problems, since few processes are controlled only by one variable and multiple limitations act in different conditions (Bloom et al., 1985) - for example, water availability may dominate variation in growth rates in deserts and light fluxes in rainforest understories. Such considerations may not be important in many small-scale studies but ought to be taken into account when translating a model from one location to another. These issues may be resolved by employing multivariate and generalised (for Chapter 1 Introduction 13 example polynomial or logistic) regression techniques. Regression in its classical frequentist form also makes strict assumptions about the normality of the data and the error. The assumption that the error distribution is homogeneous regardless of the value of X may well not be valid for many situations (e.g. increased variance in height with stem diameter, Chapter Four). In addition, if systematic variation in the error structure is present, methods which account for this should be used, such as weighted least squares or an appropriate transformation. Another assumption of standard regression is that there is no error in the independent variable X, which is unrealistic in many ecological applications where measurement error is likely in both variables. Reduced major axis line fitting (RMA – also known as standardised major axis line fitting, SMA) has become commonly used in ecology because, unlike OLS, it assumes error in X as well as Y (which is often true of biological measurements), although it fails to address the distinction between measurement/sampling error and equation error ("natural variation" or "intrinsic scatter", Warton et al., 2006). A key difference between OLS and RMA is that RMA is a symmetric method (it produces the same line of best fit regardless of which variable is the predictor and which the response) whereas OLS regression is asymmetric. The choice of one or the other may therefore depend on not only the measurement error but on consideration of whether the aim is finding the relationship between two variables or predicting one from the other (Warton et al., 2006). 1.4.2 Philosophical problems of frequentist hypothesis testing There are some drawbacks of classical regression that cannot be addressed using related frequentist methods. For example, when parameterising a model using frequentist methods, parameter values are assumed to be fixed, with no underlying variation, which may be unrealistic for ecological processes. This is a consequence of the frequentist philosophy of statistics, which assumes that parameters are fixed and that data is drawn randomly from some underlying population distribution. Therefore, when a p-value is calculated for a linear regression (typically for the null hypothesis of the slope equalling zero), it represents the probability of obtaining, on repeating the experiment, the observed data or data more extreme given that the model is true (the model being the null hypothesis of no relationship). Similarly, a 95% confidence interval in frequentist analysis does not represent the range over which there is a 95% probability that the parameter value lies; rather it is the range over which, if the same experiment was repeated a very large number of times, the regression parameter estimate would fall within 95% of the time (Gelman et al., 2004). By testing against null-hypotheses of 'no effect', frequentist statistics assume no knowledge or relevant information about a system such as previous experiments or observations. This is an inefficient use of data, since it means that tests are carried out to disprove null-hypotheses that may have been falsified before. Chapter 1 Introduction 14 1.4.3 Frequentist likelihood-based approaches to parameterisation Well-recognised drawbacks and limitations of classical null-hypothesis frequentist approaches have led to increased interest and uptake of frequentist likelihood-based approaches to model parameterisation. These methods provide increased flexibility for parameterisation, and more useful and intuitive methods for model comparison (see below), because instead of testing a null and alternative hypothesis and reporting a significance, likelihood methods compare different models and select the best (defined by comparing the support given to each model by the data). Suppose we wish to parameterise a model M which depends on parameters θ to describe the relationship between variables X and Y. Likelihood-based methods differ in their approach to hypotheses from classical frequentist-based ones (such as OLS) because they select parameter values by maximising the probability of the model given the data, P(M(θ,X)|Y), and by calculating and maximising the likelihood (L) of the data given the model (although in practice to ease computation, methods often minimise the log-likelihood, which is equivalent): L(Y|M(θ,X))  P(M(θ,X)|Y) (Eqn 1.1) Thus, likelihood-based methods rely on the specification of the likelihood of a model and apply an iterative optimisation strategy (such as simulated annealing) to search for the set of parameters θ that maximise the likelihood (called the maximum likelihood estimators, MLEs). As the number of iterations of the optimisation increases, the MLEs converge to the true parameter values, and the sampling distribution of the MLEs is asymptotically normal for all parameters (Gatti, 2005), so confidence intervals can easily be calculated if parameter distributions are specified. However, these are frequentist confidence intervals so suffer from the misconceptions discussed earlier (see above), and likelihood methods still treat parameters as being fixed in the underlying model. Likelihood-based methods also allow for evaluation of covariance between parameters through examination of the Hessian matrix (the inverse of which approximates the variance-covariance matrix), which can be used to derive the multidimensional curvature of the log-likelihood parameter surface. Likelihood methods are substantially more computationally demanding than classical regression methods, and can be time inefficient if the likelihood space has multiple local maxima. 1.4.4 Bayesian inference Bayesian methods are fundamentally different from frequentist methods, both classical and likelihood-based (as described above), and both philosophically and practically, in ways that make them attractive to ecologists trying to solve real-world problems (e.g. Dennis 1996; Hilborn & Mangel 1997; Ellison 2004; Hobbs & Hilborn 2006; Kruschke 2010a,b). Bayes’ rule is the only statistical method for quantifying the probability of competing hypotheses being true given available data (Hobbs and Hilborn, 2006). During optimisation, frequentist methods iterate across parameter values to find the MLEs, whereas Bayesian methods integrate the likelihood of all possible values of each parameter, given all others. Chapter 1 Introduction 15 Although Bayesian methods have led to some controversy, it is not Bayes’ theorem itself that is in doubt, even for the most ardent frequentist, but merely how it is interpreted and applied in parameter estimation methods. Bayes’ theorem states that, for a given model M which depends on parameters θ and data X to describe data Y, then: (Eqn 1.2a) assuming fixed data Y, this leads to the unnormalised posterior density: (Eqn 1.2b) which is more frequently written simply as: (Eqn 1.2c) In Bayesian terminology, the term P(θ |Y) is called the joint posterior probability distribution, and gives the distribution of parameter values that best fit the data, which is the distribution of interest but which in many cases cannot be directly sampled from. The term P(Y|M(θ,X)) or simply P(Y| θ), when regarded for fixed Y and variable θ, is the likelihood function, and P(θ) is the prior of model M dependent on parameters θ (often denoted π(M(θ)) or simply π(θ)). For multiple parameter models, Bayesian methods seek the marginal posterior distribution for each parameter conditioned on all others. Thus, for a model with two parameters, equation 1.2c takes a slightly different form (Gelman et al., 2004): (Eqn 1.3a) Finding the conditional posterior distribution given the data for a single parameter, say , requires integrating over all other parameters: d d (Eqn 1.3b) It is rare to be able to directly sample from the right hand side of equation 1.3b (e.g. using the Gibbs sampler, see below) because this requires the prior distributions of all parameters to be conjugate – that is, of the same family – so numerical integration techniques are often required (e.g. MCMC sampler, see below). Monte Carlo simulation algorithms such as the Gibbs sampler and the Metropolis-Hastings algorithm are extremely useful tools for parameterisation. These methods work by taking a random walk through parameter space to estimate the conditional posterior distribution . In a multiparameter model, parameters are not sampled from the full conditional distribution, but rather individual parameters are chosen and sampled from the conditional distribution given all other parameters, , so-called alternative conditional sampling. Sampling then continues for many iterations until the Markov chain has converged to the joint posterior of all parameters. When the prior distributions of the parameters are conjugate (that is, the conditional distribution is known and can be directly sampled from) this is done directly using the Gibbs sampler. If is not known then a rejection method such as the Metropolis or Metropolis-Hastings algorithm may be used. Chapter 1 Introduction 16 The Metropolis algorithm works as follows. At each step in a random-walk, a parameter is selected and a new value θ* drawn from some pre-defined distribution. The likelihood of the new parameter set is calculated and compared with the likelihood of the previous parameter set θ (the likelihood ratio). Then the new parameter set is accepted with probability equal to the ratio of the posterior distributions (the posterior odds): Accept θ with Probability = min (Eqn 1.4) That is, the posterior odds are equal to the prior odds multiplied by the likelihood ratio (Gelman et al., 2004) and in practice the right hand side of equation 1.4 is calculated during optimisation. Such Bayesian methods not only sample parameter values that are better according to the likelihood than the previous sample (as frequentist likelihood methods do), but they also accept some parameter sets with lower likelihood if the likelihood is close, thereby sampling (asymptotically) directly from the posterior distribution of parameter values. By sampling directly from the posterior, Bayesian methods provide intervals of belief for parameter values that differ from the frequentist confidence interval: a Bayesian 95% credible interval is exactly the interval within which it is believed (given the data) that there is 95% probability that the true value of the parameter lies. Some Markov chain methods require individual parameter distributions to be specified (e.g. Gibbs sampling, a method which allows the posterior distribution to be sampled from directly), but others do not, which can be an advantage when there is no particular reason to choose a certain distribution. When fitting multiple parameter models using environmental predictors, Bayesian methods provide a substantial advantage for ecologists, because such variables are often partially correlated (e.g. denser forests are often found in wetter places). This means that the parameters associated with their effects are also likely to be partially correlated (Kruschke, 2010a), whereas frequentist-fitted confidence intervals can be misleading because such methods are unable to account for high levels of correlation among estimated parameters (Straume and Johnson, 2010). Bayesian methods provide an easy solution to such a problem because, instead of providing an estimate of the mean and variance of each parameter separately from all others, they provide parameter estimates in sets, each dependent on the other (estimating one parameter requires integrating across all others). These joint parameter estimates, drawn from the joint distribution of the model parameters, may therefore be used to make model predictions in a way that robustly accounts for correlation within a dataset (and can be used in a straightforward way for future research). In Chapters Two and Four, I demonstrate how MCMC methods allow the parameterisation of models with partially correlated variables (such as drought length and annual precipitation) on forest processes, and how the posterior distribution samples produced by the MCMC runs easily allow for the effect of one variable to be examined whilst controlling for variation in all others. Although frequentist approaches such as generalised linear modelling do allow the variance and covariance of parameters to be estimated, the approach is less Chapter 1 Introduction 17 intuitive and flexible, and requires specification of parameter distributions, which are not easily well chosen. The use of priors, representing prior belief in a parameter set or model, is often cited as an argument against Bayesian methods, since priors imply prior belief, which is subjective, and therefore Bayesian methods cannot be objective statistical tests. There are two main arguments as to why this is not a good reason to reject Bayesian methods. Firstly, with all but very small amounts of data, the strength of the prior can easily be set to be weak (or ‘uninformative’) so that information from the data far outweighs the prior's influence on the final estimated parameter values (Kruschke, 2010a), and the influence of prior strength on parameter estimation may easily be examined (e.g. Dietze, et al. 2008). Secondly, prior beliefs may be highly logical in nature, for example the height of a particular species of tree should be positive and less than 50m. Intelligent use of priors can get the most out of scarce data when there is good knowledge of a system from previous experiments. For example, in forest dynamics, good data on processes for rare species may be extremely difficult to collect, but through hierarchical modelling, strength can be ‘borrowed’ from other species to constrain processes in order to take sensible parameter values via constraint through priors (see below for further discussion). In addition, there is nothing inherently Bayesian about using prior information, and within a frequentist likelihood framework knowledge may be accumulated from one experiment to another (Hobbs and Hilborn, 2006). 1.4.5 Parameterisation without sufficient data There are many times when parameterisation of particular ecological models may be difficult due to insufficient data to constrain the desired model. Parameterising species-specific models can pose difficulties when species are rare and there is not enough data to constrain models for each individually. Hierarchical Bayesian modelling (HBM) provides a solution to this problem by fitting both individual species' models and a 'universal' model relationship for all species together, essentially reducing the number of free parameters by constraining the models of less common species to be more similar to the population average (Gelfand et al., 1990; Gelman and Hill, 2007). Parameters for the ‘species-average’ model may simply be fitted as averages of all species or they may be fitted as functions of species’ traits, therefore allowing for systematic variation in the average model, where species with similar traits have more similar parameters. For example, Dietze et al., (2008) use HBM to construct both species-specific and an across-species allometric relationship based on species traits of shade tolerance and wood properties. This approach not only constrains the value of rare species’ parameters to be informed by those of more common ones with similar trait values, but also allows for future estimation of parameters for unobserved species based on these traits alone, and without having to collect field measurements. The advantages of the hierarchical approach have been recognised in many areas of ecology and HBM techniques have been used, for example, to model species’ distributions accounting for spatial random effects (Hooten et al., 2003), to estimate tropical tree Chapter 1 Introduction 18 species’ growth rates (Feeley et al., 2007) and to evaluate scaling models (Price et al., 2009; Coomes et al., 2011). Maximum likelihood, MCMC and HBM methods all require the calculation of a likelihood – that is, the probability of the data given the model – but there are times when such a calculation is impossible or impractical, for example when the process we wish to model is either unobservable or too difficult or expensive to observe directly. Approximate Bayesian Computation (ABC) methods present a mathematically rigorous solution to such problems, because they may be used to fit models to data when the likelihood cannot be formulated or is computationally prohibitive to analyse (Tavaré et al., 1997; Pritchard et al., 1999; Tanaka et al., 2006; Sisson et al., 2007, 2009). ABC methods may be used to estimate parameters for models of fine-scale complex processes for which only coarse- scale aggregated data are available. Instead of comparing candidate parameter sets using likelihoods, ABC methods use simulation models to create simulated datasets, then summarise the simulated data and compare it with the real summarised data. These techniques typically take a truly Bayesian yet likelihood-free approach to parameterising models by incorporating both summary data and prior knowledge of unobserved processes to estimate parameter values. Over the last decade these likelihood-free ABC methods have increasingly been applied to data in areas such as epidemiology, for example to estimate disease transmission dynamics when only data on clusters of cases with identical genotypes are available (Tanaka et al., 2006), and population genetics (Foll et al., 2008), but have barely reached ecology (but see Jabot and Chave, 2009). In Chapter Five, I demonstrate the power of these methods to estimate annual recruitment rates using count data of small trees and carefully constructed priors to describe juvenile growth and mortality. 1.4.6 Models selection: methods for testing ecological hypotheses All models are wrong, and models to describe simulate ecological and environmental processes make explicit assumptions that are known to be wrong, yet they are necessary to reduce the overwhelming level of complexity in the natural world. Questions in ecology often arise from a desire to find the best model to describe a process, and pose two or more competing models formulated to represent competing ecological hypotheses to be compared (Hobbs and Hilborn, 2006). Complex models with more parameters inevitably leads to better fits to training data, but overfitting can cause problems with covariance between parameters, and leads to overcomplicated model structures that are too difficult to analyse. Sensible model comparison and selection is therefore vital to ensure that patterns in processes are described in a way that is sufficient to capture the variation of interest, whilst taking care not to create models whose complexity makes them too difficult to understand. Model selection in classical frequentist statistics is, in its most basic form, null hypothesis testing. Simple ‘no effect’ versus ‘some effect’ p-value significance tests do not provide much information about the best model structure when presented with a set of competing models. ANOVA and F-tests can be used to compare nested models, and to perform stepwise regression to construct the Chapter 1 Introduction 19 best model by analysing the amount that residual variation changes with the inclusion (or exclusion) of explanatory variables. Maximising fit using adjusted R 2 or Mallow’s Cp are both methods of selecting the best model fitted using OLS regression, which penalise models with higher numbers of parameters and can be used to compare non-nested models. The likelihood ratio test (LRT) compares the likelihood of two competing nested models and selects the more complex one only when its likelihood is significantly greater according to a χ2 statistic. It is the most commonly used null-hypothesis approach (Johnson and Omland, 2004) and, since it compares nested models, may be used with a stepwise selection approach to construct the best model. Stepwise algorithms are particularly important when there are a large number of models in the candidate set, because comparing all models may be impractical. However, the selected best model using such an approach may depend on the order in which the models were tested, and if not all possible models have been tested may therefore not be the best overall model. LRTs are unsatisfactory for all but the simplest situations for a number of reasons. They can only be used for nested models and rely on the choice of significance level to accept or reject a model (Posada and Buckley, 2004), when in reality we are often interested in comparing competing non- nested models and wish to know not just which is the best model to represent the data, but also how much better it is. Some stepwise methods of LRTs are so sensitive that they may even result in different models being selected as the best depending on their initial configuration (Pol, 2004). Information criteria bypass these problems and provide much more flexible approaches to model comparison. The Akaike information criterion (AIC; Akaike, 1974) is the most common and is derived from the Kullback-Leibler information distance (K-L; Kullback and Leibler, 1951), which defines the distance between two models or the information lost when a poorer model is used to approximate a better one (Burnham and Anderson, 2002). K-L cannot be used directly because it relies on knowledge of the 'true' model, but AIC is an asymptotically unbiased estimator of the expected relative distance (the K-L distance). AIC is defined using the maximum likelihood (L(Y|M1( ˆ ,X))), whereˆ are the maximum likelihood estimators) for model M1 with m1 parameters give as: AIC= -2 log(L(Y|M1(ˆ ,X))) + 2m1 (Eqn 1. 5) The term -2log(L(Y|M(θ,X))) is sometimes called the deviance. AIC values for a set of models M1,...MN are calculated and compared, and the model with the minimum AIC is chosen. It is important to know the relative support for each model tested in the data. Absolute AIC values have no meaning on their own, and it is the difference between the AIC for the best model and each of the others that is used to determine the relative support for each, Δi=AICi – AICMIN (Eqn 1.6) Models with Δi<2 are judged to be as well supported as the best model, and those with Δi>10 have no support (Burnham and Anderson, 2004). AIC is difficult to apply for multilevel or hierarchical models, because the number of parameters is not well defined for these situations. Instead, the Chapter 1 Introduction 20 deviance information criterion (DIC; Spiegelhalter et al., 2002) is used to perform model comparison for hierarchical models, and is calculated from posterior samples from an MCMC chain: DIC=- 2 log(L(Y|M( ,X))+ 2pD (Eqn 1.7) where  is the mean parameter value, the term - 2 log(L(Y|M( ,X))) represents the mean deviance and pD is the effective number of parameters (calculated by comparing the mean of the deviances from the posterior chain and the mean deviance). DIC was constructed to be a generalised version of AIC, and in a non-hierarchical model DIC and AIC should be approximately the same (Wagenmakers and Farrell, 2004). The Schwarz or Bayesian information criterion (BIC; Schwarz, 1978) is formulated in a similar way to AIC in that it also uses the likelihood and the number of parameters to compare different models, but BIC also considers the number of data points used and penalises models with more parameters more strongly (so it overfits less in comparison). For a model M1 with m1 parameters fitted to n data points, BIC is defined using the maximum likelihood L(Y|M1(ˆ ,X)) as: BIC= -2 log(L(Y|M1(ˆ ,X))) + m1log(n) (Eqn 1.8) AIC and BIC can often be used in the same situation, but since both may be considered Bayesian and both may be derived from non-Bayesian approaches, selecting one over the other does not depend on being a Bayesian practitioner or not (Burnham and Anderson, 2004). There are conditions under which AIC will outperform BIC and visa versa, which can be demonstrated using simulated data with known model structure and parameter values (e.g. Acquah, 2010). BIC is likely to choose a simpler model than AIC, and unlike AIC is both consistent as sample size increases and accounts for parameter uncertainty (Wagenmakers and Farrell, 2004). However, the derivation of BIC assumes that the 'true' model is amongst the model set being tested (Burnham and Anderson, 2004), which may be unrealistic for ecological applications. Both AIC and BIC can be used to produce weights for a set of models. Akaike weights are calculated for each model from the difference in its AIC value relative to the whole set, and can be interpreted as the probability of each model conditioned on the set, thereby giving insight into the relative confidence in the best model (Wagenmakers and Farrell, 2004). BIC weights are also calculated using the difference in BIC value for each model relative to the whole set, and can incorporate prior belief in each model (although commonly priors are set as equal for all models). They represent the posterior model probability; that is, the probability that each model is correct given the data, and therefore also show the level of confidence in the best model relative to the others. 1.4.7 Bayesian methods for forest dynamics modelling Bayesian approaches provide a straightforward way of comparing competing hypotheses for the four processes typically included in forest dynamics models (growth, allometry, mortality and recruitment) described in section 1.3. For many forest types, the extent to which these processes are dependent on Chapter 1 Introduction 21 stem size, climate and competitive environment, and the amount of variability between different species, is not well studied. Moreover, incomplete or insufficient inventory data and high species diversity means that defining such dependencies is not necessarily straightforward. In this thesis I present applications of Bayesian methods for model selection and parameterisation for these processes, using forest inventory data from two regions: the Eastern US and Spain. I fully parameterise models for all four processes for the major tree species in Spain, and demonstrate how these models may be incorporated into a model of whole-stand dynamics. 22 Chapter 2 Eastern US mortality 23 2 Influences of forest structure, climate and species composition on tree mortality across the Eastern US Abstract Few studies have quantified regional variation in tree mortality, or explored whether species compositional changes or within-species variation are responsible for regional patterns, despite the fact that mortality has direct effects on the dynamics of woody biomass, species composition, stand structure, wood production and forest response to climate change. Using Bayesian analysis of over 430,000 tree records from a large eastern US forest database we characterised tree mortality as a function of climate, soils, species and size (stem diameter). We found (1) mortality is U-shaped vs. stem diameter for all 21 species examined; (2) mortality is hump-shaped vs. plot basal area for most species; (3) geographical variation in mortality is substantial, and correlated with several environmental factors; and (4) individual species vary substantially from the combined average in the nature and magnitude of their mortality responses to environmental variation. Regional variation in mortality is therefore the product of variation in species composition combined with highly varied mortality-environment correlations within species. The results imply that variation in mortality is a crucial part of variation in the forest carbon cycle. 2.1 Introduction An understanding of tree mortality is central to any predictive understanding of forest dynamics. The long-term dynamics of woody biomass are regulated by the difference between gains through individual growth and losses through mortality. This makes tree mortality a crucial determinant of the forest carbon cycle, the future of which is a major source of uncertainty in Earth System Model predictions of future climate (Sitch et al., 2008). Moreover, differences in mortality rates among species appear to be major determinants of ecological succession (Kobe et al., 1995; Purves et al., 2008), the geographical ranges of species (Loehle, 1998; Purves, 2009), stand structure (e.g. stem size distributions: Coomes et al. 2003; Muller-Landau et al. 2006), and responses of forests to climate change and disease (van Mantgem and Stephenson, 2007; Kurz et al., 2008). However, we currently Chapter 2 Eastern US mortality 24 have little quantitative information about the nature, magnitude or causes of geographical variation in tree mortality. The simplest approach to making predictions about mortality in a changing world would be to correlate stand-level mortality obtained from permanent plot data with climatic variables, and use these relationships to predict changes under future climate scenarios. The problem with this approach is that it neglects the effects of species, individual size and competition, factors that individually have been shown to strongly affect mortality at the scale of the individual tree, with potentially serious consequences for landscape-level predictions. In order to predict the impacts of changing climate on forest-level mortality, it is therefore important to isolate the effects of these factors because they are likely to show complex, semi-independent changes in the future. For example, in much of the temperate zone, many forest stands are successional and regenerating, undergoing directional change in species composition independent of any changes in the environment (Rhemtulla et al., 2009; McMahon et al., 2010). Additionally, species are unlikely to disperse rapidly enough to track their optimal climatic conditions under rapid anthropogenic climate change, leading to combinations of species composition and environment that do not occur currently (He et al., 1999; Morin and Thuiller, 2009). Tree-level mortality patterns can also be confounded by external actions: harvesting can create various novel combinations of basal area, size distributions and species composition (e.g. de Graaf et al., 1999; Villela et al., 2006), and pests and pathogens are often highly species-specific (e.g. sudden oak death: Rizzo and Garbelotto, 2003; Lovett et al., 2006). To estimate the individual effects of each factor, it is necessary to study factors simultaneously in order to tease apart their individual effects, otherwise the apparent effect of one is likely to be confounded by the others (e.g. apparent differences in species' average mortality rates might reflect differences in the average environments occupied by those species: Caspersen and Kobe, 2001). Here we use the Eastern USA Forest Inventory and Analysis (FIA) dataset to parameterise, for each of 21 common US tree species, a logistic regression model that assigns an annual probability of mortality to an individual tree given its size, species identity, competitive environment (plot basal area) and physical environment. We estimate the nature and relative magnitude of the different factors affecting tree mortality and parameterise a model that could be useful in predicting potential responses of US forest carbon stocks to climate change (e.g. Joyce, 1995). Here we report: (1) how each factor affects the mortality rate of individual trees; (2) whether, and how, species differ in their underlying mortality rates and responses to size, competition and the environment; and (3) differences in the environmental dependency of forest stand-level vs. species-level mortality, which determine the level of model complexity required to accurately predict forest mortality in a changing environment. Chapter 2 Eastern US mortality 25 2.2 Materials and methods 2.2.1 Forest Inventory data We used the pre-1999 USA Department of Agriculture Forest Inventory and Analysis (USDA FIA) dataset containing tree-level data for 182 species from a network of plots distributed across the Eastern USA (Smith, 2002). The data comes from forest inventory plots which were surveyed in the 1980s and again in the 1990s, although the interval between surveys differs between states between 1 and 21 years (93% of survey intervals were between 6 and 15 years). Surveys were taken using a two- phase sampling procedure known as double sampling for stratification. In the first phase random points were chosen on aerial photographs and classified by land cover and forest type, and in the second a random subsample of each class were selected and established as field plots. Five or more points were chosen within each plot, around which several sub-plots were established and sampled using variable radius sampling, whereby the effective subplot size differs according to tree size (for more details see Purves et al., 2004). Species, size (diameter breast height, DBH) and status (alive, dead from harvesting, dead from natural causes) were recorded for each tree sampled, along with plot basal area (m 2 ha -1 ). The FIA survey was designed specifically to allow accurate estimates of average forest characteristics such as species composition and average tree size through scaling from the tree, through the stand, to the regional level (Smith, 2002; Purves et al., 2004). Before analysis began, the dataset was filtered to include only those dead trees that we could be certain were not removed by human activity, and to remove various kinds of errors in the data (e.g. false mortality events corresponding to subplots that were measured in the first, but not the second, survey). The model was parameterised for 21 of the most common species, using 438,401 individual tree records in total, accounting for around 60% of all trees in the reduced dataset. Due to the high number of possible predictors being considered, only species with over 10,000 individuals in the data set were used for parameterising the model. Of these, two species (Ulmus americana and Abies balsamea) were known to have suffered severely from disease and pests during the survey period. Other species are likely also to suffer a variety of impacts from diseases which are part of the mortality patterns studied here. However, the disease impacts on Ulmus americana and Abies balsamea are known to be so severe, episodic, and localised, that in our opinion it was better to exclude both species from the analysis. Since these factors were not included as predictors of mortality in our model we did not include these species in the model fitting. We also did not consider the effects of other disturbances, both natural (e.g. fire and hurricanes) and human, on the observed mortality in the dataset. Such disturbances are likely to have had a marked effect on current species composition (Russell et al., 1993; Fuller et al., 1998; Bürgi et al., 2000) and demographic rates (Foster, 1988), but are likely to be complex and interacting and, combined with a lack of a detailed Chapter 2 Eastern US mortality 26 land-use history, the quantification of such disturbances and evaluation of their effects may be unachievable in many areas (Motzkin et al., 1999). 2.2.2 Environmental data Since little is known about the geographical variation in tree mortality we had little information to judge which climatic factors might correlate with mortality. However, there have been many studies linking growth with a wide variety of climatic variables; for example, solar radiation, (Rolland, 1993; Vaganov et al., 2009), precipitation and drought (Pacala et al., 1994; Hanson et al., 2001; Miao et al., 2009), temperature (Matala et al., 2005), severe frost (Liu and Muller, 1993) and wind speed (Kronfuss and Havranek, 1999). Since many studies link individual rates of mortality within a species as a function of growth (e.g. Kobe et al., 1995; Wyckoff and Clark, 2002; Muller-Landau, Condit, Chave, et al., 2006; Wunder et al., 2006) there is good reason to believe that mortality also varies with many different climatic variables. Our approach was therefore to assess which of these variables were most closely correlated with observed mortality patterns, rather than to attempt to generate hypotheses, in order to determine which were most important within our data. We assigned environmental factors to each tree using two sources of environmental data, both available on a 0.5° x 0.5° degree. The first source was the CRU05 climatology product (Climatic Research Unit, University of East Anglia: New et al., 1999) which provides monthly averages for many climate variables including temperature, precipitation, radiation, frost frequency, vapour pressure, cloud cover and wind speed (monthly average refers to the average over the period 1961 – 1990). We took the mean of each climate variable rather than climate observed over the survey period associated with each tree (which differs from tree to tree). From the CRU05 data we calculated the additional metrics of minimum temperature, degree days and average warm season (as opposed to annual) precipitation. The second source of environmental data was the Vegetation/Ecosystem Modelling and Analysis Project (VEMAP) (Kittel et al., 1995), a multi-institutional project to develop a database of climate, soils and vegetation on a 0.5° latitude/longitude grid across the United States for use with ecosystem physiology models. From this source, we took only the data on US soil, which included over 20 different metrics including soil depth, and measures of soil texture. In addition the FIA provided data on soil texture for each inventory plot, divided into five classifications from xeric (normally low or deficient in available moisture), through mesic (normally moderate but adequate available moisture) to hydric (normally abundant or overabundant moisture all year) (FIA-DB, 2008). The classification of each FIA site into one of these five soil classes is intended to be independent of the climate (e.g. rainfall) at that site. To avoid convergence problems during parameter estimation, we applied principal component analysis (PCA) to the 14 different environment variables (both from the VEMAP and CRU05 data) to remove highly correlated variables. Among highly correlated variables, the variable with the highest weighting in the principal components was retained and the rest discarded. This left four CRU05- Chapter 2 Eastern US mortality 27 derived climatic variables (radiation, yearly precipitation, mean annual temperature and maximum wind speed) to be included as possible mortality predictors, plus one FIA soil texture classification associated with each tree. We normalised each factor (i.e. subtracted the mean value and divided by the standard deviation) to allow for a simple comparison between the magnitudes of effects of each of the factors. We also checked that plot basal area was not highly correlated with the remaining climate variables. 2.2.3 Model description Tree mortality is a difficult property to estimate because unlike growth, it has only 2 possible outcomes from each re-measured tree (survived or died), and typical tree mortality rates are low (on the order of 0.1 to 2% year -1 ), such that large sample sizes and / or long re-measurement periods are required. Moreover this dataset contained varying re-measurement intervals, meaning that a simple 'proportion dead' would not have been informative (Purves et al., 2008). We therefore chose to parameterise a model describing the annual probability of death for each individual tree i, P(mortality, i). Since P(mortality, i) must lie between 0 and 1, we used a logistic transformation P(mortality, ) (Eqn 2.1) where ki (which can vary from ± ) is a function of the predictor variables. We included different combinations of the predictor variables: DBH (continuous); soil type (discrete, ranging from 1-5); plot basal area (i.e. FIA inventory plot) (continuous); and environmental variables (all continuous) as follows: (Eqn 2.2) where is a constant parameter, and f1 is a function of the first predictor variable (e.g. DBH) , f2 is a function of the second (e.g. precipitation), and so on. Initial analysis indicated that the relationship between DBH and mortality was U-shaped, corresponding to high mortality in small trees, low mortality for medium sized trees (typically 25-40 cm) and increasing mortality in larger trees. To describe this relationship we tried several different model equations and found the best fit to the data using the following functional form si e dbh dbh (Eqn 2.3) where 1 and 2 are parameters. In keeping with the qualitative pattern visible in the initial assessment of the size-dependency of mortality, equation 2.3 allows the initial decrease in mortality vs. size for small trees to be steeper than the increase in mortality for size for larger trees whilst giving high flexibility to the shape of the response. For each environmental variable V (i.e. climate and soil measures) we considered two alternative functional forms: ,V i V if V (linear) (Eqn 2.4) 2 ,V i V i V if V V   (non-linear) (Eqn 2.5) Chapter 2 Eastern US mortality 28 where Vi is the value of environmental variable V associated with tree i, and  and  are parameters. We used the same functional forms to include the effects of plot basal area B (m 2 ha -1 ): ,B i V if B (linear) (Eqn 2.6) 2 ,B i B i B if B B   (non-linear) (Eqn 2.7) where Bi is the plot basal area B associated with tree i, and  and  are parameters. Although we chose to use a quadratic functional form, we did not constrain the shape further so that, within a species' range, it could predict shallow or steep monotonic curves, as well as U-shaped (or hump- shaped) responses. Since we had no strong evidence for a particular across-species response for any of the environmental variables we felt that a quadratic functional form would be sufficiently complex to capture essential patterns without being too complex. Together, equations 2.1-2.7 allow for a very large possible number of models, with a wide variety of numbers of parameters, depending on which predictor variables are included, and depending on whether each variable is included using a linear or non-linear (quadratic) functional form. We allowed each parameter in any given model to be either species-specific (e.g. in equation 2.4 this would give us 21 separate parameter values, one for each species, each of which is unaffected by data from other species) or global, that is, shared among species (e.g. in equation 2.4 there would be a single value for all trees regardless of their species). To avoid having to fit all possible models, we used a selection procedure that compared models with major differences in their predictor variables (see Model selection, below). 2.2.4 Parameter estimation We used Bayesian methods based on Metropolis-Hastings Markov Chain Monte Carlo sampling (Gelman et al., 2004) to estimate values and confidence intervals for each of the parameters in each model. These methods were chosen because they allow for simple, efficient estimation of parameters, including confidence intervals. However, we did not use informative priors, so the outcome of the analysis can be expected to be similar to the outcome of a Maximum Likelihood analysis using the same data and models. The first step of the analysis was to define, for a given candidate model M, the log-likelihood of the inventory data (referred to here as X), given a particular set of parameters (referred to here as ) values for model M :       1 mortality, if tree survived | , ln 1 1 mortality, if tree died Si i i S P i i l X M P i i             (Eqn 2.8) equation 2.8 represents a sum, over all trees i, of the logarithm of the probability of the observation for i (survived or died), given the model structure M and parameter set , where Si is the survey interval (years) for tree i. Chapter 2 Eastern US mortality 29 We used non-informative uniform priors on all parameters so the MCMC algorithm (see below) needed to refer to the log-likelihood only. However, for numerical reasons we imposed upper and lower limits on the allowable values of all parameters, i.e., a prior probability of 0 on parameter values outside of the allowable range. We set the allowable range much wider than the plausible values, and also checked the posterior distributions to make sure the tails of the posterior distributions were a long way from the edge of the allowable range. The next step was to estimate values for the parameter set in model M, given the definition of the log-likelihood (equation 2.8). We did this using an adaptive Metropolis MCMC algorithm (Gelman et al., 1999, 2004), which returns random samples from the posterior distribution of . At each iteration, a particular parameter pk is chosen and altered by adding a random value from a normal distribution where vk is specified for each parameter. The likelihood of the data given the new parameter is calculated and the parameter change is 'accepted' based on the ratio of the new likelihood and the previous likelihood: P acceptanceofnewparameterset min The variance vk for each parameter was tuned during a 'burn-in' period to achieve an optimal parameter acceptance rate of 25% (Gelman et al., 2004) so the samples returned from the MCMC can be said to have efficiently sampled the posterior of each parameter. We implemented the MCMC algorithm by initializing each parameter value at a random point close to the middle of the allowable range, allowing a suitable burn-in period (between 25,000 and 1,000,000 iterations) for the algorithm to reach quasi-equilibrium, then recording every 100 th sample of (to avoid auto-correlation) from a post burn-in period of between 50,000 and 250,000 iterations (the number required depended on the speed of model convergence and the number of parameters). This provided us with a set of between 500 and 2,500 samples of for each model M that we parameterised. From these samples, we calculated the mean, and 95% confidence interval, of each parameter p within . For the best-fit model we re-ran the model four times with differing starting parameter values and found the results were unchanged. 2.2.5 Model selection As metrics to compare alterative models, we calculated, for each model M that we parameterised, the Akaike Information Criterion (AIC; Akaike, 1974) and the Bayesian Information Criterion (BIC; Schwarz, 1978). Both criteria reward models for providing a better fit to the data, but penalise models according to the number of free parameters that they contain, thus allowing for model selection from sets of models that differ in model complexity. However, the AIC penalises complexity less strongly than the BIC, so it is useful to compare the two criteria. Simple likelihood-ratio based comparisons would not have been appropriate since the models were, in general, non-nested (Hilborn and Mangel, Chapter 2 Eastern US mortality 30 1997). Both criteria require an estimate of the maximum likelihood, for which we used the maximum value of the log-likelihood encountered by the MCMC algorithm in the post burn-in period. Given the high number of possible mortality predictors, the options of functional forms presented by equations 2.2-2.7 and the choice of species-specific or global for any parameter, there was a very large set of possible models M. We wished to select an appropriate best model from this set, but without having to examine every possible combination of possible predictors. To do this we used the procedure outlined in the next three paragraphs. First, we established which of the possible predictor variables was the best single predictor of mortality by parameterising all possible mortality models featuring one predictor variable (referred to here as 1-d models). This set of models was still relatively large (28 different models), since the predictor variable in question could be included using a linear or non-linear function, and with species-specific or global parameters (see equations 2.4-2.6). We also tested some of the closely correlated alternative climate predictors in this way, but none gave a better fit than the set we had already chosen. Comparing the AIC and BIC values associated with each model allowed us to determine whether, considered in isolation, each predictor variable was best described using species- specific vs. global parameters, and a linear vs. non-linear functional form (see equations 2.4-2.6). This analysis suggested that all predictor variables were best described using non-linear, species-specific functional forms. Therefore we decided to retain, within the larger set of all possible models, only those models that included non-linear functional forms. Further, comparing the maximum likelihood of the different 1-d models allowed us to rank the predictor variables in descending order of importance (meaning importance considered in isolation). The rank was: size >> radiation > yearly precipitation > mean annual temperature > plot basal area > maximum wind speed > soil type. Since size (DBH) was by far the best single predictor of mortality, we decided at this point to discard, from the large set of all possible models, any models not including DBH as a predictor variable. Second, we sought, within the remaining set of models, the best set of environmental variables to include in the model. Since radiation was the best single environmental predictor, we tested each additional environmental predictor to find the best two-predictor combination, using species-specific responses, giving a model of the form: k = α + β1(DBH)exp(β2(DBH)) + β3(radiation) + β4(radiation) 2 + β5(environment) + β6(environment) 2 We found that adding in yearly precipitation gave the best fit. We repeated this procedure to find the best three and four parameter models, and finally checked that including all five predictors (radiation, yearly precipitation, mean annual temperature, maximum wind speed and soil type) gave a better fit than the other models, using AIC and BIC. These steps gave us for types of predictor variable: the constant (equation 2.2), DBH (equation 2.3), the set of five non-linear environmental effects (equation 2.5) and the non-linear competition effect (plot basal area: equation 2.7). To determine the final model form we generated a Chapter 2 Eastern US mortality 31 set of models which allowed us to test whether each type of predictor should be species-specific or shared, and whether the extra model complexity added by including environmental and competition effects in the simple size model was justified by the improvement in fit. We tested models using every combination of species-specific or shared effects for each type of predictor, as well as every combination with or without environment and competition effects (36 models in total). The full list of different models tested are shown in Table A.1 (in Appendix A page 121), along with AIC and BIC scores. The score of the best model was a very large improvement on the next best, although it is worth noting that models without environmental effects performed significantly worse than those without plot basal area as a predictor. 2.2.6 Parameter significance The majority (74%) of parameters' 95% posterior distributions did not include 0, indicating significant effects. None of the posterior distributions for the constant or size parameters (αj, β1j and β2j in equation 2.4) included 0, while the least significant deviations from zero were seen for soil type and maximum wind speed parameters, and for the species Liquidambar styraciflua, Thuja occidentalis and Nyssa slyvatica. In principle many additional parameterisations could be used to eliminate some effects for some species (i.e. remove terms associated with species-parameters with posteriors including zero), but we considered that this extra computational effort could not be justified in terms of increased scientific understanding. 2.2.7 Interpretation of the results In order to compare the different mortality rates predicted for each species we calculated a single ‘baseline’ mortality of each species as the predicted mortality of a tree of standard si e growing in a standard environment (we used both the mean environment, taken over the study region, for all variables together with a ‘mesic’ soil texture; and the species' own median environment). We chose to use 20 cm as the standard stem diameter because it is approximately the size of a canopy tree in this region (Purves et al., 2008). In order to visualise geographical patterns in observed mortality rates, we calculated a mortality rate for each plot ("plot-averaged mortality") by fitting a single-parameter logistic model to the data, and used the coordinates of each to create a regional mortality map. We visualised geographical patterns in predicted mortality rates by creating simulated datasets which were identical to the original dataset except that whether each tree died or not was determined using the model's posterior parameter values. We then used the simulated data to calculate a model-predicted mortality rate for each plot. For each tree i we calculated its annual mortality rate based on the model equation generated from the randomly chosen parameter set. From this we determined the probability it died, Pi, over the whole survey period, and then assigned it as dead with probability Pi and alive with probability 1- Pi within the simulated dataset. We generated 100 simulated datasets in this way, using Chapter 2 Eastern US mortality 32 different parameter sets randomly drawn from the joint posterior of the parameters, and combined their results using likelihood profile methods to predict a single model-predicted mortality rate for each plot. Using multiple randomly chosen samples from the joint posterior instead of simply the mean value of each parameter accounts for any co-variance in the parameters and the effect this would have on the predictions. By comparing the predicted and observed maps we were able to examine how well our predictions fitted the observed mortality and in which regions there were mismatches. We also wanted to create maps showing how mortality varies regionally in response to variation in species identity, stand structure (stem size and plot basal area) and environmental conditions, whilst controlling for variation in the other factors. We devised an approach to do this, based on creating simulated datasets in various different ways which selectively removed variation in the predictors which were not of interest. For example, to analyse the mortality patterns arising from variation in stand structure, we generated new 100 simulated datasets in which the tree alive/dead column was predicted from our model by using size and plot basal area, but assuming all trees were Acer rubrum (the most common and wide-ranging species in the dataset) and every tree experienced the same environmental conditions (the region's average). Similarly, to analyse mortality patterns arising from species composition, we retained species information but assumed all trees had the same size (20 cm DBH), basal area (the average density) and environmental conditions (the region's average) when creating the dead/alive column of the simulated datasets. Finally, to analyse mortality patterns arising from variation in environmental conditions we retained environmental information but assumed all trees were A. rubrum, and had the same size (20 cm DBH) and were in plots of average density. Our maps are an imperfect way to partition spatial variation but this method does allow us to partition variation in mortality due to each factor by selectively controlling for variation in the others. Had we chosen a different size of tree or a different set of environmental conditions, we would expect to have seen similar spatial variation in mortality rates but a different overall level of mortality. Since different species responded in different ways to changes in environment and stand structure, we also calculated variation in mortality due to these factors but using Pinus taeda (the most common gymnosperm species) instead of A. rubrum. However, this species has a much smaller range than A. rubrum so we only considered variation in mortality in the region in which the species is found. We were also interested in seeing how mortality varied along the range of each predictor, both for all species together ("forest-averaged mortality") and for each individual species ("species- averaged mortality"). We generated estimates of how observed mortality varied along the range of each predictor by binning the raw data according to the predictor of interest into equal sized bins (i.e. each containing the same number of stems) and found the best single annual mortality rate for the whole bin in the same way as before, using a single parameter logistic model. We did this both for the raw data (for just the 21 species for which we parameterised the model) and for all the data (including Chapter 2 Eastern US mortality 33 the rare species). In order to compare this to the model predictions for all species together (the forest- averaged mortality) we created 100 sets of simulated data as before (i.e. data of the same form as the original dataset but with alive/dead status based on our model predictions), ordered and binned these according to the variable of interest and calculated a single mortality rate for each bin, and a 95% confidence interval on this rate. Thus the forest-averaged mortality accounted for simultaneous changes in species composition and size structure across whichever gradient was being considered, and could be compared to the observed data. Finally, for each species we were interested in how mortality varied with changes in the variable of interest alone, but since the predictors (size, environment and stand basal area) all co- varied along each gradient we calculated the median conditions in which each of the species was found. For each model predictor we created 100 simulated datasets using parameter values randomly chosen from the joint parameter posterior distributions. In these datasets, each tree was given the median condition of its species (apart from the predictor of interest) and was assigned as dead or alive based on its predicted annual mortality rate. For example, in order to examine the sole effect of temperature change on mortality we re-assigned each tree the median size, precipitation, radiation, maximum wind speed, soil type and stand basal area in which its species was found in the original dataset, and kept only the temperature information for each individual tree and then created the 100 simulated datasets as before by selecting 100 parameter sets at random from the joint posterior. This gave us a spread of mortality vs. temperature functions for each species, where the spread represents parameter uncertainty (variation in parameters causing variation in probability of mortality) and sampling (random variation in whether lived or died given the probability of mortality). This allowed us to consider only the effect of temperature on that species mortality, whilst modelling the species in a reasonable environment. 2.3 Results 2.3.1 Model selection Using AIC and BIC, we found that the 7 best performing models all included species-specific environment effects, even when other predictors were not species-specific, or when plot basal area was not included. Plot basal area was only found to be a worthwhile predictor if its effects were species specific but did not benefit the model if the effect was shared among species. Models with non species-specific constant or size effects performed well, but the model with all predictors included as species-specific performed substantially better than all the others, according to both AIC and BIC. Therefore in our final model the function k (equation 2.1) took the form: Chapter 2 Eastern US mortality 34 kj = αj + β1j(DBH)exp(β2j(DBH)) + β3j(radiation) + β4j(radiation) 2 + β5j(precipitation) + β6j(precipitation) 2 + β7j(mean annual temp) + β8j(mean annual temp) 2 + β9j(max wind speed) + β10j(max wind speed) 2 + β11j(soil type) + β12j(soil type) 2 + β13j(plot basal area) + β14j(plot basal area) 2 (Eqn 2.9) where j is the species and the j and js were the parameters estimated (so a different function kj was estimated for each species). The MLEs, Bayesian means and confidence intervals for the parameters for each species of the best fit model (equation 2.9) are given in Table A.2 (Appendix A page 122). The predicted trends in mortality were close to the observed patterns across all predictor variables included in the model (Fig. 2.1, A.1-A.3 in Appendix A pages 126-127) suggesting that the structure of the model was appropriate for capturing mortality patterns within these data. 2.3.2 Species-mortality relationships Species showed very different baseline mortality rates, even when other effects were factored out (Table 2.1), and as a consequence plot-level mortality was found to be highly sensitive to species composition. To illustrate this point we compared species mortality rates calculated at the median environment of each individual species. These mortality rates differed widely: the highest, for Populus tremuloides, was 80 times larger for than for the lowest, Quercus prinus (Table 2.1). In addition to the differences in baseline mortality, species showed contrasting responses to environmental variation. For example, the model predicts substantial species differences in the direction and magnitude of responses to hypothetical increases in temperature and precipitation (Fig. 2.2). 2.3.3 Size-mortality relationship The relationship between size (DBH) and mortality was U-shaped for all species (Fig. 2.1a: p << 0.001 for all species). The highest mortality rates were found for the smallest trees and the lowest rates for trees of 18-37 cm DBH. The rate at which mortality decreased with size in saplings and increased with size in larger trees varied considerably among species, from relatively flat (e.g. Thuja occidentalis) to dramatic (e.g. Acer saccharum). However, species with higher minimum mortality consistently showed both higher sapling mortality and higher mortality at larger stem sizes (Spearman's Rank Correlation p<0.05 for both trends). Predicted forest-averaged mortality was U- shaped mortality vs. size (i.e. when all data were grouped together, the model applied to each individual and the total average mortality calculated). However, the upturn in forest-averaged mortality in large trees was less pronounced as larger size classes became increasingly dominated by species with low mortality rates. Chapter 2 Eastern US mortality 35 Table 2.1 Predicted baseline annual mortality rate (deaths tree -1 year -1 ) calculated for each species in the mean environment of the dataset for 20 cm DBH trees, and at each species' median environmental conditions (that is the conditions at which the highest number of individuals within the dataset were found), using the best fit model. 95% confidence interval for the mortality rates are also given. Mortality in forest mean environment Mortality in each species' median environment Species Annual mortality rate 95% CI Annual mortality rate 95% CI Acer rubrum 0.0035 (0.0034, 0.0038) 0.0022 (0.0020,0.0023) Acer saccharum 0.0108 (0.0104, 0.0112) 0.0052 (0.0049,0.0054) Betula papyrifera 0.0012 (0.0010, 0.0014) 0.0009 (0.0008,0.0011) Carya spp 0.0011 (0.0009, 0.0012) 0.0026 (0.0023,0.0028) Fagus grandifolia 0.0020 (0.0018, 0.0022) 0.0017 (0.0014,0.0019) Fraxinus americana 0.0016 (0.0014, 0.0018) 0.0008 (0.0007,0.0010) Liquidambar styraciflua 0.0040 (0.0037, 0.0042) 0.0048 (0.0045,0.0052) Liriodendron tulipifera 0.0005 (0.0004, 0.0006) 0.0009 (0.0008,0.0010) Nyssa sylvatica 0.0180 (0.0170, 0.0187) 0.0323 (0.0311,0.0336) N. sylvatica (biflora) 0.0044 (0.0040, 0.0049) 0.0098 (0.0090,0.0104) Populus tremuloides 0.0017 (0.0015, 0.0018) 0.0407 (0.0399,0.0414) Quercus alba 0.0016 (0.0015, 0.0017) 0.0013 (0.0012,0.0014) Quercus nigra 0.0094 (0.0088, 0.0102) 0.0068 (0.0063,0.0074) Quercus prinus 0.0004 (0.0003, 0.0005) 0.0005 (0.0004,0.0006) Quercus rubrum 0.0062 (0.0059, 0.0064) 0.0035 (0.0033,0.0038) Quercus stellata 0.0392 (0.0384, 0.0399) 0.0083 (0.0078,0.0088) Quercus velutina 0.0114 (0.0110, 0.0119) 0.0072 (0.0068,0.0075) Pinus echinata 0.0157 (0.0151, 0.0163) 0.0054 (0.0051,0.0057) Pinus taeda 0.0020 (0.0018, 0.0022) 0.0054 (0.0052,0.0057) Pinus virginiana 0.0047 (0.0042, 0.0052) 0.0282 (0.0271,0.0299) Thuja occidentalis 0.0114 (0.0110, 0.0119) 0.0011 (0.0010,0.0013) Chapter 2 Eastern US mortality 36 2.3.4 Environment-mortality relationships Of the several environmental factors included in the model, temperature and precipitation are particularly important in this region because they are likely to change substantially, and perhaps rapidly, under anthropogenic climate change (IPCC, 2007). Forest-averaged mortality was U-shaped against annual mean temperature. The minimum mortality, which occurred at a temperature of around 8-10 °C, was 6-9 times lower than the rate at low or high temperatures (mean annual temperature <5 or >15 °C) (Fig. 2.1b). This observed pattern was robust to whether all species, or only the 21 common species, were considered (Fig. 2.1b, green and orange lines respectively), and was reproduced by our model (Fig. 2.1b, compare green and purple lines). This occurred despite the fact that the observed forest-averaged mortality pattern across the temperature gradient was not always reflected in the species-averaged responses of the particular species present at those temperatures. In the range 10-15 °C, both the forest-averaged mortality and the species-averaged mortality for the majority of the species increased with temperature (Fig. 2.1b, grey lines). However, forest-averaged mortality decreased with increasing temperature below 10 °C and increasing mortality with increasing temperature above 15 °C. In contrast, species-averaged mortality for the majority of species found in these temperature ranges showed the opposite trend. Analogous mismatches in the response of particular species vs. the forest average were also found for precipitation Fig. 2.1c) and radiation (Fig. 2.1d). Forest-averaged mortality rates decreased with increasing precipitation up to a threshold of around 800 mm yr -1 and showed no clear trend thereafter (Fig. 2.1c), but individual species showed both increasing and decreasing mortality in the driest part of the range. At higher precipitation levels the forest-averaged mortality pattern was less clear, with some species showing increasing species- averaged mortality with higher precipitation (producing an overall U-shaped response to precipitation), and some a flat response. The opposite effect was found in the relationship between mortality and radiation, with a strong trend for increasing forest-averaged mortality up to a threshold point of about 200 W m -2 (Fig. 2.1d), after which the response was much flatter. For most species we found that species-averaged mortality vs. basal area was hump-shaped (p<0.05 for 16 of 21 species), with 50% of species showing maximum mortality in stands of 10-37 m 2 ha -1 (Fig. A.3 in Appendix A page 127). The inclusion of less common species raised the observed forest-averaged mortality rate, but otherwise left the patterns unchanged (Fig. 2.1 and Fig. A.1-A.3 in Appendix A page 126-127). Chapter 2 Eastern US mortality 37 Figure 2.1 Observed and predicted forest-averaged and species-averaged annual mortality rates (deaths tree -1 yr -1 , log scale) plotted against (a) DBH (cm), (b) mean annual temperature (°C), (c) total annual precipitation (mm/year), and (d) solar radiation (W m -2 ). Each panel shows the observed trends in mortality calculated using data from all species (orange) and from the 21 common species (green), and the predicted curves for 21 common species individually (grey) and together (purple). Individual species mortality rates are shown vs. changes in the predictor variable of interest alone, i.e. with all other predictor variables held at the median for that species. Error bars on the predictions (grey and purple) are 95% confidence intervals calculated from an error propagation procedure that accounted for parameter uncertainty. Error bars on the observations for the whole forest including rare species (orange) and 21 species combined (green) are 95% confidence intervals for mortality rates in the data (see Table A.2 in Appendix A page 122 for parameter values). 2.3.5 Geographical variation in mortality The model reproduces most of the geographical patterns in plot-averaged mortality observed in the FIA dataset (compare Fig. 2.3d and Fig. 2.3e) with a high correlation seen between observed and predicted mortality in plots with more than 10 stems (Fig. A.4 in Appendix A page 127: r 2 = 0.89). Since the model reproduced geographical variation well, we were able to decompose the variation into (a) (c) (d) (b) Chapter 2 Eastern US mortality 38 the separate effects of stand structure (stem-size distributions and plot basal area), environment and species (Fig. 2.3a-c). According to this decomposition, variation in species composition and environmental conditions were much more important than variation in stand structure in determining geographical patterns in plot-averaged variation in mortality. High observed plot-averaged mortality in the southeast is reproduced by considering only the environmental conditions of the region, but not when only stand structure or species composition are considered (Fig. 2.3c). In particular, several species common in the southeast (e.g. Nyssa sylvatica, A. rubrum and Quercus nigra) showed strongly increasing species-averaged mortality with the higher average temperatures, but this effect doesn't appear when only species composition is considered since temperatures in the region are much higher than these species' median environments. High plot-averaged mortality in the west is driven primarily by species composition (Fig. 2.3b); whilst variation in stand structure has relatively little impact on plot-averaged across the region (Fig. 2.3a). We checked whether our conclusions were dependent on our choice of species (i.e. A .rubrum) by creating the equivalent maps using the most common gymnosperm species, P. taeda (Fig. A.5 in Appendix A page 128). We again found that variation in environmental conditions resulted in higher variation in mortality than variation in stand structure. However, not all variation predicted by the model was explained by a simple sum of the three components, indicating strong interactions between them. For example, both stand structure and species composition (Fig. 2.3a-b) predict higher plot-averaged mortality in the northeast than is predicted by the model or is observed (Fig. 2.3d-e), indicating an interaction with environmental conditions (Fig. 2.3c), which predict lower plot-averaged mortality in the area. The largest differences between model predictions and observations of plot-averaged mortality were all in plots with less than 100 stems. Differences were mostly due to underestimated plot-averaged mortality by the model, particularly in the furthest northwest and southeast of the region where many plots were dominated by species too rare across the whole region to be included in our analysis (see Fig. A.6 in Appendix A page 129). 2.4 Discussion 2.4.1 Size and stand structure We found that size (DBH) was the single variable with the greatest effect on mortality rate at the level of the individual tree, with trees of intermediate size exhibiting mortality rates much lower than smaller, or larger, trees. This U-shaped relationship between size and mortality appears to be a common feature of forests, whether from sub-boreal (Umeki, 2002), temperate (Buchman et al., 1983; Monserud and Sterba, 1999; Coomes and Allen, 2007b) or tropical (Chao et al., 2008) regions. It Chapter 2 Eastern US mortality 39 seems likely that this feature results from two opposing effects: (i) mortality is often high when trees are small because they are competitively inhibited by taller neighbours, but with higher light levels show an increase in growth rate and reduction in mortality (Kobe et al., 1995); and (ii) a general increase in mortality in larger individuals due to senescence and/or increased exposure to strong wind and other disturbance agents (Busing and Pauley, 1994; Yang et al., 2003; Coomes and Allen, 2007b). This explanation is supported by the fact that species exhibit their minimum mortality rates at around the size they enter the canopy (around 20 cm DBH, corresponding to a height of around 20 m for a typical Eastern US deciduous tree; Purves et al., 2008): once in the canopy, individuals are less affected by competition for light with neighbours. U-shaped mortality has potentially major implications for understanding forest structure and the forest carbon cycle, because larger trees contain a disproportionate fraction of above-ground woody biomass, such that any increase in their mortality has a large effect on carbon storage (Coomes et al., 2003). However, despite size being the most important single predictor of mortality at the tree scale, variation in stand size structure had almost no effect on geographical variation in plot-averaged mortality (Fig. 2.3a). This may be simply because stand structure does not vary systematically across the region, otherwise geographical variation in size distributions would result in geographical variation in plot-averaged mortality. However, the precise way in which the dynamics of size distributions might interact with climate change and/or changes in tree harvesting to induce future changes in plot-averaged mortality remains largely unexplored. Figure 2.2 Predicted changes in species' average annual mortality rate (calculated at each species’ median size and environment) when subjected to a hypothetical 2°C temperature increase ( ) and a 20% increase in annual precipitation ( ), shown plotted against the current average mortality rate without this change. Chapter 2 Eastern US mortality 40 2.4.2 Within vs across species variation in mortality along climatic gradients The mismatches we found between species-averaged and forest-averaged mortality-environment correlations imply that, under climate change, forest-averaged mortality will change in ways that cannot be anticipated by examining the current relationship between observed mortality and climate. Given that mortality is highly dependent on species identity, size and environmental factors, it is important to include all these factors in predictive models of climate-change effects. For example, consider the response of carbon stocks in the coldest regions of the Eastern US to a scenario of increased temperature. Forest-averaged mortality is currently greatest in the coldest locations, suggesting that warming should decrease mortality rates, and increase carbon stocks (Fig. 2.1b). In contrast, species that currently dominate cold regions had higher species-averaged mortality rates in warmer areas, implying that the warming might increase mortality in cooler regions dominated by these species. Although warming-induced mortality increases have been observed in other temperate forests (van Mantgem and Stephenson, 2007; van Mantgem et al., 2009), even this extrapolation must be viewed with caution, because it ignores any simultaneous changes in species composition. At the forest-averaged level, wind speed did not effect mortality yet several species-averaged mortality rates showed a strong correlation with it (Fig. A.1 in Appendix A page 126). The effect may be confounded with other variables, for example trees may experience higher mortality with higher wind speed in low density areas where there is little protection from neighbouring trees (Taylor and MacLean, 2007). At the forest-averaged level, we found that mortality increased with radiation (Fig. 2.1d). A similar pattern of increasing mortality in higher light conditions has been found for oak seedlings in the Mediterranean (Gómez-Aparicio et al., 2008) and linked to higher desiccation risk. Although several species followed this pattern, many others showed the opposite trend of decreasing mortality with increasing radiation, in agreement with many other studies linking light to survival (e.g. Chen and Klinka, 1998; Kobe, 1999; Gratzer et al., 2004). Chapter 2 Eastern US mortality 41 Figure 2.3 Maps of estimated annual forest-level mortality across the Eastern United States illustrating the contributions of each of the components of the model (a-c), the full model results (d), and the observed mortality for the 21 common species (e). (a) Variation in forest structure alone (stem size and plot basal area), illustrated by removing environmental effects and modelling just the most common species (A. rubrum). (b) The effect of variation in forest species composition alone, illustrated by removing environmental variation, and stand structure variation (i.e. modelling a 20 cm DBH tree). (c) The effect of variation in environment alone, illustrated by modelling A. rubrum without stand structure variation (i.e. modelling a 20 cm DBH tree) across the region. Full model results (d) are strongly affected by species-environment interaction, and closely match the observed geographical pattern of average mortality (e). (a) (b) (c) (e) (d) Chapter 2 Eastern US mortality 42 2.4.3 Species-mortality responses to changing climate Our results suggest that species show contrasting responses to changing environmental conditions, and these mortality responses were strongly non-linear which suggests that individuals within a species may respond at different rates to a change in conditions, depending on where they sit within the species range. Changes in mortality have been correlated with changing temperature and precipitation levels in the USA in other studies (Voelker et al., 2008; van Mantgem et al., 2009), and since many parts of the Eastern USA are predicted to experience increases in temperature and precipitation under climate change (IPCC, 2007), we examined changes in mortality under scenarios of blanket increases in temperature and precipitation only (Fig. 2.2). We found that the largest changes were seen in the species with the highest mortality rates, implying that under these climate change scenarios the largest changes in carbon dynamics might be seen in highly-disturbed landscapes where fast-growing species dominate. Such changes in mortality could have repercussions for forest structure and species composition, but any consequences would need to be understood in the context of compounding effects of species-specific changes in growth and recruitment rates ( b e et al., 2007; McMahon et al., 2010), and frequency of disturbance events, such as pest and pathogen outbreaks, which may change with climate change (Dale et al., 2001). However, since observed wood anatomy and demographic rates within a species may have adapted to local climatic conditions (Esteban et al., 2010), the future response of mortality to rapid climate change may follow different patterns to the correlations between current climate and mortality documented here. 2.4.4 Limitations Although this work presents strong evidence for marked variation in mortality with a variety of different factors, we recognise several shortcomings in terms of a lack of inclusion of external disturbance factors, forest management and history, which are all likely to affect mortality. It is also important to note a significant limitation of the study, namely that the data used cover a single survey period only (1980s-1990s), so particular quantitative results are dependent on conditions in this period and must be treated with caution. This raises the possibility that some of the patterns reported here reflect particular episodic events that may not be representative of mortality patterns averaged over the longer term. However, our four main conclusions (that mortality is U-shaped against DBH, hump- shaped against plot basal area, and species exhibit both different underlying mortality rates and different responses to changes in environmental conditions) are robust unless: (a) over longer periods temporal variation completely or nearly removes all effects of species, size or environment on mortality, (b) the apparent effects of the different predictor variables on mortality uncovered here were caused entirely by temporally varying factors not considered by this study, for example pests and pathogens (Lovett et al., 2006), forest management practices or extreme weather events (Batista and Platt, 2003). Fortunately, national forest inventories are beginning to provide re-surveyed data Chapter 2 Eastern US mortality 43 covering more than one time interval (e.g. FIA-DB, 2008). In principle, this kind of data can be used to estimate the magnitude of inter-decadal variation in tree mortality directly. These limitations are important and call for caution in interpreting the results given here, and/or in utilising our models of mortality (Table A.2 in Appendix A page 122). More importantly, these limitations, together with the marked correlations between climate and mortality documented here, call for further research into tree mortality and its potential contribution to the response of the terrestrial carbon cycle to climate change. 2.4.5 Conclusion We found large and statistically significant differences in mortality among species not only in baseline mortality rates (Table 2.1), but also in their responses to environmental variation (Fig. 2.1, A.1-A.3 in Appendix A pages 126-127), along with marked effects of individual size and plot basal area. Importantly, both species composition and stand structure are likely to continue to undergo directional changes over decadal timescales, independent of any effects of climate change. Therefore, projections of future forest carbon dynamics will be in error unless they incorporate the effects of projected changes in species composition and stand structure. The good news is that recent decades have seen the appearance of a variety of simulation models that can make accurate predictions of forest dynamics, whether within the context of forest community ecology (e.g. Purves et al., 2008) or silviculture (e.g. Mitchell, 1969), as well as Dynamic Global Vegetation Models which in principle can predict forest responses to changing CO2 concentrations (e.g. Bonan et al., 2003). These models, together with the large forest inventory databases that are rapidly becoming available for many of the world’s forests, suggest that believable predictions of future forest dynamics and the forest carbon balance which incorporate ecological processes are within reach. Author contributions Designed the study, analysed the data and performed statistical analysis: Emily Lines with support from Drew Purves and David Coomes. Wrote the paper: Emily Lines with supervisory support from David Coomes and Drew Purves. 44 Chapter 3 Introduction to Spanish forests 45 3 A brief introduction to Spanish forests The second part of this thesis concentrates on modelling forest processes in Spain. In this chapter I therefore provide a brief description of Spanish forest ecology to introduce the context for the following chapters. 3.1 Introduction Mainland Spain, spanning from 36°N to 44°N and 9°W to 3°E, has an extremely varied landscape containing a wide range of soil and climatic conditions which produce many different vegetation types. It has been identified as part of the Mediterranean basin biodiversity hotspot (Myers et al., 2000), making it an important place in terms of conservation to investigate the response of individual tree and whole-forest dynamics to changing environmental conditions. Although the predictions of climate models for future temperature and precipitation levels are conflicting for many regions of the world, most of them agree that Spain will become hotter and drier under future climate change (IPCC, 2007), allowing a more straightforward investigation into the effect of climate change on Spanish forests than in many other cases. Figure 3.1 Maps of a) average annual temperature (°C) and b) annual precipitation rate (mm) in the period 1940-1996 (MMA). Forests, as defined by the UN, cover around over 18 million ha or 36 per cent of mainland Spain, making it the most heavily forested country in Southern Europe (Fig. 3.2), with around 14 per cent of forest in formally protected areas (FRA, 2000). Vegetation types in Spain are controlled to a large extent by the strong climatic gradient across the country, from wet Atlantic conditions in the Chapter 3 Introduction to Spanish forests 46 north to arid Mediterranean conditions in the south, along which temperature and precipitation regimes vary substantially (Fig. 3.1). Although much of the country is characterised by seasonally dry Mediterranean conditions, northern and mountainous parts of central Spain have much higher levels of rainfall, and are accordingly the most densely forested regions. In the south, precipitation varies greatly with season, with Atlantic fronts bringing higher precipitation in winter (Millán et al., 2005) and the lowest levels in summer. Precipitation can also be highly variable between years, with the south and east coasts experiencing periodic autumn deluges (>200 mm in 24 hours, Grove and Rackham, 2001). Forests, both natural and managed, are found at altitudes from sea-level to over 2000 m and three-quarters of Spanish forests are found in hilly and mountainous areas, on slopes greater than 20 per cent. Woody-plant dominated landscapes in Spain include shrub land (matorral), savannah (dehesa) and as well as more familiar forest forms (Grove and Rackham, 2001). Many Mediterranean tree species, notably Quercus coccifera and Q. ilex, can take a shrub form, creating matorral ecosystems in areas which are dry, where soils are poor, or where there is intense pressure from grazing animals or frequent fires. Savannah (wood-pasture) systems exist in areas with low rainfall (400-700 mm/year; Grove and Rackham, 2001) and are often dominated by the evergreen oaks Q. ilex and Q. suber (holm and cork oak) at densities of 20-40 mature trees per ha (Joffre et al., 1988), although some Pinus species can also form dehesas. Dehesas have a long history of management for acorn production, livestock grazing, wood and charcoal production, and cork harvesting. Oaks have commonly been pruned and pollarded to maximise crown spread and therefore acorn production (Plieninger et al., 2004). Figure 3.2 Distribution of the approximately 90,000 forest plots in the Second Spanish Forest Inventory (IFN2) representing the distribution of Spanish forests. Chapter 3 Introduction to Spanish forests 47 3.2 Climate, topography and tree species distributions The very high diversity of Spanish forest types make it both an attractive and a challenging system to study. The Iberian peninsula’s forests are dominated by a wide variety of species (Fig. 3.3), primarily determined by the high variation in climate and topography, as well as a long history of human activity. Spain contains both temperate forests in the northern and central parts, and Mediterranean- type forests in the south. In the early Holocene Pinus forests dominated in many parts of Spain, but a wetter and warmer period in the mid Holocene (around 7000-5000 BP) led to increased dominance of angiosperm species such as Quercus, Fraxinus and Betula (Carrión et al., 2001; Rubiales et al., 2008; Pérez-Obiol et al., 2011). Since the mid Holocene, increasing aridification and more frequent fires (possibly causing changes in soil quality) combined with increase human activity, has led to an increased dominance by evergreen Quercus species over Pinus and deciduous species in the south of Spain (Carrión et al., 2000, 2001; Valbuena-Carabaña et al., 2010; Pérez-Obiol et al., 2011). The Mediterranean Basin is a biodiversity hotspot containing 25,000 species of plant (50 per cent of which are endemic), including 100 tree species; compared to just 6,000 plant and 30 tree species in the whole of northern and central Europe (Scarascia-Mugnozza et al., 2000). Successional dynamics and species composition in Mediterranean communities are qualitatively different to those in temperate zones because the distributions of species in the Mediterranean are more affected by water limitation than light availability (Zavala et al., 2000). The northern part of Spain lacks a distinct dry summer season, and temperate forests dominate. Conifer-dominated forests form a large part of all Spanish forests, accounting for about 40 per cent of forested land, with about 25 per cent of forests dominated by Pinus species (Barbéro et al., 1998). Temperate conifer forests are found in the northern and central mountainous regions in areas with Atlantic climates that have regular long frosts and cool summers with no drought, and on slopes above 1500m (Garcia Viñas et al., 2006). They contain species such as the mountain pines Pinus sylvestris, P. uncinata and P. nigra, along with some fir (Abies alba). Temperate broadleaf forests are also found in the north in areas without drought, but are generally on lower slopes. Fagus sylvatica, a species that doesn't tolerate drought, dominates many montane forests in the north of Spain, representing the southern range edge of its distribution (Jump et al., 2006). Deciduous oak species such as Quercus petraea and Q. robur are commonly found on lower slopes in northern and central parts (Benito Garzón et al., 2008). The southern part of Spain is characterised by a Mediterranean climate with mild autumns and winters, during which most of the rainfall occurs, and hot dry summers, along with a highly varied topography and nutrient-deficient soils (Zavala et al., 2000). Deciduous species are found in the south of Spain, but are restricted to riparian areas and moisture-retaining soils. Oak species such as Q. pyrenaica and Q. faginea are strongly limited by water availability (Benito Garzón et al., 2008), Chapter 3 Introduction to Spanish forests 48 whilst even in arid areas species such as Ulmus minor and Populus nigra are able to survive alongside rivers and in areas with a high water table (Tabacchi et al., 1996; Grove and Rackham, 2001). Two evergreen oak species, Q. ilex and Q. suber, dominate in areas too dry for deciduous oaks, but are outcompeted by pine species in the very driest parts of southern Spain as well as in very frequently disturbed areas (Rodà, 1999; Zavala et al., 2000). For example, Q. ilex forms a dense forest in regions with 500-600mm rainfall/year but is sparse in areas with 400-450 mm/year, whereas Pinus halepensis plantations may survive in areas with as low as 300 mm/year precipitation (Rodà, 1999). Figure 3.3 Forest species dominance across Spain (adapted from Brus et al., 2011). The Mediterranean pines, Pinus pinea, P. halepensis (Aleppo pine) and P. pinaster, are more drought resistant than the mountain pines and Mediterranean oak species. The relative abundance of oaks and pines is strongly related to water availability and topography (Zavala et al., 2000). Mixed stands are found in areas with droughts lasting between 4 and 8 months per year, but pines also form monospecific stands in drier regions. Along gradients of altitude and slope aspect, higher densities of light-demanding pines are found on warmer south-facing or lower slopes, with higher densities of oak found on cooler north-facing or higher slopes. Although in many parts of Spain pine-dominated stands are the early successional stage ending with broadleaf dominance, for example P. halepensis to Q. ilex and P. pinaster to Q. suber, in the very driest regions broadleaves are unable to establish and pine communities represent a stable or climax vegetation (Barbéro et al., 1998). Chapter 3 Introduction to Spanish forests 49 3.3 Adaptation to drought The Mediterranean region has an environment characterised by soils that are commonly nutrient-poor (Thompson, 2005) and a long drought in the summer that limits growth in the warmest period of the year. Trees living in the region are adapted to tolerate drought and low nutrient levels through strategies such as physiological acclimation (e.g. altering photosynthesis rates), having longer-lived and thicker sclerophyllous leaves, smaller leaf area index and higher root-to-leaf area ratio (Bréda et al., 2006). A high proportion of rainfall in the Mediterranean occurs in winter, meaning winter deciduous species with a more limited photosynthetically-active period may be at a disadvantage (Thompson, 2005). Evergreen woody-vegetation, often with small, thick, sclerophyllous leaves, is therefore often dominant in areas with the longest droughts (Joffre et al., 1999). Sclerophyllous angiosperms (e.g. Q. ilex and Q. suber) and conifer species have been found to have higher levels of stomatal control than deciduous species and may therefore be better able to avoid water loss during drought than deciduous species (Duhme and Hinckley, 1992; Mediavilla and Escudero, 2003), although there is evidence this is not the case in all environments where deciduous and sclerophyllous species coexist (e.g. Damesin and Rambal, 1995). Both angiosperm and gymnosperm species adopt deep and wide rooting systems to maximise water availability and access to deep reservoirs during drought (Joffre et al., 1999; David et al., 2004; Moreno et al., 2005), which, along with stomatal response, reduces the risk of cavitation and leaf damage by avoiding low water potentials. Deep rooted trees may benefit their surrounding vegetation because they can reduce the water stress experienced during drought by increasing upper soil layer moisture through hydraulic lift (Filella and Peñuelas, 2004). Sclerophylls tend to have low levels of nutrients in their leaves (Turner, 1994), which is an advantage in low-nutrient conditions due to higher nutrient use efficiency (Aerts, 1995), but may be less responsive than deciduous species to increases in nutrient levels, for example after fires (Valladares et al., 2000). Sclerophyllous angiosperms have lower photosynthetic capacity than deciduous species leading to lower growth rates, and are therefore outcompeted by deciduous species in areas with shorter droughts or higher nutrient levels (Thompson, 2005). Sclerophylls did not evolve under current Mediterranean conditions (Herrera, 1992) and the mechanism for the dominance of sclerophylls over deciduous species or vice-versa is not always a straightforward product of drought length or intensity (Blumler, 1991). Chapter 3 Introduction to Spanish forests 50 3.4 Adaptation to forests fires Forest fires are frequent occurrences in many parts of Spain, with an average of more than 14,000 ha per year in the period 1980-2000 (Civil Protection and Environmental Accidents Unit, 2001). Forest fires play a substantial role in influencing vegetation composition and dynamics and have done so for at least the last 5000 years (Carrión et al., 2001). Many Mediterranean tree species have developed specific adaptations to aid survival and regeneration following fire, both to avoid damage (by growing thicker bark) and to regenerate quickly post-fire (through resprouting and reseeding), and post-fire dynamics are strongly dependent on the ability of species to regenerate (Gracia et al., 2002). When burnt, both Q. ilex and Q. suber are able to resprout from stem and basal buds (Pausas, 1997; Broncano et al., 2005), while Q. suber is also protected from milder fires by its thick bark (Pausas, 1997). Many shrublands are dominated by these resprouters along with others such as Q. coccifera and Arbutus unedo (Pausas et al., 2008). Individuals of resprouting species recover faster in the short term than reseeders because they maintain live biomass underground, but light-demanding conifer reseeders tend to dominate in frequently disturbed stands and after intense fires (Zavala et al., 2000). P. halepensis and P. pinaster have adapted to the effects of intense crown fires by reproducing at a young age and by having serotinous cones that allow rapid seed dispersal and regeneration after a fire (Tapias et al., 2004; Calvo et al., 2008). Levels of serotiny is, however, highly variable between species and populations, and may also be related to the length and severity of drought in the area (Tapias et al., 2001). P. halepensis has thin bark and is easy killed by fire, whereas P. pinaster has a thicker and more protective bark. P. pinea is not serotinous but reduces fire damage through a thicker, more insulating and more resistant bark. It also escapes more intense crown fires because it is naturally found in areas of poor sandy soil, which typically do not support dense flammable understories (Grove and Rackham, 2001; Rigolot, 2004; Tapias et al., 2004). Other species such as the mountain species P. nigra and P. sylvestris are not serotinous and are unable to either survive intense crown fires or regenerate after them, although P. nigra has a thick bark and is able to survive low-severity understory fires (Pausas et al., 2008). Although there were forest fires in Spain in pre-human times (Grove and Rackham, 2001), humans have managed the forests with fire for so long that it is difficult to establish what effect a natural fire regime would have had on species distributions. For example, P. halepensis burns very fiercely but reproduces so profusely (Pausas et al., 2004) that more frequent burning due to human activity may have increased its range over less tolerant species (Agee, 1998). It has also been suggested that the dominance of Q. ilex over deciduous oaks in the subhumid south of Spain could be attributed to human-induced fires lit to create clearings from around 4000 BC (Reille and Pons, 1992). Chapter 3 Introduction to Spanish forests 51 3.5 Land use change, threatened species and climate change All Mediterranean forests have a long history of human activity and management and Spain is no exception. Fire regimes, herbivory, and a long history of human land use and disturbance have influenced current species' distributions (Zavala and Marcos, 1993; Zavala et al., 2000). Anthropogenic effects have influenced vegetation composition and structure since at least 10,000 yr BP and driven major changes since the late Holocene (3,000 yr BP), particularly through forest clearing at low altitudes (Ramil-Rego et al., 1998). The combination of human activity and climatic changes, for example warming after the last ice age, has changed forest species composition, for example causing the reduction in Pinus sylvestris seen in north-western Spain in the Holocene (Rubiales et al., 2008). In the latter half of the 20th century a large amount of agricultural land in Spain was abandoned resulting in increases in forest cover (e.g. Barbero et al., 1990; Poyatos et al., 2003), which may have interacted with fire regimes to cause changes to forest structure and dynamics (Viedma et al., 2006; Chauchard et al., 2007). In the last few decades, the abandonment of agricultural land combined with new EU policies have led to the re-forestation of substantial areas of land (3.2 million ha in the period 1940-1980 and 450,000 ha in 1994-1999; Campos et al., 2005). Planted forests account for around 2.7 million ha, around 40 per cent of which is introduced species such as Pinus radiata and Eucalyptus spp. Many forest types in Spain are considered threatened. For example, Abies pinsapo (Spanish fir) is found in only three areas in the high mountains of southern Spain, and has had its range greatly reduced by fire and human activity (Esteban et al., 2010). The oaks of the dehesas of south-west Spain provide important ecosystem services such as preventing soil erosion, enriching soil nutrients, and providing habitats and food for many animals; but a decrease in their use for pastoral agriculture combined with a lack of natural regeneration in many areas has led to aging and degraded stands, threatening ecosystem functionality (Joffre et al., 1999; Plieninger et al., 2003, 2004). The impact of climate change on Spanish forests is not clear, as some studies have predicted positive and others negative impacts. For example, climate change is expected to have a detrimental effect on the distributions of many Iberian species, and under future warming, mountain conifers are predicted to be limited to the few high altitude areas into which they are able to disperse (Benito Garzón et al., 2008). Individual tree growth has been shown to have been affected by increased water stress in recent decades (Andreu et al., 2007), however some species may also benefit from a longer growing season (Sabaté et al., 2002). Chapter 3 Introduction to Spanish forests 52 3.6 The Spanish Forest Inventory The data used in this thesis have come from the second and third Spanish Forest Inventories (Inventario Forestal Nacional IFN2 and IFN3; MMA, 1996, 2007), national databases which have already been used to investigate patterns in forest dynamics (e.g. Montoya et al., 2008) but have not yet been applied to construct and parameterise an individual-based model for Spanish forest dynamics. These database contains details of the species, size, height, volume, growth, damage and survival of over a million trees of over 40 species, monitored at 10-year intervals from 90,000 plots distributed over a 1 km 2 grid across all the currently forested areas of Spain (Fig. 3.1). The plots each contained four concentric circular subplots of radius 5, 10, 15 and 20m, within which all trees larger than the minimum subplot tree diameter were measured (7.5, 12.5, 22.5 and 42.5 cm respectively). The surveys include details of tree height, crown size and depth for several randomly chosen trees within each plot, and plot-level data on altitude, slope, land use, major tree species, forest type and cover, and management has also been recorded. Further details of the inventory are given in Chapters Four, Five and Six. Chapter 4 Aboveground allometry 53 4 Predictable changes in aboveground allometry of trees along gradients of temperature, aridity and competition Abstract Trees are often observed to get shorter and narrower-crowned in dry regions and at high elevations. We explore how this pattern is driven by two opposing factors: competition for light makes it advantageous to extend branches to their biomechanical limit, whereas in cold or arid conditions it is advantageous to have shorter branches, thereby reducing the length of the hydraulic transport system and embolism risk. Using data from the Spanish forest inventory of 700,000 trees of 26 species, we quantify how environmental conditions influence the scaling of height (H) and crown diameter (CD) with stem diameter (DBH). We compare our predictions with those of Metabolic Scaling Theory (MST), which suggests allometry is invariant of environment. We fit DBH-H and DBH-CD functions using Bayesian methods, allowing comparison of within and across-species trends in allometry along gradients of temperature, precipitation, drought and competition for light (i.e. the basal area of taller trees). Competitive environment had a strong influence on aboveground allometry, but all trees were far shorter than predicted by biomechanical models, suggesting that factors other than biomechanics are important. Species that dominate in arid and cold habitats were much shorter (for a given diameter) than those from benign conditions; but within species heights did not vary strongly across climatic gradients. Our results do not support the MST prediction that DBH -H and DBH-CD allometries are invariant, or that biomechanical constraints determine height allometry. Rather, we highlight the role of hydraulic limitations in this region. The fact that intraspecific adjustment in DBH-CD-H allometry along environmental gradients was far weaker than across-species changes may indicate genetic constraints on allometry which might contribute to niche differentiation among species. Chapter 4 Aboveground allometry 54 4.1 Introduction Aboveground allometry - the scaling of tree height and crown width with stem diameter - has a strong influence on plant performance, but the way in which it changes along environmental gradients is poorly understood. The biomechanical limitation hypothesis suggests there is strong selective pressure for trees to extend to their critical buckling height in order to project their leaves above those of neighbours and maximize competitiveness in the battle for light (McMahon, 1973; Chave et al., 2005; Mäkelä and Valentine, 2006; Dietze et al., 2008; Kaitaniemi and Lintunen, 2008). In contrast, the hydraulic limitation hypothesis suggests that resistance to sap flow within the plant vascular system limits height growth, particularly in arid regions, and that plants compensate by adjusting their morphology, allocation patterns, and wood anatomy (Ryan and Yoder, 1997). In this paper we investigate how aboveground allometry varies along environmental gradients. It is well known that vegetation becomes shorter and more open in dry regions and that cavitation risk is reduced at the sacrifice of increased hydraulic resistance per unit pipe length (Pockman and Sperry, 2000; Martínez- Vilalta et al., 2002). Similar patterns are observed up altitude gradients (Coomes and Allen, 2007a). In this study we investigate whether aboveground allometry also changes systematically, or whether trees growing in arid and cold environments are simply miniaturised versions of those growing in benign environments. Whilst allometric studies are numerous, most have been undertaken using a small sample of trees at single locations, and applying the functions to different sites, species or scales is challenging (Chave et al., 2005). Surprisingly little is known about how scaling relationships vary within and among species along environmental gradients at regional scales (Wang et al., 2006; Méndez-Alonzo et al., 2008). The need for more accurate and generalisable allometric relationships which incorporate climatic variation has been highlighted by the sensitivity to allometric scaling of individual based simulation models such as ALLOCATE (Tilman, 1988), SORTIE (Pacala et al., 1996) and PPA (Purves et al., 2008), as well as dynamic global vegetation models (Scheiter and Higgins, 2009). Moreover, ecotypic variation in tree allometry is expected due to local adaptation and plastic responses (e.g. due to crowding) (Weiner, 2004). The lack of studies quantifying inter- and intra- specific allometric variation across the landscape may reflect the paucity of allometry datasets spanning wide environmental gradients and the under-use of computational methods to effectively separate co-varying correlates. We investigate how aboveground allometry of Iberian tree species varies with wood traits, climate and competitive environment using data extracted from 48,000 forest plots in the Spanish Forest Inventory. The large amount of data contained in the inventory, combined with the high variation in climatic and competitive environments in Spain enable us to make strong inferences about both inter- and intra-specific variation in allometry along gradients, as well as comparing the two. The Chapter 4 Aboveground allometry 55 height (H) vs. stem diameter at breast height (DBH) dataset comprises over 700,000 stems of 26 native species, and the crown diameter (CD) vs. DBH dataset comprises 225,000 stems of 14 species. Our first hypothesis is that asymmetric competition for light has strong influences on aboveground allometry, particularly in wetter regions of Spain where forests are denser and aboveground competition more intense. It is well established that trees adjust aboveground allometry in response to their local competitive environment: trees growing in dense young stands tend to invest more into height growth than into diameter growth or crown expansion (Henry and Aarssen, 1999). Individuals in dense stands restrict their crown development in response to competition for light from neighbours (Bragg, 2001), whereas open-grown trees develop much wider crowns (Muth and Bazzaz, 2003). Yet we know of no studies that have compared the responses of multiple species to competitive effects in the context of environmental gradients. Our second hypothesis is that tree heights never exceed the theoretical height above which they are unable to avoid buckling under small horizontal displacement of the top of the trunk, but that trees growing in neighbourhoods with many taller competitors may attain heights close to the theoretical limit. The critical buckling height of a tree height is given by the Euler–Greenhill formula: 3 2 3 1 D G E CH crit         (Eqn 4.1) where Hcrit (m) is a function of trunk diameter (D: m), the green wood density (ρ: kg/m 3 ) and modulus of elasticity (E: N/m 2 ), with G being the gravitational force (9.8 N/kg), and C a constant of proportionality (0.792) (Greenhill, 1881; Niklas, 1994). Maximizing height growth is essential for plants growing under strong competition for light. In lowland Malaysian forests, King et al. (2009) found that some Dipterocarp trees grew very close to their Hcrit in sheltered understory conditions. However, growing close to the critical value is risky, and several studies have found that trees have heights 2–5 times lower than the theoretical maximum (e.g. McMahon, 1973; van Gelder et al., 2006). Our third hypothesis is that water shortage reduces tree height and crown diameter relative to DBH, and that hydraulic constraints are more important than biomechanical ones in xeric conditions. Water shortage has a direct effect on aboveground allometry, because plants reduce height and branch growth in order to reduce cavitation within xylem tissues (Bréda et al., 2006). Water supply may indirectly affect allometry by changing competitive interactions within forests: there is evidence that asymmetric competition for light is more intense in mesic forests because they have greater leaf area indices and capture more light than xeric forests (Coomes and Grubb, 2000), so we predict a shift towards height growth in mesic forests, where the battle for light is most intense (Henry and Aarssen, 1999). Our final hypothesis is that the height and crown diameter attained by a tree will decrease in colder areas. Long or severe frosts may cause embolism and branch loss (Lemoine et al., 1999) and have been found to limit forest height (Wang et al., 2006). There may also be an altitude–temperature Chapter 4 Aboveground allometry 56 effect on allometry, since growth and biomass have been found to decline in higher, cooler areas (e.g. Coomes and Allen, 2007) and DBH-H and DBH-CD allometry may be affected by stronger winds (Bruchert and Gardiner, 2006). By testing these hypotheses we intend to demonstrate the influences of competition, aridity and temperature in regulating aboveground allometry, with potentially important implications for carbon storage and forest dynamics. 4.2 Materials and Methods 4.2.1 Inventory data and climate variables Data on height (H) and DBH were taken from the second Spanish Forest Inventory (IFN2) (MMA, 1996) for over 700,000 stems of 26 native tree species (see Appendix B page 131), sampled from over 48,000 plots arranged systematically on a 1 km grid across Spain. For each tree a variety of attributes including position within the plot, species, two measurements of DBH (perpendicular to each other) and H (to the nearest 0.5 m) were taken. For a subset of the database (around 150,000 stems of 14 species of silvicultural interest), two measurements of crown diameter were recorded for around four trees per plot. Spanish forests have a long history of human management, so for five species that are commonly coppiced/pollarded (see Appendix B page 131) we used a smaller database of trees that showed no signs of cutting at the time the inventory was taken. For both DBH and crown diameter we used the average of the two measurements taken in the inventory in the model. The IFN2 plots were sampled using a sampling technique whereby trees of different sizes were measured in concentric plots of varying plot radius. All trees of DBH > 7.5 cm were measured in a plot of radius 5 m, all of DBH > 12.5 cm in a plot of radius 10 m, all of DBH > 22.5 cm in a plot of radius 15 m and all of DBH > 42.5 cm in a plot of radius 25 m. Estimates of plot basal area are therefore calculated using only basal area of trees over 7.5 cm DBH. Estimates of precipitation, temperature (annual and seasonal for both) and drought length were taken from data layers extracted from Gonzalo Jiménez (2008). These were created by interpolating data from Spanish weather stations (from the State Meteorological Agency AEMET) recorded between 1951 and 1999. Drought length (months) was calculated using the Gaussen- Bagnouls method (ombrothermic curves; Bagnouls and Gaussen, 1957). Plot altitude and slope were measured during the inventory. We removed highly correlated predictor variables from the climate data before models were fitted (excluding those with |r| >0.7), leaving three climate variables to consider in the analysis (mean annual temperature, mean annual precipitation and drought length). We calculated three different possible predictors of the effect of competition from neighbouring trees on allometry (plot basal area (m 2 /ha), basal area of larger trees (those larger in Chapter 4 Aboveground allometry 57 diameter than the target tree) (m 2 /ha) and tree density (stems/ha)) and compared models fitted with each of them to find the best by comparing model fit using Akaike and Bayesian Information Criteria (AIC and BIC; Akaike, 1974; Schwarz, 1978). 4.2.2 Species-level wood traits and critical buckling heights We compared observed heights with Hcrit values predicted by the elastic similarity model. These comparisons were made for the 14 species with more than 1000 samples in the dataset, and for which we found green wood density and modulus of elasticity (Young's modulus) data in the literature (see Appendix B page 131). We calculated the theoretical maximum attainable height curve using equation 4.1. We characterised species' shade tolerance using Ellenberg scores (Ellenberg, 1988). 4.2.3 Statistical models Hierarchical Gibbs Sampler To test whether wood density, elastic modulus or shade tolerance affect allometric relationships we constructed a hierarchical normal Gibbs sampler to fit species-trait constrained parameters to traditional power-law allometric equations, yij = α xij β to describe the relationship between DBH of a given tree i of species j (xij) and H and CD (yij), following the methods used by Dietze et al., (2008) (see Appendix B page 131 for details). We fitted this for a subset of 14 species for which we used data for both wood density and modulus of elasticity. We recognised that DBH-H relationships are non- linear on log-log axes when a very wide range of plant sizes is used (e.g. Niklas and Spatz, 2004) but since all our trees were >7.5 cm DBH and Enquist et al., (2007) found power laws to be adequate for trees greater than about 5 cm DBH, we determined that a power law was appropriate. MCMC Metropolis algorithm We used a Markov Chain Monte Carlo (MCMC) Metropolis algorithm to fit models of DBH-H and DBH-CD allometric relationships and examine how they varied with the climate and competition. Justification for the choice of these methods is discussed in Appendix B page 131. Since we had information on the environmental conditions of each sampling plot we were able to quantify both inter- and intra-specific variation along gradients. We constructed an adaptive MCMC Metropolis algorithm (Lee, 1997; Gelman et al., 1999), which searches parameter space efficiently, to find best fit values and credible intervals for model parameters. The MCMC algorithm searches parameter space and returns not only a best-fit value for each parameter given the data but also estimates its distribution. For a set of starting parameter values θ for each model M tested, the algorithm calculates the predicted H (or CD) hij for each tree i with recorded H (or CD) yij and then calculates the log-likelihood of the data (X) given the model and parameters: Chapter 4 Aboveground allometry 58     2 2 1 | , ln exp 22 ij ij i h y l X M                    (Eqn 4.2) At each iteration the algorithm selects a parameter to alter and recalculates the likelihood (equation 4.2). If the new parameter improves the likelihood then it is accepted by the algorithm. If not, it is accepted with probability of the ratio of the new and old likelihoods. The algorithm has two periods: burn-in and sampling. During the burn-in period (between 500,000 and 5,000,000 iterations of the algorithm for our models) the algorithm alters its parameter search range ("jumping distance") to achieve an optimal acceptance ratio of 25% (Gelman et al., 1999). After the burn-in period, the jumping distance is fixed (independently for each parameter). During sampling, parameter values are recorded every 100 iterations and the resulting parameter samples are taken as samples from the distribution of each parameter. We fit traditional power-law allometric equations of the form yij = α xij β to describe the relationship between DBH (xij) and H(yij), and a linear relationship between DBH and CD. We chose to use a power-law rather than asymptotic relationship as we wanted to compare our results with the predictions of the elastic similarity model (a power function). The models were fitted separately for each species and took the forms:  212 ,~ 3 bijij xbNy (Eqn 4.3a)  2232 ,~ ijij xbbNy  (Eqn 4.3b) where (in equation 4.3a) yij is the H of each tree i of species j of DBH xij, and (in equation 4.3b) yij is the CD of each tree i of species j of DBH xij. The standard deviation of residual errors around the fitted power function is well known to increase with increasing DBH, so we modelled it as:. ijxbb 10  (Eqn 4.3c) We fitted different models and compared them using information criteria. We generated models in which parameter values were either the same for all species or species-specific. We compared models in which b2 and b3 of equations 2a and 2b were constants (i.e. not varying with environment) with models in which they were linear functions of the environmental predictor variables:  iieaab 02 ;  iieccb 03 (Eqn 4.4) where ei is a climatic or competition variable. We normalised all environmental variables to allow easy comparison of their effects and fit the model given in equations 4.3a and 4.3b with parameters b2, b3 as: b2 = a0 + a1(competition index) + a2(annual precipitation) + a3(drought length) + a4(average temperature) (Eqn 4.5) Chapter 4 Aboveground allometry 59 b3 = c0 + c1(competition index) + c2(annual precipitation) + c3(drought length) + c4(average temperature) (Eqn 4.6) We fitted the full 12 parameter model for all species together, and with species-specific parameters. Details of the specifications of the MCMC model are given in Appendix B and Table B.4 (page 137). We used AIC and BIC to select the best model. All models were fitted using an adaptive Metropolis algorithm written in C (compiled using MS Visual Studio 2008). 4.2.4 Interpreting the results The results of the MCMC algorithm are presented by predicting the change in H and CD for a fixed DBH (15 cm) across the central range of each species along each predictor (i.e. between ± one standard deviation of the mean environment in which the species is found). We chose 15 cm as the standard size but we also investigated the across-species correlations using 10 cm and 20 cm DBH, and found that patterns remained consistent across these sizes (Table B.3 Appendix B page 137). 4.3 Results 4.3.1 Height and crown allometries varied among species and along environmental gradients The scaling exponent of a simple across-species DBH-H model (equation 4.3a with constant b2 and b3) was 0.609 (95% credible interval 0.606–0.611) but this model has much less statistical support than a model including different curves for each species (AIC >> 1000). Thus, there is strong support for the hypothesis that DBH-H allometry varies among species (Fig. 4.1 and see Fig. B.1 Appendix B page 138). There were clear differences between angiosperms (17 species) and gymnosperms (9 species): when species were fitted separately, the mean scaling coefficient (b2 in equation 4.4) of the angiosperms was much higher than the mean of the gymnosperms (300 vs. 155 respectively, t-test T = 4.7, P<0.05), whilst the mean scaling exponent (b3 in equation 4.4) was lower (0.40 vs. 0.58 respectively, t-test T = 4.5, P<0.05), and all species showed increases in variation in height with size (b1 in equation 4.3c). This indicates that, in general, gymnosperm species were relatively short at small diameters but caught up with, then overtook, angiosperms at larger diameters (Table B.2 and Fig. B.1 Appendix B pages 136 and 138). However, there was no support for a trade- off between exponents and coefficients because despite a negative association when all species were considered together (Spearman's P<0.05), there was no correlation when angiosperm and gymnosperm species were considered separately. For both H and CD, species-specific models which included environmental predictors (equations 4.5 and 4.6) were more strongly supported (i.e. had much lower AIC and BIC values) than models lacking environmental predictors, except in the case of Pinus uncinata. However, along all the gradients we considered, across-species patterns were stronger than within-species variation (Fig. Chapter 4 Aboveground allometry 60 4.2), and species did not, in general, change height rank along gradients. Climate and site dependency of the coefficient of DBH-H relationships has been found previously (López-Serrano et al., 2005; Wang et al., 2006), but we also found dependency in the exponent, indicating a compounding effect of climate on species' allometry and height growth. DBH-CD allometry varied strongly among species (Fig. 4.3) with mean CD for individuals of 15 cm DBH ranging between 220–590 cm, and angiosperm crowns were significantly wider than gymnosperms (t-test T = 3.0, P<0.05). Neither H nor CD allometry were significantly related to wood density, modulus of elasticity or shade tolerance (hyperparameter 95% credible intervals contained 0, tested using a Hierarchical Gibbs Sampler, Appendix B and Table B.1, page 135). Figure 4.1 Fitted parameters (coefficient b2, exponent b3) for the simple height-DBH scaling relationship (equation 4.3a) with 95% credible intervals for each of the 26 species, and histograms of the mean parameter values. Angiosperm species (closed circles) had, in general, lower exponents and higher coefficients than gymnosperm species (open triangles). Most species' exponents lay below the theoretical 2/3 exponent (dotted line) predicted by the elastic similarity model. Chapter 4 Aboveground allometry 61 4.3.2 (H1) Tree heights were greater and crown diameters narrower in competitive neighbourhoods All 26 species were taller (at 15 cm DBH) in stands with more large neighbours. Of the three indicators of competitive intensity – total stand basal area, basal area of larger trees (m2/ha) and tree density (stems/ha) – we found that basal area of larger trees was by far the best predictor of the effect of competition on allometry (AIC >> 1000), suggesting that asymmetric competition is a more important determinant of allometry than symmetric competition. Among the angiosperms, light demanding species had a stronger response to changing neighbourhood crowding (Pearson's correlation between change in height (at 15 cm DBH) across the central 67% of each species' range of competitive environments and species' Ellenberg scores: P<0.05), but there was no relationship for gymnosperms. There was correlation between the heights of species (at 15 cm DBH) and their mean competitive environments (Table B.3 Appendix B page 137): species that tended to have more large neighbours were relatively tall for a given diameter (Fig. 4.2 and Fig. B.2 Appendix B page 139). Most species showed an effect of crowding in their crowns, with 11 of the 14 having narrower crowns in areas with more larger neighbours (Fig 4.3 and Fig. B.3 Appendix B page 140) and this narrowing of crown diameter was greater for gymnosperms than angiosperms (t-test T = 3.9, P<0.01). There was no correlation between the standardised crown diameter of species (i.e. predicted value for a 15 cm tree) and the mean competitive environment (mean basal area of larger trees) of that species (Table B.3 Appendix B page 137). 4.3.3 (H2) The elastic similarity model provided an upper limit on height at a given DBH and trees in dense forests will grow to their critical height Critical buckling height Hcrit depends on the ratio of modulus of elasticity to wood density, E/ρ (equation 4.1). We found E/ρ varied among species (0.011–0.027, Table 4.1) resulting in large variation in predicted Hcrit curves (Fig. B.1 Appendix B page 138). The upper boundary of height data approached the critical curve at small DBH values (Figs 4.4 and Fig. B.1 Appendix B page 138), and species grew closer to the critical height in warmer areas with shorter droughts and more larger trees, but these effects were not strong for all species (red lines, Fig. 4.4). Very few trees approached their critical buckling height: for small trees (<15 cm DBH) about 3% of stems had heights greater than 66% of their predicted critical height, but this was true of <0.1% of large trees (Fig. B.1 Appendix B page 138). Interestingly, there was no evidence that species with high critical buckling heights were the taller species in the dataset; if the elastic similarity model were important, we would expect to see a close correlation between 3 1       G E and predicted height at a given DBH (see equation 4.1), but we found no such correlation at 15 cm DBH (Pearson's correlation, P> 0.1) or at other diameters we tried. This result indicates that factors other than biomechanics are important in determining the relative heights of tree species in our dataset. Chapter 4 Aboveground allometry 62 Table 4.1 Values of wood density (WD: oven dry mass/fresh volume kg/m 3 ) and modulus of elasticity (MOE: green wood GPa) used in equation 4.1 to find Hcrit , and the number of samples for 14 native and not commonly coppiced species with over 1000 samples in the dataset (see Appendix B page 131 for sources of data). The species for which we could not find WD and MOE information were Arbutus unedo (A), Corylus avellana (A), Ilex aquifolium (A), Juniperus communis (G), J. thurifera (G), Olea europaea (A), Pinus nigra (G), P. pinea (G), P. uncinata (G), Quercus faginea (A), Q. pyrenaica (A), Sorbus spp. (A). 4.3.4 (H3) Tree height and crown diameter for a given DBH was lower in areas of longer drought Most species (17 of 26) were shorter (for given DBH) in areas of longer drought, and gymnosperms and angiosperms did not differ significantly in their responses (t-test T = 1.2, P>0.1). However, some species were shorter in areas of higher annual precipitation: 10 of the 13 species with the wettest average environment showed decreasing heights in wetter areas (within the dataset, annual precipitation and drought length were not highly correlated), indicating a possible non-linear height response to changes in precipitation. The species with the largest decreases in height (for given DBH) in wetter areas were generally the less shade-tolerant (Ellenberg scores: Spearman's P<0.05), although the most shade-tolerant species, Fagus sylvatica, did not follow the pattern. Species growing in Species Angiosperm/ Gymnosperm WD (kg/m 3 ) MOE (GPa) Number of samples Acer campestre A 525.21 6.00 2194 Alnus glutinosa A 439.11 8.00 3825 Betula spp. A 510.00 9.25 7238 Castanea sativa A 463.33 6.00 1162 Fagus sylvatica A 585.48 8.60 51479 Quercus ilex A 820.00 14.70 21551 Quercus petraea A 559.65 9.20 13809 Quercus robur A 559.65 9.20 26550 Quercus suber A 770.00 2.00 3450 Salix spp. A 450.00 5.40 2281 Abies alba G 353.01 9.50 3961 Pinus halepensis G 460.00 6.14 88437 Pinus pinaster G 412.07 8.20 178622 Pinus sylvestris G 421.89 8.55 145446 Chapter 4 Aboveground allometry 63 regions with long drought length and low annual precipitation invested less in height (i.e. had lower heights at 15 cm DBH) than species from wetter regions (Fig. 4.2, Spearman's P<0.01). This correlation was also observed when other diameters were used to calculate standardised heights (Table B.3 Appendix B page 137). Crown diameter (for given DBH) was responsive to changes in annual precipitation, with 12 of the 14 species having wider crowns in wetter areas, but the effects were small for most species (Fig. 4.3). The predicted mean crown diameter (at 15 cm DBH) of angiosperm species was greatest for species specialising in wetter regions with shorter drought lengths (Spearman's P<0.05: see Table B.3 Appendix B page 137), but no such pattern was observed when conifers were considered or when all species were included in the analysis. Figure 4.2 Predicted changes in height of 15 cm DBH stems of 26 species in relation to (a) basal area of larger trees, (b) precipitation, (c) drought length and (d) mean annual temperature. The symbols show height at the mean value of the environmental variables for the species – angiosperms (filled circles) and gymnosperms (open triangles) – whilst predicted variation in height is shown for ± 1 standard deviation in the environmental variable around the species' distribution mean (grey arrows). Predictions were produced using posterior estimates of the distribution of the parameters. Chapter 4 Aboveground allometry 64 4.3.5 (H4) Within-species height increased with temperature, but there was no trend in crown diameter Most species (19 of 26) were taller (at 15 cm DBH) in warmer locations. Surprisingly, the opposite effect was observed in the across-species patterns: taller species were associated with cooler regions (Spearman's rank correlation between predicted mean height and average annual temperature at the centre of the species' range was negative, P<0.05) irrespective of the diameter (Table B.3 Appendix B page 137), but this may be a temperature-drought interaction since the very warmest areas are also dry. There was no consistent pattern, either within or among species, in responses of crown diameter to temperature. Figure 4.3 Predicted changes in crown diameter of 15 cm DBH stems of 14 species in relation to (a) basal area of larger trees, (b) precipitation, (c) dry season length and (d) mean annual temperature. The symbols show crown diameter at the mean value of the environmental variables for the species – angiosperms (filled circles) and gymnosperms (open triangles) – whilst predicted variation in crown diameter is shown for ± 1 standard deviation in the environmental variable around the mean (grey arrows). Predictions were produced using posterior estimates of the distribution of the parameters. Chapter 4 Aboveground allometry 65 4.4 Discussion 4.4.1 Allometric scaling varies substantially with species Species varied greatly in their DBH-H and DBH-CD allometric relationships, in agreement with other studies reporting variation among species and functional groups (Bragg, 2001; López-Serrano et al., 2005), and we found evidence against both the original MST theory of a fixed 2/3 exponent (West et al., 1999) and the later extension of MST that exponent values vary between species but cluster around 2/3 (Price et al., 2007): for these data there was no fixed exponent to the DBH-H scaling relationship either across or within species. However, it is interesting to note that the conifers had, in general, exponents close to 2/3. Indeed many may have had exponents indistinguishable from 2/3 if we had used SMA line fitting approaches, as this method is likely to give slightly larger estimates than regression methods. Elastic similarity defines the scaling relationship between the diameter and length of individual stems, and extensions of the theory have applied the same scaling relationship to stem diameter-tree height. Since many conifers have apical dominance, a single tapering stem, the extension of elastic similarity from branch to whole tree scaling may be more reasonable. In contrast, many angiosperms have fractal-like branching, making the extension to tree height scaling less appropriate. We found no relationship between the empirical DBH-H curve and modulus of elasticity or wood density, suggesting that biomechanics had little influence on the mean height of trees in the Mediterranean (Fig. 4.4 and Fig. B.1 Appendix B pages 134 and 138). However, for many species we were able to find only one value of modulus of elasticity or wood density measured on green wood and we assumed fixed values for these within species. These wood traits are likely to vary within species with age and climatic and other environmental conditions. Further evidence of the lack of influence of biomechanics on mean height came from our analyses of the critical buckling height model: almost all empirical data were situated far below the upper constraint curves predicted by the model (Fig. B.2 Appendix B page 139). This situation contrasts with that observed in Asian tropical forests trees, where trees grow much closer to their critical height in the wettest areas where competition for light was most intense (King et al., 2009). Strong water deficits and management practices in the Mediterranean result in relatively open canopied trees, resulting in more light getting through to the subcanopy (Coomes and Grubb, 2000); our results suggest there is little impetus to battle for light by growing taller in these conditions. Chapter 4 Aboveground allometry 66 Figure 4.4 Scatter plots of height-DBH data for two common species (Pinus halepensis and Quercus robur), showing the theoretical maximum attainable height predicted by the elastic similarity model (black line) and the fitted height-DBH relationship in the species' mean environment (grey). For three of the model variables, the effect of decreasing (dotted line) and increasing (dashed) each variable by 1 standard deviation (within each species' distribution) on the predicted height is shown plotted against diameter. Chapter 4 Aboveground allometry 67 4.4.2 Asymmetric competition for light was a strong determinant of DBH-H allometry The basal area of larger trees was a better predictor of allometric variation than either total stand basal area or density, and all species had a more elongated form when surrounded by more larger trees (Figs. 4.2 and 4.4). The importance of competition from larger trees shown by our data suggests that allometric relationships and biomass allocation patterns are affected by asymmetric competition, meaning that trees may invest more in height growth and increasing light capture when under more intense competition (Berntson and Wayne, 2000). Competition for light also influenced crown size, with most species having smaller crowns in more competitive environments (Fig. 4.3), in agreement with studies showing that crown area adapts to local competition (Bragg, 2001). 4.4.3 Species were shorter in drought areas but individuals did not respond consistently Species living in areas of longer drought and lower annual rainfall were shorter than those in wetter areas. This pattern of reducing tree height with increasing aridity is well documented for many parts of the world (e.g. Devakumar et al., 1999; Méndez-Alonzo et al., 2008) and may be a result of changing vessel hydraulic structure in drought areas to reduce the risk of embolism (Corcuera et al., 2004). Wood hydraulic properties have been found to be more important for demographic rates than wood density (Russo et al., 2010), and they are not closely correlated (Zanne et al., 2009). A well- established trade-off in the plant hydraulics literature is that morphological traits which confer transport efficiency make the xylem vulnerable to cavitation. Cavitation risk is reduced when conducting pipes have thick cell walls, thick pit membrane and narrow lumens, but all these properties increase resistance to sap flow, leading to a fundamental trade-off between safety and efficiency (Tyree and Zimmermann, 2002). Smaller vessels may be more resistant to cavitation under drought conditions but have high resistivity (resistance per unit cross-sectional area) and are less efficient (Markesteijn et al., 2011). More sapwood is therefore needed to transport the same amount of water and so trees in drought areas that reduce their risk of catastrophic failure of water flow by having small vessels may do so at the cost of reduced height growth. However, tree size may affect ability to respond to drought, with larger trees more able to recover from drought (Martín-Benito et al., 2008). Within-species responses did not always agree with the across-species pattern of increasing height in wetter areas, especially for light-demanding species in areas with high annual rainfall, where an interacting impact of increased competition, not modelled in this study, may reduce any benefit of additional water (Fig. 4.2). Periods of soil moisture saturation and flooding may also act as a stressor in arid-climate forests by reducing tree height (Rodríguez-González et al., 2010). 4.4.4 Within-species height increased with temperature, but across species the trend was the opposite The observed within-species increases in height with temperature (Fig. 4.2) are in agreement with studies finding a temperature stimulus specifically to height growth (e.g. Lopatin, 2007). This Chapter 4 Aboveground allometry 68 stimulus may cause a shift in biomass allocation and change in allometry so that trees are taller for a given diameter in warmer areas, perhaps as a result of more rapid increases in photosynthetic capacity than respiration rates causing a higher rate of carbon assimilation (Way and Oren, 2010). Taller trees in warmer areas may also be caused by changing vessel structure and flow properties. It has been suggested that since warmer water is less viscous, this leads to higher flow rates within vessels causing a higher mass concentration of dry matter, but evidence of this has only been observed in gymnosperm species (Roderick and Berry, 2001). Another explanation of the pattern is that cooler temperatures are associated with higher altitudes, and tree allometry may be affected by exposure to strong winds (Bruchert and Gardiner, 2006) in the many forests found in the mountainous areas of Spain. Across-species we found that species were shorter in warmer areas, but this is likely to be a temperature-drought effect since although the model corrected for drought in intra-specific relationships, it does not account for across-species temperature-drought correlations. Areas of Spain with high average temperatures are also those with longer droughts, which affect the average height at which each species occurs. This mismatch between across- and within-species responses to temperature has implications for predictions of the response of tree allometry to climate, since the observed negative correlation between temperature and tree height was caused by the dominant across-species pattern. Taken naively, this correlation would imply that Spanish trees are likely to become shorter in response to the predicted increases in temperatures across Spain (IPCC, 2007), but this could be confounded by simultaneous changes in precipitation patterns. In the short term however, species composition is likely to remain approximately constant, so in response to simple temperature increases, heights are likely to increase in line with the within-species pattern. 4.4.5 Within-species variation in allometry along climatic gradients is smaller than across-species differences Landscape-level trends in observed tree height and crown diameter allometries were found to be primarily a result of changing species composition along environmental gradients rather than within- species changes in response to environment. The importance of aboveground architectural allometry for succession has long been recognised (Tilman, 1988; Pacala et al., 1996; Purves et al., 2008), but little consideration has been given as to whether variation in height-DBH scaling along environmental gradients could influence the outcome of competition among trees. Our results suggest that it is unlikely that aboveground allometry is a primary driver of changes in competitive interactions and community structure at the landscape level, as few species changed height rank along climatic gradients (Fig. 4.1). However, relatively small differences in allometry between species along gradients may be an important component in defining a species' niche, since species showed little ability to adapt their allometry to systematic changes in climatic conditions. Chapter 4 Aboveground allometry 69 4.4.6 Sources of unexplained variation Although we found evidence for environmental influence on allometric scaling relationships, substantial intraspecific variation remains unexplained by our model (Fig. 4.4). Our measure of the effect of competition is taken from current conditions in each plot and may not well reflect past conditions since forests in Spain are subject to fire and management. Moreover, competition was averaged over each plot and therefore doesn't take into account local heterogeneity in soil conditions and light availability, which are likely to strongly influence an individual tree's allometry. The Mediterranean climate has high inter-annual variability (Valladares et al., 2002) but we considered only long-term averages of climatic variables, despite the fact that a tree's observed allometry is a product of its entire life-history. Individual events such as extremely severe droughts or pathogen outbreaks, as well as small-scale variation in soil properties may impact observed allometry (King et al., 1999; Bréda et al., 2006) but these factors were not considered here. Breakage of branches is likely to be linked to wind conditions (Bruchert and Gardiner, 2006), so wind exposure may also be a significant predictor of allometry. Forest fires, a common occurrence across Spain, may also affect biomass allocation and observed allometry, but were not accounted for in our model. On a regional scale, changes in allometry may be confounded by phenotypic plasticity and local adaptation creating differences within species, although further work would be required to test this. 4.4.7 Conclusions Our results indicate that, whilst biomechanical constraints provide an upper limit to allometric scaling, almost all trees in Spain were substantially shorter than their critical heights. Intense competition for light and more favourable hydraulic conditions did, however, push species towards their critical buckling heights. We found strong evidence for variation in allometric scaling along environmental gradients, and our results are inconsistent with the MST proposition of a fixed exponent of 2/3 to the DBH-H allometric scaling relationship, or that the exponents cluster around 2/3 (Price et al., 2007). Interspecific variation in allometry along climatic gradients was substantially stronger than systematic variation within species, and for some predictors showed opposite trends. Intra-specific differences in height allometry were apparent along gradients of drought length, showing that water limits the height attainable by trees in Spain, but competition in mesic forests confounded the pattern. Allometric regression equations are crucial to estimating aboveground biomass and carbon stocks from forest inventories, which often lack height measurements. Carbon stock estimates are highly dependent on the formulation of such equations and the errors and assumptions they incorporate (Chave et al., 2005). In this study we demonstrate that using a large dataset it is possible to quantify for a large number of species both region-scale variation in allometry and individual species' changes in response to climatic gradients. Improvements in the accuracy of allometric equations will lead to better estimates of current biomass and carbon stocks, and climate-dependent allometric equations will improve estimates for the future. Chapter 4 Aboveground allometry 70 Author contributions Designed the study: Emily Lines with support from Drew Purves, Miguel Zavala and David Coomes. Provided the data: Miguel Zavala. Analysed the data and performed statistical analysis: Emily Lines. Wrote the paper: Emily Lines with supervisory support from David Coomes, Miguel Zavala and Drew Purves. Chapter 5 ABC recruitment model 71 5 Inferences from aggregated count data using Approximate Bayesian Computation: deriving juvenile recruitment rates from inventory data Abstract Approximate Bayesian Computation (ABC) is a powerful likelihood-free approach to model fitting which may be used to parameterise models of unobserved processes by simulating data. In this study we demonstrate a practical application of ABC by fitting models of juvenile recruitment, growth and mortality to sapling count data obtained from the Spanish forest inventory, which is typical of many national inventories in providing a rich source of data for large tree demographic rates, but little information on small trees, despite the importance of the juvenile life-stage for forest succession and dynamics. Using a sequential Monte Carlo (ABC-SMC) approach, we find that ABC methods provide an effective approach to deriving unobserved juvenile recruitment rates from small tree count data provided by the inventory. We utilised priors for growth and mortality rates derived from the more detailed adult inventory data. Recruitment rates varied among species, with conifers having significantly higher rates than angiosperms in open areas, but that their rates also decreased the most under dense canopies. The change in recruitment rates from open to dense plots was significantly correlated with the species’ shade tolerance, as was the effect of increased canopy cover on both growth and mortality rates. Recruitment rates derived using this approach were in keeping with what is known about the species' life histories, suggesting that ABC may be a valid approach for tackling this type of problem. We suggest that ABC methods may have many applications for ecological modelling, particularly in situations where data collection is expensive, difficult or impossible (for example to derive historical dynamics), as well as to parameterise models whose likelihoods are prohibitively complex to compute. Chapter 5 ABC recruitment model 72 5.1 Introduction Complex, computationally-intensive process-based simulation models are vital tools in many areas of ecological research and parameterisation of demographic processes is a major part of building realistic models. Methods such as Markov Chain Monte Carlo (MCMC) and simulated annealing allow the parameterisation of complex functions from data but require the computation of a likelihood function, for which the calculation of the probability of the data given a model is needed. For many situations this may be too computationally intensive to be practical, for example where the probability of a model relies on summing over very large numbers of hidden states (Beaumont, 2010). For other models the data needed to accurately describe processes may be unavailable or impractical to collect, leading to unsatisfactory approximations or forced simplification of model structure. Approximate Bayesian Computation (ABC) methods are a significant advance for fitting models to data when the likelihood cannot be formulated or is computationally prohibitive to analyse (Sisson et al., 2007). They provide a means of estimating parameters to describe fine-scale complex processes but for which only coarse-scale, aggregated data are available, for example to estimate disease transmission dynamics when only data on clusters of cases with identical genotypes are available (Tanaka et al., 2006). These techniques typically take a truly Bayesian, yet likelihood-free, approach to parameterising models by incorporating both summary data and prior knowledge of unobserved processes to estimate parameter values where traditional methods present no mathematically rigorous approach. Over the last decade, these likelihood-free ABC methods have increasingly been applied to data in areas of biology such as epidemiology (e.g. Tanaka et al., 2006) and population genetics (e.g. Foll et al., 2008), but have barely reached ecology (but see Jabot and Chave, 2009;) and we know of no examples using ABC methods combined with large-scale ecological data to estimate unobserved demographic processes. Modelling the performance of juvenile trees using data collected in permanently marked inventory plots provides an excellent opportunity to test ABC methods in an ecological context. Juvenile responses to competition form a central component of predictive models of forest succession and dynamics, because the juvenile stage of the life-cycle is a critical filter (Shibata and Nakashizuka, 1995; Kobe, 1996), yet permanent plot datasets rarely contain detailed information on juveniles. For example, the Spanish Forest Inventory (IFN) (MMA, 1996, 2007) is typical of many datasets originally collected for timber stock evaluations in that it systematically sampled millions of mature trees in thousands of locations across the country, but contains no information on the performance of individual trees < 7.5 cm in diameter, with only categorical and count data being recorded. This lack of detailed data for many species is a major stumbling block to the development of a forest dynamics model for Spain. Chapter 5 ABC recruitment model 73 In this study we infer the underlying dynamics of juvenile recruitment (defined in our model as the annual rate of establishment of stems of 1 cm DBH), growth and survival from large-scale aggregated juvenile data and examine their response to changes in competitive environment. Accurate representation of dispersal and recruitment limitation is vital to produce realistic simulations of succession and spatial dynamics (Ribbens et al., 1994). Models of recruitment within simulation models often estimate rates at the stand level (Porté and Bartelink, 2002), and vary in their treatment of seedling dispersal and recruitment. The simplest models assume that seed is always available for regeneration (Clark et al., 1998), whilst others treat recruitment as a function of asymmetric competition for light and shade tolerance (Busing, 1991) or simply a function of total stand basal area and basal area of conspecifics (Kolbe et al., 1999). Many do not explicitly model the steps of seed dispersal, germination and establishment together but combine them in one process (Price et al., 2001; Busing and Mailly, 2004). On the other hand, spatially-explicit individual-based forest simulation models typically model recruitment using a seed dispersal kernel, where seedlings establish in a location with probability related to the distance to conspecific adults, and the number of seedlings established per adult varies with species, adult size and shading from adult trees (e.g. Busing, 1991; SORTIE, Pacala et al. 1996; TROLL, Chave, 1999). The presence and density of conspecific adults and competition for light are well-recognised determinants of seedling recruitment (e.g. Gómez-Aparicio et al., 2006), but many other factors may affect seedling dynamics. For example, small scale spatial heterogeneity and microsite quality are important for seedling establishment (Nathan and Muller-Landau, 2000), although such effects are difficult to measure and quantify. Canopy gaps and competition from understory shrubs (Beckage et al., 2000), soil moisture, drought and precipitation (Lloret et al., 2004; Gómez-Aparicio et al., 2008; Urbieta et al., 2008; Mendoza et al., 2009), facilitation through protection from water stress and direct sunlight by 'nurse' plants (Lookingbill and Zavala, 2000; Gómez-Aparicio et al., 2004, 2008; Smit et al., 2008; Plieninger et al., 2010), and fire frequency (Lloret et al., 2003) have all been found to affect seedling recruitment. Improved knowledge of recruitment processes is a pressing need given the widespread concern about low regeneration rates in Spanish forests (e.g. Plieninger et al., 2010; Urbieta et al., 2011). However, most studies of recruitment rates in Spanish forests have focussed on a very small number of species in a few locations. The national scope of the Spanish Forest Inventory allows the examination of recruitment rates of many species across the whole region. Using a simple simulation model, we use ABC methods to infer best-fit parameters for each process for different species by comparing simulated with observed data, and choosing those which best predict the data to within a defined ‘tolerance’ level. We believe that the approach presented here is applicable to estimating functions for use in individual based simulation models when individual- based data are not available, and that the use of ABC methods will enable the parameterisation of ecological models for which data is difficult or impossible to obtain. Due to the small amount of data Chapter 5 ABC recruitment model 74 available for several of the species (Table 6.1) we examined only the effects of two commonly used variables, conspecific adult density and competition for light (represented by canopy area of tall trees), on recruitment rates, and fitted recruitment as: Recruitment (# new stems /ha/year) = p0(conspecific adults)exp(-p1(competition for light)) However, we believe that the method used here could be used to model more complex processes affecting recruitment for the most common species and/or through the use of prior information from other studies. Our function predicts that recruitment increases with conspecific adult canopy area (which is proportional to basal area) and decreases with total canopy area - which is a measure of competition for light. 5.2 ABC Algorithms Conceptually, ABC algorithms are as easy to understand as any basic parameter space-searching algorithm. To illustrate the idea of ABC we first describe a simple ABC rejection algorithm (Tavaré et al., 1997; Pritchard et al., 1999; Beaumont et al., 2002). Suppose we wish to fit a given model M which depends on some unknown parameter set p. Without observed data (y0) we cannot use a traditional likelihood method to estimate the parameter set p. However, with one or more observed summary statistics of data S(y0) we can still infer values of the parameter set by sampling from a distribution that approximates the posterior. For example, given a simulation model M and summary statistic(s) S, a simple ABC rejection algorithm would be as follows: BASIC ABC REJECTION ALGORITHM: 1. Sample a candidate model parameter set from the prior π(p). 2. Generate a simulated dataset using the model M, . 3. From the simulated data , calculate the summary statistic(s) . 4. Using a given metric d and tolerance level ε, compare the observed (S(y0)) and predicted summary statistic(s). If , accept parameter set , if not, reject the parameter set and return to step 1. (Eqn 5.1) Each accepted simulated dataset generated in this algorithm is an independent sample from the distribution , and if is small enough, this will approximate the true posterior π(p|y). This method is likely to be extremely slow to converge without a good knowledge of the prior; many improvements have been suggested to improve speed of convergence and to ensure efficient searching of parameter space (Marjoram et al., 2003; Del Moral et al., 2006; Sisson et al., 2007; Toni et al., 2009). Chapter 5 ABC recruitment model 75 In this study we use an ABC Sequential Monte Carlo (SMC) approach first proposed by Sisson et al. (2007) and later amended to correct a bias by Beaumont et al. (2009). Although random walk ABC-MCMC algorithms have been described (e.g. Marjoram et al., 2003), parameter chains risk spending long periods of time in areas of low probability and, since samples may be highly correlated, very long chains are likely to be needed (Toni et al., 2009). The main advantage of SMC methods is that they bypass the risk of collecting a set of highly correlated samples by repeatedly sampling from a series of approximations to the posterior with decreasing tolerance levels, within which samples are not correlated (Sisson et al., 2007). ABC-SMC works as follows: at each iteration t from t=1 to t=T, N independent particles (parameter sets) are sampled from the distribution with a defined decreasing vector of tolerances ε1> ε2>...> εT 0. Particles are sampled from weighted samples (with weights ) from the previous distribution, ensuring that particles that better approximate π(p|y) are re-sampled more often, and those that are a poor fit are discarded. The version of the algorithm we used is taken from Beaumont (2009) (but is very similar to those given by Sisson et al., 2009; Toni et al., 2009). We constructed the algorithm (using C code) as follows: ABC-SMC ALGORITHM: 1. When t=1, for i=1...N a. Sample a particle from the prior, , and generate until , b. Set all weights equal, as = 1/N, c. Set to be twice the empirical variance of particles . 2. For t=2...T a. For i=1...N i. Sample particle p* from the previous distribution with weights , ii. Perturb the particle according to a transition kernel, p*~N(p*, Σt-1), and generate y**~f(y|p**), iii. If , set , otherwise return to 2ai. b. For i=1...N Calculate the weight of each particle according to: where is the multinormal density with variance Σt-1. c. Set Σt to be twice the empirical variance of particles . (Eqn 5.2) Chapter 5 ABC recruitment model 76 5.3 Materials and Methods 5.3.1 Data The inventory data used for this study came from the second and third Spanish Forest Inventory (IFN2 and IFN3; MMA, 1996, 2007), sampled from over 70,000 remeasured plots arranged systematically on a 1 km 2 grid across Spain. The IFN plots were sampled using a variable radius technique, whereby trees of different sizes were measured in concentric plots of varying size. All trees of DBH > 7.5 cm were measured in a plot of radius 5 m, all of DBH > 12.5 cm in a plot of radius 10 m, all of DBH > 22.5 cm in a plot of radius 15 m and all of DBH > 42.5 cm in a plot of radius 25 m. Although no individual sapling data were recorded, densities of large juveniles (heights > 130 cm and DBH in the range 2.5-7.5 cm) were recorded within the central 5 m radius plot in IFN3. We selected plots for recruitment modelling carefully so as to fit only recruitment which arises during succession, i.e. we removed data that could have arisen from pulses of recruitment following large scale disturbances such as fire, as we did not have information on the occurrence or impact such events. Our criteria were: 1) plots must be thinning (stand density decreasing and mean stem size increasing over the time period of the two surveys)- around 26% of the remeasured plots, 2) juveniles must have been recorded as being of 'natural' origin in the inventory, and 3) plots must have at least one adult (DBH>7.5 cm) of the species of interest in IFN2 plot (largest 25 m radius plot). There was no record of whether the recorded stems were resprouts (a feature of some Spanish tree species; Grove and Rackham, 2001) or saplings, so we were unable to differentiate between these two cases. We selected the 14 species with at least 50 plots matching these criteria to parameterise the model, comprising seven conifers (Pinus pinea, P. halepensis, P. pinaster, P. sylvestris, P. uncinata, P. nigra, Juniperus thurifera) and seven angiosperms (Quercus faginea, Q. ilex, Q. suber, Q. petraea, Q. pyrenaica, Fagus sylvatica and Castanea sativa). Number of plots selected for each species are shown in Table 5.1. Histograms of the number of large juveniles in each selected plot for each of these species are shown in Fig. 5.1. We expected recruitment to be proportional to conspecific adult density (potential parent trees) and be negatively affected by aboveground competition for light. We chose to use crown area to represent both these processes, which has been used to represent asymmetric competition within forest dynamics models (e.g. Bohlman and Pacala, 2011; Caspersen et al., 2011; Coomes et al., 2012). We calculated total crown area and total basal area of adults (stems > 7.5 cm DBH) of each plot and each species for both inventories using adult data, and used these as measures of conspecific density and competition (Caspersen et al., 2011; Coomes et al., 2012). For each plot we defined two values, the crown cover of adults of the species of interest (CAIsp, m 2 /ha) and of all adults on the plot (CAIpl), using species-specific crown width allometric equations derived from data collected on a subset of IFN2 sites (see text and Tables C.1-C.3, Fig. C.1 in Appendix C page 141). We also calculated basal Chapter 5 ABC recruitment model 77 area of each species (BAsp, m 2 /ha) and all species on the plot (BApl). In order to compare recruitment rates among species we took shade tolerance scores from Niinemets and Valladares (2008), and when species-specific values were not available we took genus averages. Figure 5.1 Histograms of the recorded number of large juveniles in the central 5 m radius circular subplot of each selected plot for each of the 14 species. Total numbers of plots used for each species are given in Table 5.1. 0 10 20 30 0 5 10 15 20 25 30 35 Castanea sativa 0 2 4 6 8 10 12 14 0 20 40 60 80 Fagus sylvatica 0 2 4 6 8 10 0 5 10 15 20 25 Juniperus thurifera 0 2 4 6 8 10 12 0 50 100 150 200 250 Quercus faginea 0 5 10 15 20 25 30 0 50 100 150 200 250 300 350 Quercus ilex 0 2 4 6 8 0 20 40 60 80 Quercus petraea 0 10 20 30 40 50 0 20 40 60 80 Quercus pyrenaica 0 2 4 6 8 0 20 40 60 80 100 120 Quercus suber 0 5 10 15 20 0 50 100 150 200 250 300 Pinus halepensis 0 5 10 15 20 25 30 0 50 100 150 200 250 Pinus nigra 0 10 20 30 40 0 50 100 150 200 250 300 Pinus pinaster 0 2 4 6 8 0 50 100 150 Pinus pinea 0 2 4 6 8 0 50 100 150 Pinus sylvestris 0 2 4 6 8 10 12 0 10 20 30 40 Pinus uncinata Density of stems with 2.5 cm < DBH < 7.cm F re q u e n c y Chapter 5 ABC recruitment model 78 5.3.2 Prior information for growth and mortality rates Many combinations of recruitment, growth and mortality rates may combine to give the observed juvenile density patterns, yet not all are reasonable given prior knowledge of demographic processes. We used information from larger trees in the inventories to construct priors for parameters for growth and mortality rates of juveniles. We fit species-specific growth and mortality functions to data from trees between 7.5 and 10 cm DBH that had been remeasured in the inventory (see Tables C.4 and C.5 in Appendix C page 149). For both processes, we tested several alternative models and used Akaike Information Criteria (AIC: Akaike, 1974) to select the best models for the largest number of species (see Appendix C page 141). The best fit models were size-independent models but included the canopy area of taller trees in the plot (CAIh) as a measure of competition: Annual growth rate (cm/year) = p2/(1+ p3CAIh) Annual mortality rate (stems/year) = logistic (p4+ p5CAIh) (Eqn 5.3) where p2- p5 are fitted parameters (see text and Tables C.4-C.7 and Figs. C.2 and C.3 in Appendix C page 141). These functional forms (Fig. 5.2) were used within the simulation model, and their parameter values were estimated during the ABC-SMC algorithm with strong priors defined by the values estimated from the adult data. As the inventory data contained only trees > 7.5 cm DBH, and we simulated juveniles of < 7.5 cm DBH only, when including these rates within the simulation model we took CAIh = CAIpl, the total canopy area of trees in the plot > 7.5 cm DBH (so it took the same value for all simulated juveniles). 5.3.3 Simulation model In order to estimate juvenile recruitment, growth and mortality rates we constructed a model which simulated the size structure and density of juveniles given a set of parameter values. The model was based on a simple forest stand dynamics model (the PPA, Purves et al. 2008) which predicts the fate of cohorts of individuals of the same species and age rather than individual stems, and therefore does not contain spatial references. We simulated cohorts Ci (introducing a new one at each time step) within a 5 m radius circular plot to make comparison with our data sample, over T years (time steps), and recorded their densities deni,t (#stems / 5m radius plot) at each timestep and diameters DBHi,t (cm), producing a predicted size distribution with corresponding density of juveniles in the 5 m radius plot. The simulation model ran as follows: Chapter 5 ABC recruitment model 79 SIMULATION MODEL: For each time step (t=1...T) For each plot (p=1...P) 1. For all existing cohorts (i=1...N) a. Kill off a proportion according to the mortality rate: deni,t=(1-P(mortalityi,t-1))x deni,t-1 b. Increase the size of trees in the cohort according to growth rate: DBHi,t = DBHi,t-1 + growth 2. Create a new cohort of DBH = 1 cm and density according to the recruitment rate. 3. N=N+1. (Eqn 5.4) Mortality and growth rates were as in equation 5.3. We fitted recruitment rates as: Recruitment (#stems 1cm DBH/ha/year) = p0 CAIspexp(-p1CAIpl) (Eqn 5.5) which allowed the recruitment rate within the central 5 m radius plot to vary with the crown area of conspecifics (CAIsp) in the larger 25 m radius plot, and included an effect of shading and competition effect by total plot crown area (CAIpl). Although the prior distribution for p1 was set to be U(0,15) (i.e. the prior was positive), the ABC-SMC was able to sample negative values for this parameter (for example to represent a facilitation effect of aboveground competition for species found in areas of intense drought). We chose to use crown area rather than the more common basal area (or some function of it, e.g. Ribbens et al., 1994) as it had provided a better fit to the data as a predictor in the growth and mortality functions (see Tables C.4 and C.5 in Appendix C, page 149). Since both CAIsp and CAIpl changed in each plot between the two inventories for the first part of the simulation model we used values calculated from IFN2, but varied these during the final 10 time steps (corresponding to an average 10-year time interval between inventories) using a simple linear relationship between the measurements calculated from IFN3 data. This meant that for the first period of the simulation model the conspecific canopy and aboveground competition was held constant but in the second part it varied according to the observed canopy area dynamics between the two inventories. 5.3.4 Assessing the algorithm on test data: method We initially tested the ability of the method to recover parameter values from simulated test data created using the structure of the inventory but replacing juvenile counts with counts simulated with known parameter values. To create the simulated data we took the inventory data for a common species - Pinus sylvestris - and our calculated values of CAIpl and CAIsp for each plot in which it was found. We simulated juvenile densities using the simulation model described above. We specified growth, recruitment and mortality models as given in equations 5.3 and 5.5. We ran the model for 200 iterations (each representing a year) to ensure that in all plots there were stems of at least 7.5 cm Chapter 5 ABC recruitment model 80 DBH. We simulated two different datasets, with and without variability in the parameters. For the dataset without variability we simply simulated data using fixed parameter values (p0- p6 in equations 5.3 and 5.5) as follows: p0=2; p1=0.8; p2=0.45; p3=1.75; p4=-5.6; p5=1.28 (Eqn 5.6) To create variability within the data, during the creation of the simulated data, at each time step and for each cohort we drew values of the parameters from specified normal distributions: p0~N(2,0.2) 2 ); p1~N(0.8,(0.08) 2 ); p2~N(0.45,(0.045) 2 ); p3~N(1.75,(0.18) 2 );p4~N(-5.6,(0.56) 2 ); p5~N(1.28,(0.13) 2 ) To test whether the ABC algorithm was able to reproduce the parameters, we fitted the ABC-SMC sampler to the simulated data. We set uniform priors (π(.)) on the recruitment parameters (p0-p1): π(p0)~U(0,5); π(p1)~U(0,2). We set normal priors on the growth and mortality parameters (p2- p5) which were informative but randomly chosen: we drew a mean value for each parameter from normal distributions with mean values as given in equation 5.6 and standard deviations of 0.05. Standard deviations for the priors were set as 5% of the selected mean. Priors were used as initial sampling distributions for all parameters. The choice of metric used to compare predicted and observed densities can affect the convergence of the algorithm (Beaumont et al., 2009), and we therefore tested two metrics, d1 and d2, to compare the predicted and observed densities (Pden and Oden): d1=Σplots|Pden- Oden|; d2=Σplots(Pden- Oden) 2 ; (Eqn 5.7) We ran 10-20 iterations of the ABC-SMC algorithm using 100 particles on the four data-metric combinations. We set tolerance levels as a multiple of the number of plots simulated and decreased levels by 5% at each iteration, and continued to decrease the tolerance level until the algorithm was unable to find parameter values satisfying it. 5.3.5 Performance of the algorithm on test data and choice of metric For both sets of simulated test data the ABC-SMC algorithm was able to recover the parameters of interest, although as expected parameter values were better recovered for the data without variation, with much tighter ranges in fitted values (Fig. 5.3). However, we did find that parameterisations run using metric d1 (equation 5.7) were much better able to recover the true parameter value than those using d2, (Fig. 5.3), for both sets of simulated test data (with and without variation in parameter values). We therefore chose to use metric d1 to fit parameters with the real data. 5.3.6 Implementation of the ABC-SMC algorithm on the inventory data We used the fitted growth and mortality parameters (given in Tables C.6 and C.7 in Appendix C page 150) as prior means for the parameters p2-p5, (equation 5.3), with standard deviations set as 2% of the prior value, and the prior distribution was used as the initial sampling distribution. Priors for the parameters p0 and p1 (equation 5.5) were set as U(0,15) with initial sampling distribution U(0,5). We Chapter 5 ABC recruitment model 81 simulated cohorts from 1 cm DBH and summed densities of all cohorts with DBH values between 2.5 and 7.5 cm to give the same form as the count data presented in the inventory. We used metric d1 (equation 5.7) to compare predicted and observed densities, as this had performed better in the simulated data example. Starting and final tolerance levels varied between species according to how well the model fitted the data. Tolerances were defined as a multiple of the number of plots for each species, with starting values in the range 0.9-0.1, and reduced by 25% at each simulation step (t in equation 5.2). We used the means of the predicted values of parameters p0 and p1 (equation 5.5) to compare recruitment rates among species. Table 5.1 The fourteen species included in the analysis and the number of plots used, as well as their shade tolerance scores (from Niinemets & Valladares, 2008). The predicted average recruitment rates (number of 1cm DBH stems ha -1 year -1 ), the predicted annual growth (cm year -1 ) and predicted annual mortality rate (stems stem -1 year -1 ), all calculated in a fixed environment (0.25 ha ha -1 conspecific adult crown area (CAIsp) and 0.5 ha ha -1 total adult crown area of the plot (CAIpl)), are also shown (see equation 5.5 for rate equations). Species Number of plots Shade tolerance Recruitment rate in fixed environment Annual growth rate in fixed environment Annual mortality in fixed environment Castanea sativa 58 3.15 4.740 0.258 0.035 Fagus sylvatica 150 4.56 0.899 0.210 0.005 Juniperus thurifera 70 1.61 1.238 0.075 0.006 Pinus halepensis 600 1.35 3.009 0.177 0.019 Pinus nigra 583 2.1 5.427 0.193 0.013 Pinus pinaster 488 1.35 1.038 0.262 0.042 Pinus pinea 178 1.35 0.240 0.249 0.026 Pinus sylvestris 691 1.67 1.619 0.264 0.010 Pinus uncinata 103 1.2 5.148 0.176 0.009 Quercus faginea 294 2.88 0.093 0.122 0.007 Quercus ilex 489 3.02 0.159 0.131 0.008 Quercus petraea 100 2.73 0.165 0.165 0.004 Quercus pyrenaica 131 2.88 1.142 0.140 0.015 Quercus suber 148 3.02 0.182 0.143 0.012 Chapter 5 ABC recruitment model 82 5.4 Results from inventory data 5.4.1 Convergence of the ABC-SMC algorithm Convergence of the parameters varied among species, with the 95% credible intervals smallest for recruitment parameters of the species with more data (Fig. 5.4). It is difficult to interpret the effect on recruitment rates from individual parameter values alone, as CAIpl and CAIsp will always at least partially covary (as CAIsp ≤ CAIpl). All species showed decreases in growth and increases in mortality with increasing CAIpl, with angiosperms showing less response than conifers (Fig. 5.2). Growth and mortality parameters, which were strongly constrained by priors, converged much more strongly with 95% credible interval limits on average just 3% from the prior value (Fig. 5.4), however when we re- ran the model with much larger standard deviations on the growth and mortality rates we found that the convergence of the recruitment rates was substantially reduced (data not shown). 5.4.2 Average recruitment rates and the effect of canopy cover Both average recruitment rates (Table 5.1) and how competition affected species (Fig. 5.5) varied across species, with most species showing a strong negative response to increased whole-plot density (CAIpl). One species, C. sativa, showed almost no reduction in recruitment rates in the most dense plots and had the highest recruitment rates in all but the most open areas (Fig. 5.6). In open, low density plots, angiosperm species had lower average recruitment rates than conifer species (Fig. 5.6a) though the difference was weak (t test p<0.1). The two evergreen Mediterranean oak species (Q. ilex and Q. suber), along with the semi-deciduous Q. faginea had low recruitment rates in all environments. On average the Mediterranean species had lower recruitment rates in the fixed environment of 0.25 ha ha -1 CAIsp and 0.5 ha ha -1 CAIpl (0.85 vs 2.73, Table 5.1) but the difference was only weakly significant (Spearman's p<0.1) and disappeared in higher density plots. Among species, we found a significant positive correlation between the mean fitted values of the two recruitment parameters (p0 and p1 in equation 5.5) (Spearman's p<0.05), so that species with higher recruitment rates in open areas were also more negatively affected by increased competition from canopy cover overhead. This corresponded to a split in recruitment rates under different plot conditions between angiosperm and conifer species. In low density, monospecific plots (CAIsp = CAIpl = 0.25 ha ha -1 ), conifer species had higher recruitment rates than angiosperms (Fig. 5.6b), but this difference disappeared in more dense plots as the conifer species had much stronger negative responses to increases in CAIpl than angiosperm species (Fig. 5.6b). There was a significant negative correlation between shade tolerance and the change in recruitment rate from low to high density stands (Pearson's p<0.05 between shade tolerance and the difference between predicted rates in a plot with CAIsp = CAIpl = 0.25 ha ha -1 and in a plot with CAIsp = 0.25 ha ha -1 and CAIpl = 1 ha ha -1 ), with the most shade intolerant species suffering the largest drops in recruitment rate with higher canopy cover. Chapter 5 ABC recruitment model 83 Figure 5.2 Predicted growth and mortality rates used as priors for the simulation mode (equation 5.5, tables C.2 and C.3) for the 14 species across the range of competitive environment (crown area of taller trees CAIh) in which they are found. Species are split into four groups according to type and geographical distribution: Mediterranean conifer (Pinus pinea, P. halepensis, P. nigra, P. pinaster, Juniperus thurifera), temperate conifer (P. sylvestris, P. uncinata), Mediterranean angiosperm (Quercus faginea, Q. ilex, Q. suber) and temperate angiosperm (Q. petraea, Q. pyrenaica, Fagus sylvatica, Castanea sativa). Figure 5.3 Estimated parameter values for simulated test data without and with variation in the data, fitted using two different metrics (equation 5.7). True values of each parameter are shown by the dotted red line. Metric 1 was better able to recover the true parameter values. p0 d1[x:y, 1] 1.8 1.9 2.0 2.1 2.2 0 2 4 6 8 10 12 p1 d1[x:y, 2] 0.6 0.7 0.8 0.9 1.0 0 2 4 6 8 p0 d3[x:y, 1] 1.6 1.8 2.0 2.2 0 2 4 6 8 p1 d3[x:y, 2] 0.6 0.8 1.0 1.2 1.4 0 2 4 6 8 10 p0 d2[x:y, 1] 1.8 1.9 2.0 2.1 2.2 0 1 2 3 4 5 6 7 p1 d2[x:y, 2] 0.6 0.7 0.8 0.9 1.0 0 2 4 6 8 10 p0 d4[x:y, 1] 1.6 1.8 2.0 2.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 p1 d4[x:y, 2] 0.6 0.8 1.0 1.2 1.4 0 1 2 3 4 5 6 Without variation With Variation P ro b a b ili ty D e n s it y M e tr ic 1 M e tr ic 2 Parameter value Chapter 5 ABC recruitment model 84 Figure 5.4 Fitted mean values and 95 % credible intervals for each of the six parameters (equation 5.3 and 5.5) estimated in the ABC-SMC algorithm for each of the 14 species (Species abbreviations: P. sy = Pinus sylvestris, P. un = P. uncinata, P. pa = P. pinea, P. ha = P. halepensis, P. ni = P. nigra, P. pr = P. pinaster, J. th = Juniperus thurifera, Q. pe = Quercus petraea, Q. py = Q. pyrenaica, Q. fa = Q. faginea, Q. il = Q. ilex, Q. su = Q. suber, F. sy = Fagus sylvatica, C. sa = Castanea sativa). p0 p1 p2 p3 p4 p5 Chapter 5 ABC recruitment model 85 Figure 5.5 Predicted recruitment rates (number of new 1 cm stems ha -1 year -1 ) for each of the 14 species plotted along the range of plot canopy area (CAIpl) in which they are found, with CAIsp as a) a fixed proportion of CAIpl, and b) fixed at CAIsp = 0.25 ha ha -1 . Species are split into four groups according to type and geographical distribution: Mediterranean conifer (Pinus pinea, P. halepensis, P. nigra, P. pinaster, Juniperus thurifera), temperate conifer (P. sylvestris, P. uncinata), Mediterranean angiosperm (Q. faginea, Q. ilex, Q. suber) and temperate angiosperm (Q. petraea, Q. pyrenaica, Fagus sylvatica, Castanea sativa). 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2 4 6 8 10 (a) Canopy area of conspecifics (CAIsp) is fixed at 50% of total canopy area (CAIpl) CAIpl (ha/ha) R e c ru it m e n t ra te ( 1 c m s te m s /h a /y e a r) Mediterranean Conifer Temperate Conifer Mediterranean Angiosperm Temperate Angiosperm 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2 4 6 8 10 (b) Canopy area of conspecifics (CAIsp) is fixed at 0.25 ha/ha CAIpl (ha/ha) R e c ru it m e n t ra te ( 1 c m s te m s /h a /y e a r) Mediterranean Conifer Temperate Conifer Mediterranean Angiosperm Temperate Angiosperm Chapter 5 ABC recruitment model 86 5.4.3 Relationships between recruitment, growth and mortality Average recruitment rates in the fixed environment (0.25 ha ha -1 CAIsp and 0.5 ha ha -1 CAIpl) did not correlate with growth or mortality rates, though species with higher growth rates had significantly higher mortality rates (Spearman’s p<0.05, Table 5.1). We found a significant negative correlation between parameters p0 and p4 (equations 5.3, 5.5), suggesting that in open areas the species with the highest recruitment rates also had the lowest mortality rates. Across species and rates the effect of competition was not consistent, with no correlation between either p3 (the effect of CAIpl on growth) or p5 (the effect of CAIpl on mortality) and p1 (the effect of CAIpl on recruitment), nor was there a correlation between the change in growth or mortality rates from low to high density plots with the change in recruitment rates. There was however a positive correlation between p3 (the effect of CAIpl on growth) and p5 (the effect of CAIpl on mortality) (equation 5.5), so that species whose growth rates dropped most in more dense plots had the largest increases in mortality rates (Spearman’s p<0.05). Among species, the effect of CAIpl on both growth and mortality was significantly related to shade tolerance, with the most shade tolerant species being least affected by increases in canopy cover (Pearson’s p<0.05, negative correlation between shade tolerance and both p3 and p5). Figure 5.6 Boxplots showing differences between conifer and angiosperm species in a) predicted recruitment rates in a low density plot (CAIsp = CAIpl = 0.25 ha ha -1 ), and b) their predicted changes in recruitment rates from low to high density plots (the predicted rate in CAIsp = CAIpl = ha ha -1 plot minus the predicted rate in CAIsp = 0.25 ha ha -1 and CAIpl = 1 ha ha -1 plot). Conifer Angiosperm 0 1 2 3 4 Predicted recruitment rates in low density plots R e c ru it m e n t ra te ( 1 c m D B H s te m s /h a /y e a r) t test p.value = 0.066 Conifer Angiosperm 0 2 4 6 8 10 Change in recruitment rates from high to low density plots R e c ru it m e n t ra te ( 1 c m D B H s te m s /h a /y e a r) t test p.value = 0.051 (a) (b) Chapter 5 ABC recruitment model 87 5.5 Discussion 5.5.1 Parameterisation and predicted recruitment rates We have demonstrated the ability of the ABC parameterisation framework to infer dynamic rates using summarised data and partial knowledge of a system. By utilising the extensive data of the Spanish forest inventory we were able to calibrate a recruitment model for a much larger number of sites and competitive conditions than could have been examined had we collected more detailed data. Since many forest inventories do not include detailed data for small trees we suggest that this method may prove useful for constructing recruitment models in other parts of the world. We found large variation in recruitment rates among species, and that, in less dense stands at least, conifer species had higher recruitment rates than angiosperms, in agreement with comparisons between Mediterranean pine and oak regeneration levels (Urbieta et al., 2011). We found that for almost all species (with the exception of C. sativa) canopy density strongly negatively affected juvenile recruitment, growth and mortality rates, but that the effect was correlated with the reported shade tolerance of the species, suggesting that all aspects of juvenile dynamics are affected by aboveground competition. However, facilitative effects from neighbouring trees and shrubs are known to aid seedling survival and growth in the Mediterranean, by preventing desiccation by reducing water stress and protecting from high levels of irradiance (Gómez-Aparicio et al., 2006; Quero et al., 2006; Smit et al., 2008; Mendoza et al., 2009), with the strongest facilitation benefits reported for deciduous and Quercus species and the weakest for evergreen and Pinus species (Gómez- Aparicio et al., 2004, 2006). Therefore we might expect a positive interaction between the effect of drought length and canopy cover on recruitment for some species in the areas with the longest drought. However, the small amount of data we had for many species in this study prevented a full investigation of how large an effect facilitation might have on this process. The high predicted recruitment rates and lack of negative response to increased canopy cover for C. sativa may be a result of historical management of the species, which has been a very commonly planted and coppiced species in the Mediterranean (Grove and Rackham, 2001; Giudici and Zingg, 2005). This species also had some of the highest observed sapling densities (Fig. 5.1) so much of the data may be a result of multi-stemmed regeneration from stools rather than seedling establishment. 5.5.2 Other factors affecting Mediterranean juvenile recruitment The low amount of data for several of our species meant that we only included the effects of conspecific adults and competition for light on intraspecific variation in recruitment rates. However, several studies have identified environmental factors affecting variation in seedling establishment and performance in the Mediterranean which may be relevant to our study area. Fine-scale site variables such as soil water content and light availability are important determinants of germination, but Chapter 5 ABC recruitment model 88 responses are complex; for example both seasonal drought and waterlogging may negatively affect establishment of Mediterranean oaks (Rey Benayas, 1998; Urbieta et al., 2008). At a larger scale, climate and topographic factors affect regeneration, though responses are different among species. For example, high rainfall has been found to increase regeneration rates of the deciduous Q. pyrenaica but decrease rates of the evergreen Q. ilex (Plieninger et al., 2010), and temperature has been found to be important for differential regeneration rates between species (Gómez-Aparicio et al., 2009), whilst in southern France grazing pressure has been found to be a primary determinant of regeneration (Chauchard et al., 2007). Climate change and in particular warming is likely to cause an upwards altitudinal shift in species’ distributions, and indicators of this have been found in higher recruitment rates on higher slopes in mountainous areas for several species in mountainous areas of Spain (Camarero and Gutiérrez, 2007; Peñuelas et al., 2007). 5.5.3 Implementation of ABC Whilst the form of the data in this study made the use of ABC methods necessary to derive annual recruitment rates, the method presented here is inefficient for parameter estimation compared to traditional likelihood-based techniques, if available. The multiple elements involved in ABC methods mean that they are not necessarily straightforward to implement, and this must be considered when specifying the algorithms. Since the algorithm relies on the design of an underlying simulation model, the careful construction of this to describe the underlying processes of interest is paramount to the success of the parameter estimation. Methods of model selection for ABC have been suggested, for example by calculating the posterior probability of a given model over a set of other models (Toni et al., 2009; Leuenberger and Wegmann, 2010) in a method analogous to traditional Bayes’ factors. One such method of model comparison approximates Bayes factors using a simple rejection algorithm (similar to equation 5.1) where model indicators are treated as categorical parameters and the ratio of acceptance rates of one model over another gives the Bayes factor (Grelaud et al., 2008). This approach may be made more efficient if it is implemented using an SMC approach (Toni et al., 2009). However, when used to compare a set of models, it can also result in less informative models being given a Bayes factor of 0, since a low tolerance level could exclude them from being selected at all. Model parameterisation itself may be dependent on the choice of summary statistic, and many studies have suggested methods of selecting the best summary statistics (Joyce and Marjoram, 2008; Wegmann et al., 2009; Jung and Marjoram, 2011). However, it likely to be difficult or even impossible to know if a selected set of summary statistics is sufficient (Marjoram et al., 2003) and the optimal statistics are likely to be highly dataset specific (Nunes and Balding, 2010). Although it may be intuitive to think that adding more summary statistics would improve model performance, studies have shown this is not always the case as acceptance rates can be dramatically reduced when they are based on both informative and uninformative summary statistics (Blum, 2010; Fearnhead and Prangle, Chapter 5 ABC recruitment model 89 2010). The use of multiple summary statistics may also make the choice of tolerance levels difficult, as convergence of the model to reproduce multiple statistics well may not be possible with variable data. The credible intervals on posterior parameter estimates arising from ABC simulation models are likely to be inflated due to a loss of information from summarising the data (Csilléry et al., 2010). By far the most computationally expensive element of the parameterisation algorithm was the simulation model which produced new juvenile count data for each candidate parameter set, making practical application of ABC methods at least partially dependent on its efficiency. One major advantage of ABC-SMC is that, within any iteration of the model, parameter samples are independent and so simulations can be run easily in parallel on multiple CPUs or clusters. Although we constructed our own code in C to implement the algorithm, statistical packages for implementing a range of different ABC algorithms are available, including several R packages (such as abc; Csilléry et al., 2012) and the stand alone program DIY-ABC (Cornuet et al., 2008). 5.5.4 Applications of ABC in Ecology We have demonstrated the ability of Approximate Bayesian Computation techniques to parameterise individual-based demographic models in a mathematically rigorous way with data that could not have been done using traditional likelihood-based techniques. These methods have been regularly adopted in some areas of biological research, and have potential applications in many areas of ecology, for example to infer unobserved historical processes that have led to an observed state of a system (e.g. Thornton and Andolfatto, 2006). These methods also allow parameter inference for stochastic models for which likelihoods cannot be constructed, for example parameters for the neutral model of biodiversity (Jabot and Chave, 2009). ABC methods are also likely to have applications in areas of pattern orientated modelling (POM), where emphasis is placed on simulation models reproducing ecological patterns in data. POM focuses on the most essential information in a system (Grimm et al., 2005), and ABC methods provide an analogous way of selecting the best model, as by selecting summary statistics to represent the patterns of interest, rigorous parameterisation and model comparison can be focussed on how well the models reproduce patterns. One major advantage of the structure of ABC modelling when applied to ecological situations is that they allow us to incorporate partial knowledge of the processes within a complex ecological simulation model, as we have demonstrated in this study. Whilst direct measurements of some processes may be lacking, it is unlikely that nothing is known about the direction or magnitude of any process within an ecological system. The inclusion of prior information is likely to dramatically improve the convergence and realism of estimated parameters, and uncertainty in priors can easily be incorporated and explored within the algorithm. ABC methods therefore take a truly Bayesian approach by allowing the use of partial knowledge of systems to infer unmeasured processes, and thus fully parameterise complex models that previously could not be fully specified. Chapter 5 ABC recruitment model 90 Author contributions Designed the study, analysed the data and performed statistical analysis: Emily Lines. Provided the data: Miguel Zavala. Wrote the paper: Emily Lines with supervisory support from David Coomes and Miguel Zavala. Chapter 6 Simulation model 91 6 Regional successional dynamics simulated from tree-level processes Abstract Forest simulation models have been used to understand forest dynamics, structure and productivity, and species dominance and succession but typically contain no climatic dependency in their underlying processes, meaning that they have limited use in understanding climate-driven changes in forest dynamics, or in predicting the impact of changing climate on them. In this study we parameterise a spatially-implicit individual-based forest model, the PPA, for 14 major native tree species in Spain using national inventory data. We parameterise growth, mortality and allometry subroutines of the model as functions of both competition and climate, and run simulations of forest development from open stands over the climatic conditions found in Spain. The model was able to reproduce observe patterns of Pinus dominance across Spain, and decomposing the underlying processes revealed that inter and intraspecific variation in both growth and mortality along climatic gradients determined regional shifts in species dominance. We used the model to predict long-term successional dynamics over 500 years, and found that the inclusion of competition-dependent recruitment within the model produced predictions for the eventual dominance of angiosperm species over Pinus species in most parts of Spain. Despite its simplicity, we found that including climate dependency of growth, mortality and allometry produced realistic predictions of species dominance as emergent properties of the model. We suggest that this approach could be repeated in other parts of the world using national forest inventories to improve understanding of the climatic drivers of species dominance. 6.1 Introduction Simulation models are vital tools for forest ecology research as they allow questions posed on large geographical scales and over long periods of time to be answered. Forest simulators which incorporate individual rates of tree growth and mortality have been show to be able to reproduce forest properties such as stand structure (e.g. Lindner et al., 1997), productivity (e.g. Caspersen et al., 2011; Coomes et al., 2012) species composition and successional dynamics (e.g. Pacala et al., 1996), and have been parameterised for boreal (e.g. Zhang et al., 2009), temperate (e.g. Kunstler et al., 2009) and tropical forests (e.g. Bohlman and Pacala, 2011). Chapter 6 Simulation model 92 Uncertainty in the response of forests to climate change forms a large part of the uncertainty in Earth systems models, and incorporating ecological processes through a better representation of species diversity and competition will fundamentally improve the realism of simulations (Purves and Pacala, 2008). Large-scale long-term forest datasets, combined with increasing computational processing power with which to parameterise and run simulations, mean that the development of forest dynamic models which incorporate the effects of multiple environmental drivers on demographic processes for many species is now possible in many parts of the world. In order to simulate landscape level forest dynamics it is axiomatic that forest models must include representation of the drivers of variation in forests at a landscape level. The dynamics of succession are known to be sensitive to climatic variation (Kardol et al., 2010), as are individual tree level processes of growth (Coomes and Allen, 2007a; Martínez-Vilalta et al., 2011), mortality (van Mantgem and Stephenson, 2007; Voelker et al., 2008) and recruitment (Mendoza et al., 2009). Forest simulators are usually parameterised from detailed site data, and despite often complex calculations of light availability and local competition, do not typically include climate dependency in model processes, which means that they have very limited use in understanding the role that climate plays on determining successional dynamics and species dominance on a regional scale, or in predicting future forest dynamics under climatic change. The perfect plasticity model (PPA) is a cohort-based spatially-implicit model which uses individual-tree based allometry, growth and mortality functions to describe whole-forest dynamics and which was originally developed for forests in the lake states of the US (Purves et al., 2007, 2008; Strigul et al., 2008). In order to calculate the level of competition for light within a stand the PPA assumes plasticity in each tree’s crown area position, meaning that the crown of a tree of a given height may be placed anywhere within the horizontal plane it occupies in order to maximise light capture. This means that recording the exact location of the crown (and the stem) within the stand is not necessary, making the model spatially-implicit, and dramatically reducing the level of complexity compared to spatially-explicit individual-based forest models such as SORTIE (Pacala et al., 1996). In this study we test the ability of a PPA-type forest simulation model based on climate- dependent individual-based demographic rates to reproduce observed species distributions at a regional scale. We parameterise the model for 14 major tree species in Spain using inventory data (the Spanish forest inventory; MMA, 1996, 2007) to constrain climate and competition dependent model processes and use simulations to predict early and late successional dominance across Spain. The idea that species dominance and successional dynamics on a regional scale are a product of climate is not a new one, nor is the interest in predicting the response of vegetation to changing climate through simulation modelling (Shugart and West, 1980; Hall et al., 1991; Glenn-Lewin et al., 1992). However, we know of no study which tests whether a forest simulation model can reproduce observed species' dominance at a regional scale as an emergent property from individual-based demographic rates parameterised from inventory data. Chapter 6 Simulation model 93 We test the ability of the model to reproduce species dominance patterns across the landscape using species dominance observations recorded in the inventory. We simulate successional dynamics by initialising a stand with small trees of many species, and recording how species dominance changes over time under the demographic rules driving the simulations. Pinus species are frequently the early-successional dominants in Spain, whilst angiosperms species, primarily Quercus, are often observed to be late successional (Lookingbill and Zavala, 2000; Pausas et al., 2004; Capitanio and Carcaillet, 2008; Santana et al., 2010). The inventory did not include stand age or successional stage so we used pine/angiosperm dominance as a proxy for this. We use the simulations to predict geographic variation in which Pinus species rise to dominance after 30 years of stand development, and to predict which angiosperm species dominate up to 500 years after initialising the model. These predictions are then compared with observed distributions. Forests in Spain have a long history of human management which has driven forest dynamics in some areas (Urbieta et al., 2008), and are frequently disturbed by forest fires (Grove and Rackham, 2001). As in many parts of the temperate world, forest area is increasing due to land abandonment in the second half of the twentieth century (Campos et al., 2005), meaning that much of the forested land in the region in unlikely to be in equilibrium. The simulation models do not explicitly consider the effects of fire or human management. Therefore, by parameterising the model using only competition and climate dependency of individual species' processes we hypothesise that there will be some mismatches between what we predict and the observed patterns of species dominance across the region. 6.2 Simulation model 6.2.1 Canopy area of taller trees, CAIh, as a measure of competition The simplest form of the PPA uses a binary definition of competition for light by assigning each stem as either being in the overstory (and receiving full sunlight) or in the understory, with two corresponding sets of growth and mortality rates per species. In this form, the PPA has been shown to be able to reproduce species composition, canopy structure, forest dynamics and succession in the US lake states (Purves et al., 2007, 2008). Subsequent applications of the PPA model have utilised the crown area of the plot as a continuous measure of competition, using either total crown area and an individual tree's crown position in multiple understory layers (Bohlman and Pacala, 2011), or a metric CAIh: the total crown area of trees taller than a given height h as a proportion of total area (Caspersen et al., 2011; Coomes et al., 2012). Although previous studies have varied the height at which CAIh is calculated for a given tree, either at the midpoint of the crown (Caspersen et al., 2011) or at both the top of the crown and Chapter 6 Simulation model 94 several lower points (Coomes et al., 2012), we had no information on crown depth for these data so chose to calculate CAIh from the top of the tree. The metric CAIh is defined for a given height h as: (Eqn 6.1) where CAh,i is the projected crown area of crown i at height h, N is the total number of trees in the stand and A is the ground area of the stand, for which we used 1 ha (Caspersen et al., 2011). For a tree of height h the metric CAIh is therefore the projected crown area of all taller trees as a proportion of the total crown area. A tree with a CAIh value of at least 1 would be completely in the understory, as the crown area of taller trees would fill at least the area of the stand above it, whereas a value less than one would imply that some of its crown has exposure to direct sunlight, proportional to the area of canopy filled above it. Full details of the fitting of CAIh are given in Appendix C (page 141). 6.2.2 Demographic rates and allometric equations The simulation model depends on four subroutines for recruitment, growth, allometry (tree height and crown width) and mortality. These processes were parameterised using data from the second and third Spanish forest inventory (IFN2 and IFN3; MMA, 1996, 2007), which surveyed all forested land in mainland Spain on a 1 km 2 grid approximately 10 years apart. We used Bayesian methods and tested many different model functional forms before selecting the one that best fit the data (see Appendix D page 157 for details). All processes were fitted with species-specific parameters. For the recruitment model we simply used the functions fitted in Chapter Five, which predicted the total number of new 1 cm diameter breast height (DBH) stems per hectare per year as a function of conspecific adult crown area (CAIsp, calculated as the sum of all crown areas of conspecific stems >7.5 cm DBH), and total plot crow area (CAIpl, calculated as the sum of all crown areas of conspecific stems >7.5 cm DBH), as: Recruitment (#1 cm DBH stems /ha/year) = (Eqn 6.2) where φ0 and φ1 are estimated parameters. This function describes recruitment as increasing with conspecific crown area, but being negatively affected by aboveground competition for light. Annual growth rate (stem diameter increase, cm year -1 ) was fitted as a power function of stem size (DBH) with an exponential decline in growth for large stems, with additional dependencies on drought length (DL), annual precipitation (PA) and average annual temperature (AVT) as well as the competition measure CAIh, using a size dependent standard deviation: (Eqn 6.3) where α, β and ρ0-ρ9 are parameters and α is dependent on climate and competition as: (Eqn 6.4) This equation describes growth as increasing initially with stem size but a possible decline in growth for large trees (depending on the value of ρ9), and decreasing with asymmetric competition (CAIh). Chapter 6 Simulation model 95 The climate dependency of growth is not pre-determined, since parameters ρ6- ρ8 may take either positive or negative values. Annual mortality rate (stems stem -1 year -1 ) was fitted using a logistic function (similar to the method in Chapter Two), with the same predictors as the growth function. The best fit function was: (Eqn 6.5) where k is dependent on stem size, competition and climate as: (Eqn 6.6) where τ0- τ6 are estimated parameters. This equation describes mortality as decreasing initially with stem size but the functional form also allows a possible increase in mortality in larger trees (depending on the value of τ2). The climate and competitive dependency of mortality is not pre- determined, since parameters τ3- τ6 may take either positive or negative values Growth and mortality functions were strongly size dependent but were developed from trees > 7.5 cm DBH in the inventory, which does not provide individual-level information on smaller trees. To avoid extrapolating these rates to small sizes and producing unreasonable results, for trees smaller than 7.5 cm we used the growth and mortality functions derived for juveniles in Chapter Five. Allometry, tree height and crown diameter, were also modelled as functions of climate. Tree height was fitted as a power function of stem size (DBH), using a simplified version of the model in Chapter Four, using a size dependent standard deviation: (Eqn 6.7) where ϕ0- ϕ7 are estimated parameters. Crown diameter (used for calculating CAIh) and CAIh were modelled exactly as in Chapter Five, again with a size dependent standard deviation: (Eqn 6.8) where ν0- ν6 are estimated parameters. Full details of model fitting, model selection and a comparison of model predictions and observations, for all processes, are given in Appendix D, tables D.1-D.6 and figures D.1-D.3 (Appendix D page 157). 6.2.3 Predictions of successional dominance The original form of the PPA model makes analytical predictions for the early and late successional dominant species under simplifying canopy structure assumptions and assuming just two constant species specific growth and mortality rates, one for individuals in the understory and one for canopy trees (Purves et al., 2008). The predicted early successional dominant is the species which is able to grow to the tallest height in a completely open stand (i.e. in full light) and which has the largest share of the canopy at the earliest time of canopy closure. Purves et al. (2008) predict this to be the same species under the assumption of low interspecific differences in allometry and mortality rates in full light, and so the early successional species may be determined by a simple equation relating height and growth rates in light. As we used a continuous definition of the effect of asymmetric competition Chapter 6 Simulation model 96 on growth and mortality rates (in the form of the metric CAIh, in contrast to the discrete canopy/understory) we could not apply the analytical equations of Purves et al. (2008) to predict early succession in Spain. Instead we used the simulation model to predict both the tallest species in the plot and the species with the largest share of the canopy after 30 years’ growth in full light. The original PPA model's predictions for the late-successional dominant species in a given area is the species with the tallest canopy closure in an equilibrium monoculture, that is, the tallest species which is able to regenerate under its own canopy to replace lost adults and maintain a closed canopy forest. However, we chose to test the predictions of our model for late successional dominance by simulating dynamics for each plot with all species found in that plot. We set each species with the same initial density as before and simulated dynamics for 500 years, recording the dominant species (the one with the largest share of the canopy) every 100 years. 6.2.4 Simulation steps The model creates and follows cohorts of stems the same size of each species within a stand throughout their lives. At each one-year time step it creates cohorts of 1 cm DBH stems, the density of which (in stems/ha) is determined for each species by the crown area of large (>7.5 cm DBH) conspecifics and the crown area of all large trees within the stand, using equation 6.2. For each time step the simulation model then applies the species-specific growth function (equation 6.3) to each cohort’s DBH, and reduces the density of each cohort according to the species-specific mortality function (equation 6.5). The simulator then recalculates height and crown width for each individual in each cohort (using species-specific equations 6.7-6.8), and its CAIh value as well as the total crown area of large trees for each species and for the whole plot according to each cohort’s density and si e. In order to prevent stems growing to a size unobserved in the data we restricted each stem to grow to no more than 10% larger than the largest stem observed in the inventory for that species (see Table 6.1). The inventory contained too few large trees to detect a senescence effect on mortality rates, despite this being found in many other systems (e.g. the Eastern US, Chapter Two; and forests in sub-boreal Umeki, 2002, temperate, Buchman et al., 1983; Monserud and Sterba, 1999; Coomes and Allen, 2007b or tropical, Chao et al., 2008). To avoid unrealistically large long-lived individuals surviving in the stand we increased the mortality rates of stems larger than the largest observed for their species by 25%. To aid computational speed we also removed cohorts of very low density (<0.1 stems/ha) from the simulation model at each time step. We simulated dynamics in forest stands on a grid of cells of varying climate to reflect the observed climate in Spain. We simulated stands with varying average annual temperature (in steps of 0.2°C) and annual precipitation values (in steps of 20 mm) across the range found in Spain, at drought length values of 0.5, 1.5, 2.5, 3.5, 4.5 and 5.5 months drought (5783 gridcells in total). To avoid extrapolating rates far outside of the species ranges we calculated species’ climatic ranges using the inventory data (see Fig. 6.1, Table 6.1), and only initialised plots with species found within those Chapter 6 Simulation model 97 climatic conditions (within 0.1°C temperature, 10 mm precipitation, 1 months drought), and only simulated for plots where more than one species was found (so removing plots where the predicted dominant species would be pre determined, see Table 6.1, Fig. 6.1). To predict early successional dominance we initialised plots with a cohort of 500 1 cm DBH stems per hectare of each chosen species and applied the growth, allometry and mortality functions as described above for 30 years, but assumed no effect of competition for light (i.e. CAIh=0 for all cohorts during the simulation), and no additional recruitment into the stand. To predict long-term successional dynamics we ran the simulator for 500 years with 500 initial 1 cm DBH stems for each species in each gridcell, and to provide a continuous size distribution in the early stages of the model we added an extra 100 1 cm DBH stems of each species into the stand in each of the first 10 years. Table 6.1 The 14 species in the analysis, the maximum DBH of that species observed within the data and the upper limit drought length the species was simulated in (all were simulated in no drought). Species Maximum DBH (cm) Maximum simulated drought length (months) Pinus sylvestris 44.9 1.99 Pinus uncinata 43.9 0.10 Pinus pinea 46.1 5.03 Pinus halepensis 33.8 5.00 Pinus nigra 40.6 2.69 Pinus pinaster 44.2 3.73 Quercus robur 55.1 2.00 Quercus petraea 55.0 1.05 Quercus pyrenaica 52.9 3.04 Quercus faginea 52.6 3.46 Quercus ilex 60.5 4.35 Quercus suber 61.9 4.20 Fagus sylvatica 54.7 1.05 Castanea sativa 83.3 3.27 Pinus sylvestris 44.9 1.99 6.2.5 Observed species dominance In IFN3 the dominant species was recorded in the field for each inventory plot, and we used this to test the model's predictions of early and late successional dominance. We selected plots from the data for which one of the 14 species in the study was recorded as dominant. We split the data by climate to create gridcells and determined average dominance within each gridcell as the species most often recorded as dominant in the inventory plots whose climatic range lay in the range of the gridcell. We Chapter 6 Simulation model 98 split the data by drought length in to six groups (0-1, 1-2, 2-3, 3-4, 4-5 and 5+ months drought), by annual precipitation on a 20 mm grid and by average annual temperature on a 0.2°C grid. 6.3 Results 6.3.1 Models of growth and mortality Growth and mortality rates varied strongly with species, size, competition and climate, but the effects of size and competition were strongest (Fig. 6.2). Most species showed a strong increase in growth with size at small size, but for most species growth rates also slowed or declined in larger sizes. Mortality was also responsive to size, with almost all species showing declining rates from small to medium sized trees, with most species' mortality levelling around 20-40 cm DBH, although the mortality rates of Q. suber and F. sylvatica also rose in larger sized trees. Both growth and mortality were affected by increases in asymmetric competition (CAIh), with growth of all species declining sharply in more competitive environments. Mortality rates also increased with competition, but there were much larger differences among species, with the Mediterranean angiosperm species appearing to suffer the least from increased competition. The effect of drought length, temperature and precipitation on growth and mortality were less consistent among species, with some species' rates responding positively and others negatively to changes in all three variables (see Figs. D.1 and D.2, Appendix D page 157) Figure 6.1 Observed 95% climatic ranges of each species taken from all occurrences within the inventory data. Species were only simulated within the climatic range in which they were found in the data (P.sy = Pinus sylvestris, P. un=P. uncinata, P. pa=P. pinea, P. ha=P. halepensis, P. ni=P. nigra, P. ps=P. pinaster, Q. ro=Quercus robur, Q. pe=Q. petraea, Q. py=Q. pyrenaica, Q. fa=Q. faginea, Q. il=Q. ilex, Q. su=Q. suber, F.sy=Fagus sylvatica, C.sa=Castanea sativa). 5 10 15 20 500 1000 1500 2000 Pine species' climatic ranges P.sy P.un P.pa P.ha P.ni P.ps 5 10 15 20 500 1000 1500 2000 Angiosperm species' climatic ranges Q.ro Q.pe Q.py Q.fa Q.il Q.su F.sy C.sa Average Annual Temperature C A n n u a l P re c ip it a ti o n ( m m ) Chapter 6 Simulation model 99 Figure 6.2 Average predicted annual growth and mortality rates of the 14 species along ranges of size and competition. Predictions were calculated along the gradient of each predictor (DBH or CAIh) by varying only that predictor and holding all others at the species' average (see Table D.1 for average values), and then binning the data along the gradient to give the average. Species shown are Mediterranean pines (Pinus pinea, P. halepensis, P. nigra, P. pinaster), temperate pines (P. sylvestris, P. uncinata), Mediterranean angiosperms (Quercus faginea, Q. ilex, Q. suber) and temperate angiosperms (Q. robur, Q. petraea, Q. pyrenaica, Fagus sylvatica, Castanea sativa). 6.3.2 Observed and predicted pine dominance The observed patterns of pine species' dominance (Fig. 6.3) were strongly determined by climate, with the mountain pine species (P. uncinata and P. sylvestris, and to a lesser extend P. nigra) dominating only in the cooler and wetter areas with the shortest drought, P. pinaster dominating in the warmer and wetter regions and P. nigra and P. pinea dominating in the warm areas with lower rainfall. In the very driest areas P. halepensis dominated (Fig. 6.4). For each climatic gridcell we predicted what the early successional dominant species would be using two approaches- the tallest species after 30 years of stand development and the species with the largest share of the canopy area after 30 years. The two predictions were the same for only around 40% of the 5783 gridcells simulated. We found that the model predicted the observed patterns of pine 0.1 0.2 0.3 0.4 0.5 0.6 G ro w th r a te ( c m y r 1 ) 20 40 60 80 0.002 0.005 0.010 0.020 0.050 A n n u a l m o rt a lit y r a te ( s te m s s te m 1 y r 1 ) DBH (cm) 0.0 0.5 1.0 1.5 2.0 CAIh(ha ha 1 ) Med Pine Temp Pine Med Angiosperm Temp Angiosperm Chapter 6 Simulation model 100 dominance well across the climatic gradients (compare Fig 6.3 and Fig 6.4, which shows predictions for dominance according to the species with the largest share of the canopy), but there were some mismatches, specifically the predicted dominance of both P. halepensis and P. pinaster was lower than observed, with P. pinea and P. nigra often predicted in their place, particularly in regions of low and medium drought (compare Fig 6.3 and Fig 6.4). The predicted dominance according to share of the canopy area reproduced the observed pine dominance pattern better than the tallest species method, particularly in the areas of longest drought, where the predicted dominance according to height was solely P. pinea, whereas the predicted dominance according to the canopy share was P. halepensis in the cooler areas and P. pinea in the warmer and wetter areas (compare Fig. 6.4 and Fig. D.4, Appendix D page 157). We analysed the mechanism of changes in dominance within the model for an example change in predicted pine dominance along a temperature gradient by examining the intraspecific variation in the component rates (growth, survival and crown diameter) along with changes in temperature (at drought length = 1.5 months, precipitation = 700 mm/year, Fig. 6.5). Along the temperature gradient in this region we found that P. sylvestris dominated in the coolest areas, P. nigra dominated in the middle of the temperature range and P. pinea dominated in the warmest areas. P. sylvestris dominated almost everywhere it was found because of its high growth and survival rates, and large crown diameter. However, in the cooler areas there was concurrent dominance of P. nigra, despite its low and declining growth rate, due to its extremely high survival rate. P. pinea had a low survival rate in the middle of the temperature range but this increased substantially and, despite a simultaneous decline in growth rates, led to dominance in the warmest area. There were small areas of P. halepensis dominance in the warmer part of the range, likely due to its low mortality rate, but despite its larger crown area it by and large lost out to both P. pinea (which had higher growth rates) and P. nigra (which had higher survival rates). Despite it occurring within this climatic region, P. pinaster was unable to dominate due to its low growth and high mortality rates. Chapter 6 Simulation model 101 Figure 6.3 Observed climatic range of dominance of the six pine species studied as recorded in the inventory. Each square is a cell of 0.1°C temperature ×10 mm precipitation grid cell, representing the most often dominant species in the dataset in that range and in the stated range of drought length. 0 500 1000 1500 2000 2500 <1 month drought 1-2 months drought 0 500 1000 1500 2000 2500 2-3 months drought 3-4 months drought 0 5 10 15 20 0 500 1000 1500 2000 2500 4-5 months drought 0 5 10 15 20 > 5 months drought Average Annual Temperature C A n n u a l P re c ip it a ti o n ( m m ) P.sylvestris P.uncinata P.pinea P.halepensis P.nigra P.pinaster Chapter 6 Simulation model 102 Figure 6.4 Predicted range of dominance of the six pine species studied determined as the species with the largest canopy area after 30 years of growth under full light, split by drought length and plotted along gradients of average annual temperature and annual precipitation. 0 500 1000 1500 2000 2500 0.5 months drought 1.5 months drought 0 500 1000 1500 2000 2500 2.5 months drought 3.5 months drought 0 5 10 15 20 0 500 1000 1500 2000 2500 4.5 months drought 0 5 10 15 20 5.5 months drought Average Annual Temperature C A n n u a l P re c ip it a ti o n ( m m ) P.sylvestris P.uncinata P.pinea P.halepensis P.nigra P.pinaster Chapter 6 Simulation model 103 Figure 6.5 Predicted growth, survival and crown diameter of pine species (predicted for DBH=10 cm, CAIh=0.5 ha/ha, drought length = 1.5 months and annual precipitation = 700 mm/year) plotted against average annual temperature, with each species shown in the range in which it was simulated in the model. Horizontal bars show the range (thin line) predicted dominance (thick line) of pine species in along average annual temperature, which overlap because they represent dominances within an annual precipitation range of 600-800 mm/year. Pinus pinaster was not predicted to be dominant in this range. Dotted lines show the edges of the ranges of dominance. Chapter 6 Simulation model 104 6.3.3 Observed and predicted angiosperm dominance Both the ranges of occurrence and dominance of angiosperm species' ranges were less segregated and less well defined by climatic variation than pines (Fig. 6.1, and compare Figs. 6.3 and 6.6). There was little pattern of dominance in the areas of shortest drought length, whereas in parts with higher rainfall F. sylvatica, Q. robur, Q. petraea and Q. pyrenaica all dominated. In areas of intermediate drought, Q. pyrenaica and a few stands of C. sativa dominated where rainfall was higher, being replaced by Q. suber in the warmest parts. In the areas of longest drought, Q. ilex dominated most stands, with Q. suber dominating a small number of stands with higher rainfall. We predicted species compositional dynamics from an initially completely bare stand over 500 years using the full climate and competitive-dependent model and examined the predicted transition in species' dominance (determined as largest share of the canopy) from pines to angiosperm species at 100 year intervals (Fig. 6.7). After 100 years of dynamics we found that most stands were still dominated by pine species, but large parts of the cooler areas of least drought were dominated by F. sylvatica. By 200-300 years the model predicted much higher levels of angiosperm dominance across the region, but in the areas of longer drought there were still large areas dominated by pine species. In areas of short drought (0-2 months) there was reduced dominance by F. slyvatica but increased dominance by deciduous species such as Q. robur and C. sativa in the warmest and wettest areas, and some dominance of Q. pyrenaica and Q. petraea in the cooler areas. In the warmest areas with lowest rainfall there was some dominance by the sclerophyllous Q. ilex and the semi-deciduous Q. faginea. In areas of medium drought (2-4) there was higher dominance by Q. robur, Q. pyrenaica and C. sativa, and dominance by both sclerophyllous oaks (Q. suber and Q. ilex) in the warmest and driest regions. In the very driest parts (4+ months of drought) only the two sclerophyllous oaks were found, with Q. suber in slightly wetter parts, Q. ilex in hottest and very driest areas and many pine dominated stands in the areas of least rainfall. After 400-500 years of the model most stands were dominated by angiosperm species, and in the areas of shortest drought (0-2 months) most species distributions had not changed substantially. However, both Q. pyrenaica and Q. faginea were no longer dominant in any area. Q. suber still dominated some stands in the hottest areas but in areas of intermediate and long drought (2+ months) it had been outcompeted, primarily by Q. ilex. In these areas there were also still many stands dominated by pine species (almost exclusively P. halepensis, data not shown), particularly in areas with the very lowest rainfall. It is difficult to directly compare the predictions of the model with the observed angiosperm dominance, since the inventory data does not give clear climatic patterns of dominance. This was especially evident in the areas of shortest drought, where both the model and the data show dominance of many different species without clear climatic differentiation. However, there were some Chapter 6 Simulation model 105 consistent patterns, such as the dominance of F. sylvatica in the coldest and wettest areas (compare Fig. 6.6 and Fig. 6.7) and the dominance of Q. ilex and Q. suber in the hottest areas with least rainfall. Figure 6.6 Observed climatic range of dominance of the eight angiosperm species studied as recorded in the inventory. Each square is a cell of 0.1°C temperature ×10 mm precipitation grid cell, representing the most often dominant species in the dataset in that range and in the stated range of drought length. 0 500 1000 1500 2000 2500 <1 month drought 1-2 months drought 0 500 1000 1500 2000 2500 2-3 months drought 3-4 months drought 0 5 10 15 20 0 500 1000 1500 2000 2500 4-5 months drought 0 5 10 15 20 > 5 months drought Average Annual Temperature C A n n u a l P re c ip it a ti o n ( m m ) Q.robur Q.petraea Q.pyrenaica Q.faginea Q.ilex Q.suber F.sylvatica C.sativa Chapter 6 Simulation model 106 6.4 Discussion 6.4.1 Growth and mortality rates varied with size, climate and competition We found that there was strong interspecific and intraspecific variation in growth and mortality. Growth for most species rose and then fell with stem size (DBH), but there were exceptions. Growth declined with all sizes in the inventory data (stems > 7.5 cm DBH) for P. uncinata and Q. ilex, and only increased with size for F. sylvatica. The size-related decline in growth has been found in many parts of the world (e.g. Mencuccini and Grace, 1996; Binkley et al., 2002; Coomes et al., 2012) and may be caused by the impact on large trees of nutrient or hydraulic limitations, increased respiratory costs, disease, herbivory or increased allocation of resources to reproduction (Zeide, 1993; Ryan and Yoder, 1997; Weiner and Thomas, 2001). Mortality rates of most species declined with stem size but two species, P. uncinata and Q. petraea showed almost no change in mortality with size. The minimum mortality rate occurred between 20 - 40 cm DBH for most species, and only two, Q. suber and F. sylvatica, showed a marked increase in mortality rates at large stem size. U-shaped size dependence of mortality, with high mortality in larger sizes has been found in many parts of the world (e.g. the Eastern US, Chapter Two; Buchman et al., 1983; Monserud and Sterba, 1999; Umeki, 2002; Coomes and Allen, 2007b; Chao et al., 2008), so we were surprised that this pattern was not more widespread in our data. Ninety per cent of the trees in the inventory were less than 40 cm DBH, and management practices may be removing large trees from the landscape in Spain before senescence effects occur in large trees (we filtered the data to remove stem loss due to management). Increases in asymmetric competition caused both decreases in growth rate and increases in mortality for all species, but there were more interspecific differences in the effect on mortality rates, with changes in rank mortality rates with increasing competition (Fig. 6.2). Growth and mortality rates for all species showed strong intraspecific variation along climatic gradients (Figs. D.1 and D.2, pages 167-168), but there were many differences among species and changes in ranks of rates across climatic gradients. Studies of growth rates along temperature, drought and precipitation gradients are common (e.g. Wickramasinghe, 1988; Corcuera et al., 2004; Coomes and Allen, 2007a; Martínez-Vilalta et al., 2011), but large-scale assessments of climate-driven intraspecific variation in mortality are more rare, though these studies typically also show strong variation in rates (van Mantgem and Stephenson, 2007; Adams et al., 2009). Strong intraspecific variation in performance along climatic gradients implies that both growth and mortality rates are important in determining species dominance, and are likely to be primary drivers of competitive interactions and species composition. This is in contrast to our findings for intraspecific variation in allometry (Chapter Four) where we found little variation in rank in allometric scaling along the same climatic gradients. Chapter 6 Simulation model 107 6.4.2 Early successional dominance was determined by both growth and mortality The patterns of observed dominance were better captured by predicting species dominance as the species with the largest share of the canopy rather than the tallest species in a region after 30 years of growth in the open. By examining the component parts of the model along an example temperature gradient we found that some shifts in canopy share dominance were due to differences in growth rates and others due to differences in mortality rates, but none appeared to be primarily driven by differences in crown area (Fig. 6.5). This implies that intraspecific variation in both growth and mortality rates along climatic gradients are important for determining shifts in the early successional dominant species, which was not simply the one with the fastest growth in full light. The importance of both growth and mortality in determining regeneration success have been observed in New Zealand forests (Kunstler et al., 2009). Here, we found that whilst a fast-growing and tall statured early successional species may be found in a particular region, an unfavourable climate may mean that too few individuals are able to survive to adult height to dominate the canopy. 6.4.3 Mismatches in pine dominance are likely due to lack of fire dynamics The most striking mismatch between the early successional predicted species and the observed dominance of pine species was the reduced dominance of both P. halepensis and P. pinaster in our predictions. Although our model captured some dominance by both species in the warmer and drier areas, in many places where they were observed to be dominant our model wrongly predicted dominance by P. nigra and P. pinea (compare Fig. 6.3 and Fig. 6.4). We suggest that this is a result of our model not simulating the effects of fire on species dominance. Both P. pinaster and P. halepensis is a highly flammable serotinous species which is adapted to reproduce quickly after intense fires by reproducing at a young age and having serotinous cones which allow rapid seed dispersal and regeneration post fire (Tapias et al., 2004; Calvo et al., 2008). P. nigra and P. pinea are not serotinous and are unable to survive intense fires or quickly regenerate after them, although both have insulating bark which protects against low intensity fire damage (Rigolot, 2004; Pausas et al., 2008). Given the high frequency of fires in the hot and dry parts Spain, P. halepensis is therefore able to dominant in areas which, without fire, might be dominated by less fire-tolerant species (Agee, 1998). Both P. halepensis and P. pinaster can burn and re-establish a stand on a cycle of as little as 15 years (typically the cycle is 20-30 years). By promoting regular fires they are able to prevent succession by shade-tolerant angiosperm species indefinitely (Grove and Rackham, 2001), particularly in the driest areas where oaks are at the edge of their distributional limit, but this depends on both the intensity and the frequency of the fire regime (Zavala et al., 2000). In order to include these effects it would not be enough to simply simulate a frequent disturbance regime; we would also require detailed data on species-specific post-fire establishment and regeneration rates from recently burned stands. Chapter 6 Simulation model 108 6.4.4 Late successional dominance dynamics The predicted late successional dominance distribution did not show the clear climatic differentiation found for the pine species, but the angiosperm species in the data also occupied much more similar observed climatic ranges than the pine species (Table 6.1 and Fig. 6.1). The model was able to predict a transition from pine to angiosperm species across the region but the time taken for this to happen varied with climate (Fig. 6.7), with slower dynamics in areas of longer drought. Some species showed much higher dominance over an intermediate time frame (e.g. dominance of Q. faginea and Q. pyrenaica after 200-300 years) but had disappeared after 500 years, suggesting that these may be recruitment limited under their own canopy. We also saw a rise and fall in dominance of Q. suber in the driest areas, where it was eventually outcompeted by Q. ilex, which may be a result of a lack of fire simulation in our model. Q. suber relies on fire to maintain dominance (Grove and Rackham, 2001), and in areas with low fire frequency Q. ilex is known to regenerate better (Curt et al., 2009). We also found that pine species, and in particular P. halepensis, continued to dominate in the areas of very low rainfall even after 500 years. The relative abundance of oaks and pines is known to be strongly related to water availability and topography (Zavala et al., 2000), and although mixed stands are found in areas with intermediate droughts, pines form monospecific stands in the driest regions. P. halepensis plantations are known to survive in areas with precipitation as low as 300 mm/year whereas Q. ilex is found only sparsely in areas of less than 400 mm/year (Rodà, 1999). 6.4.5 Conclusions: climate-driven simulation models at a regional scale In this study we have demonstrated the straightforward way in which simple forest simulators may be parameterised to include climatic dependency of processes given appropriate inventory data. All processes within forest dynamics models are known to vary strongly with climate at regional scales and large national inventories, such as the one used here, provide the opportunity to study multiple species' dominance and succession across a wide variety of environmental conditions. We believe that models such as the one presented in this chapter could be used as tools to predict changes in species composition with climate change at regional scales. However, we recognise that constraining the occurrence of species to within the range in which they were found in this study (Fig. 6.1) is unsatisfactory, and suggest that incorporating physiological-based climatic-dependency of growth and mortality could allow rates to be extrapolated outside of the range in which they were parameterised, to give new insight into forest dynamics in novel climates. Despite this drawback, this study represents, to our knowledge, the first individual-based forest simulator incorporating both species-specific ecological and climatic drivers to reproduce regional patterns of species dominance. This approach has the potential to generate ecologically reasonable predictions of changes to forest structure, productivity, dynamics and species composition for many parts of the world and is an important step towards a better understanding of the mechanisms driving observed climate-forest type patterns. Chapter 6 Simulation model 109 Figure 6.7 Predicted range of dominance of the angiosperm species over 500 years of succession. Panels show succession after 100, 200, 300, 400 and 500 years, showing the gradual replacement of pine dominance (light green) by angiosperm dominance. Each panel is split by drought length and species' dominances are plotted along gradients of average annual temperature and annual precipitation. 0 500 1000 1500 2000 2500 0.5 months drought 1.5 months drought 0 500 1000 1500 2000 2500 2.5 months drought 3.5 months drought 0 5 10 15 20 0 500 1000 1500 2000 2500 4.5 months drought 0 5 10 15 20 5.5 months drought Average Annual Temperature C A n n u a l P re c ip it a ti o n ( m m ) 100 years Q.robur Q.petraea Q.pyrenaica Q.faginea Q.ilex Q.suber F.sylvatica C.sativa Pines Chapter 6 Simulation model 110 0 500 1000 1500 2000 2500 0.5 months drought 1.5 months drought 0 500 1000 1500 2000 2500 2.5 months drought 3.5 months drought 0 5 10 15 20 0 500 1000 1500 2000 2500 4.5 months drought 0 5 10 15 20 5.5 months drought Average Annual Temperature C A n n u a l P re c ip it a ti o n ( m m ) 200 years Q.robur Q.petraea Q.pyrenaica Q.faginea Q.ilex Q.suber F.sylvatica C.sativa Pines 0 500 1000 1500 2000 2500 0.5 months drought 1.5 months drought 0 500 1000 1500 2000 2500 2.5 months drought 3.5 months drought 0 5 10 15 20 0 500 1000 1500 2000 2500 4.5 months drought 0 5 10 15 20 5.5 months drought Average Annual Temperature C A n n u a l P re c ip it a ti o n ( m m ) 300 years Q.robur Q.petraea Q.pyrenaica Q.faginea Q.ilex Q.suber F.sylvatica C.sativa Pines Chapter 6 Simulation model 111 0 500 1000 1500 2000 2500 0.5 months drought 1.5 months drought 0 500 1000 1500 2000 2500 2.5 months drought 3.5 months drought 0 5 10 15 20 0 500 1000 1500 2000 2500 4.5 months drought 0 5 10 15 20 5.5 months drought Average Annual Temperature C A n n u a l P re c ip it a ti o n ( m m ) 400 years Q.robur Q.petraea Q.pyrenaica Q.faginea Q.ilex Q.suber F.sylvatica C.sativa Pines 0 500 1000 1500 2000 2500 0.5 months drought 1.5 months drought 0 500 1000 1500 2000 2500 2.5 months drought 3.5 months drought 0 5 10 15 20 0 500 1000 1500 2000 2500 4.5 months drought 0 5 10 15 20 5.5 months drought Average Annual Temperature C A n n u a l P re c ip it a ti o n ( m m ) 500 years Q.robur Q.petraea Q.pyrenaica Q.faginea Q.ilex Q.suber F.sylvatica C.sativa Pines Chapter 6 Simulation model 112 Author contributions Designed the study, analysed the data and performed statistical analysis: Emily Lines with support from David Coomes. Provided the data: Miguel Zavala. Wrote the simulation model code: Emily Lines and Drew Purves. Wrote the paper: Emily Lines with supervisory support from David Coomes. Chapter 7 Discussion 113 7 Discussion Analysing each of the four components of a typical forest dynamics model, (growth, mortality, allometry and recruitment) individually allows the investigation of the inter- and intra-specific differences in rates, which reveals the potential relative abilities of species to adapt to changes in climatic conditions. In this thesis I have shown that the examination of the regional-scale climatic- dependency of aboveground allometric functions and demographic rates parameterised from inventory data can give insight into intraspecific variation in species performance throughout their ranges. I have also demonstrated how the combination of these processes in a forest simulator can reveal the causes of observed shifts in the ability of species to dominant the forest across the landscape. The geographical ranges of plant functional types on a continental scale are known to correlate with climate (Holdridge, 1947) and understanding the causes of observed ranges is a key question in ecological research and an important component towards understanding the possible impacts of climate change on species distributions and biodiversity (Guisan and Thuiller, 2005; Morin et al., 2007). However, empirical studies of the determinants of range shifts are fragmented and lag behind theoretical explanations (Gaston, 2009). Process-based models of species’ distributions have shown promise in explaining the limitations of species performance at the edge of their ranges, whether due to climatic stresses on phenological, reproductive or survival processes (e.g. Morin et al., 2007, 2008). However, these models typically ignore competitive effects, although there is evidence to suggest the presence of competitors can alter species’ responses to climatic variation (Leathwick and Austin, 2001). Climate is of course not the only factor affecting tree demographic processes and in this thesis I have found stem size and competitive environment to be the major determinants of variation in aboveground allometry and growth and mortality rates at the tree-level. 7.1 Tree size and competition as primary determinants of individual-level rates Stem size was the single most important determinant of growth in Spain, and of mortality in both Spain (Chapter Six) and the Eastern US (Chapter Two). At small stem sizes I observed low growth and high mortality rates, as has been shown in many other studies (e.g. Bormann, 1965; Kobe et al., 1995; Kunstler et al., 2009), but I also found strong evidence from many species for declining growth Chapter 7 Discussion 114 and increasing mortality for the largest trees (giving a U-shaped size-dependent mortality; Harcombe, 1987). Declines in stem growth rate at larger sizes may be caused by internal factors such as limitations in hydraulic flow, increases in respiratory costs, self-shading leading to a decline in photosynthetic activity or increased allocation of resources to reproduction or external forces such as herbivory and disease, and the costs in protecting against these (Zeide, 1993; Ryan and Yoder, 1997; Weiner and Thomas, 2001; Meinzer et al., 2011). The size-related decline in individual tree growth has been found in many parts of the world (e.g. Mencuccini and Grace, 1996; Binkley et al., 2002; Coomes et al., 2012), and is a factor in age-related decline of whole-stand productivity that has been observed in many studies (e.g. Ryan et al., 1997). Asymmetric competition (competition from taller trees) was found to be a determinant of both aboveground allometry and stem growth in Spain (Chapters Four and Six). Trees with more large neighbours grew taller and thinner and had narrower crowns than those in open areas, but had reduced stem growth rates when asymmetric competition was stronger. The plasticity of aboveground allometric to conditions means that trees are able to allocate resources to height growth means and so faster overtop their neighbours to reach the canopy and maximise light capture (Berntson and Wayne, 2000; Bragg, 2001; King et al., 2009). Increases in mortality with size in larger trees has been observed from field measurements in many parts of the world (Buchman et al., 1983; Monserud and Sterba, 1999; Umeki, 2002; Coomes and Allen, 2007b; Caspersen et al., 2011), and I found increases in mortality in large sizes in all species in the Eastern US but very few in Spain. Increased mortality in large trees is likely to be due to a combination of factors including the higher exposure of canopy trees to strong winds and disturbance, disease and senescence (Busing and Pauley, 1994; Yang et al., 2003; Coomes and Allen, 2007b). I found that competition was an important determinant of mortality rates in both Spain and the Eastern US, but that the effects in the two regions were different. I tested the effect of asymmetric competition on mortality rates in Spain and found that higher asymmetric competition increased mortality for almost all trees, and that its effect was stronger than that of symmetric competition, which has also been found in Norwegian forests (Eid and Tuhus, 2001). The mechanism for the negative effect of taller canopy cover is likely to be primarily the strong asymmetry of competition for light (Weiner, 1990), causing a reduction in performance in shade for many species. The magnitude of the effect however may depend on the species identities of the canopy and understory trees (He and Duncan, 2000) and the size of the suppressed tree (Kunstler et al., 2009). A more detailed study of Spanish forest mortality rates has revealed an interaction between competition and precipitation, so that in wetter sites where less light reaches the forest floor and competition for light is more intense (Coomes and Grubb, 2000), this increased asymmetric competition has an even larger impact on mortality rates (Ruiz-Benito et al. in review). Chapter 7 Discussion 115 In the Eastern US I tested the effect of symmetric competition (total plot basal area) rather than asymmetric competition, and found that it had a broadly hump-shaped relationship with mortality, with low mortality in both the lowest and highest basal area plots, although there was strong interspecific differences, with some species showing no change in mortality rates with total plot basal area. Low mortality in the least dense stands is likely to be a result of lack of competition for resources such as water and light. However, the pattern seen for several species of lower mortality in areas with high basal area is less easy to explain. It may be a result of an unmodelled interaction between competition and resources, in that the most dense forests are likely to occur on sites with high levels of soil moisture and nutrients where overall tree performance could be higher, and could also caused by inter-specific differences in response to competition. A study of the effects of plot density on mortality in Canadian forests found that the response varied between species and not all showed increases in mortality in denser plots (Yao et al., 2001). A possible future avenue of research would be to repeat this analysis using asymmetric rather than symmetric competition as a predictor, and to include interactions between competition and climate. The effects of strong asymmetric competition, found in all these rates, are in contrast to metabolic scaling theory (West et al., 1997, 1999, 2009), which predicts whole-forest dynamics from metabolic rates and ignores asymmetric competition for light (Muller-Landau et al., 2006). The critical importance of including asymmetric competition for light in understanding whole forest dynamics has been highlighted in many studies (e.g. Muller-Landau et al., 2006; Coomes and Allen, 2007b; Coomes et al., 2011). I suggest that any forest simulator concerned with species dominance at a large scale will be able to reproduce observations more accurately with the inclusion of asymmetric competition in growth, mortality and allometry, but future research could test this by quantifying the improvement in predictive ability between simulators with symmetric and asymmetric competition. 7.2 Climate dependency of tree-level processes and inventory data For the data tested in this study, models describing growth, mortality and allometry at regional scales were all improved by the inclusion of climatic dependency. Variation in climate and the influence of this on vegetation type and structure is very strong across Spain (Grove and Rackham, 2001), and this was evident in the climate dependency of individual-tree level processes I parameterised (Chapters Four and Six). Including climatic dependency along with the more commonly modelled competition- dependency of tree-level forest processes enabled the simulation model I parameterised for Spain to reproduce regional patterns of species dominance, and gave insight into the drivers behind shifts in dominance along climatic gradients (Chapter Six). Including a simple competition-dependent recruitment model (Chapter Five) meant that the simulator was able to reproduce dynamics of Chapter 7 Discussion 116 succession in Spain; the commonly observed shift from dominance by Pinus species to dominance by angiosperm species in all but the driest areas (Rodà, 1999; Lookingbill and Zavala, 2000; Pausas et al., 2004; Capitanio and Carcaillet, 2008; Santana et al., 2010). Forest dynamics models which simulate individual tree-level processes have been shown to be able to reproduce a wide variety of forest properties such as stand structure, species composition and succession and productivity, and have been used to investigate critical questions for forestry, management, conservation and carbon storage (Pacala et al., 1996; Lindner et al., 1997; Kunstler et al., 2009; Zhang et al., 2009; Bohlman and Pacala, 2011; Caspersen et al., 2011; Coomes et al., 2012). However, such models typically do not include climatic dependency in processes at a regional scale, and are often parameterised from local-scale data. At small scales the lack of climate-dependency does not detract from the ability of the models to reproduce forest dynamics, but if individual tree- based forest simulators are to be used to answer questions on the large-scale effects of climate and land use change on forest dynamics then climate dependencies must be included as drivers of processes. Forest inventory datasets, available for many parts of the world, have the potential to be used for broad-scale forest dynamics models which reproduce species-climate distributions. National forest inventories such as those used in this study typically do not contain detailed information such as soil nutrient levels, seed dispersal, or below ground dynamics which are necessary to parameterise processes for detailed individual based forest simulation models (e.g. SORTIE, Pacala et al., 1996) or the photosynthetic and respiratory measurements required for physiology-based models (e.g. HYBRID, Friend et al., 1993). The inclusion of climate and resource- driven physiological-based limitations to tree performance within existing forest simulation model frameworks has the potential to produce models which are able to both predict accurate rates in novel climatic conditions (extrapolated outside of the current species ranges) and the important ecological processes that current models are able to reproduce. Such an approach has been suggested (Wullschleger et al., 2001; Lichstein et al., 2010) and would be a major step towards the construction of forest models, currently able to reproduce local-scale dynamics, which are robustly predictive on a regional scale (Purves and Pacala, 2008). 7.3 Limitations of the use of forest inventory data for simulation models Although they provide a wealth of information on variation in tree-level processes and forest properties at a large scale, there are important limitations to using forest inventories to parameterise forest models which should be considered when drawing conclusions from such studies on the climatic dependence of forest dynamics. Chapter 7 Discussion 117 Performance of individual trees at a particular site may be strongly influenced by a variety of factors not recorded in inventory data. For example, allocation of resources to defend against and repair the damage done by herbivores reduces those available for growth (Herms and Mattson, 1992), and high levels of herbivory in seedlings have been found to increase mortality rates in future years (Eichhorn et al., 2010). Spatial heterogeneity in microsite quality is a strong determinant of juvenile performance, with understory cover, soil nutrients and canopy gaps all affecting seedling establishment and survival (e.g. Broncano et al., 1998; Kobe, 1999), and interspecific differences in performance in response to site quality influence coexistence and composition dynamics (Latham, 1992; Beckage and Clark, 2003). Large scale disturbances such as fire, earthquakes and hurricanes can alter species composition and forest dynamics (Hickler et al., 2004; Keane et al., 2004; Liedloff and Cook, 2007; Uriarte et al., 2009; Coomes et al., 2012), and forest fires, a common form of disturbance in Spain, are known to influence vegetation composition (Carrión et al., 2001). I found clear mismatches between model predictions and species composition in Spain which I suspect have arisen because of the lack of consideration of fire dynamics in the model. In particular the model fails to predict the widespread dominance of two fire-adapted early successional species (Pinus pinaster and P. halepensis) which are able to reproduce at young age and are serotinous so are able to reestablish from the seed bank quickly following fire (Tapias et al., 2001; Calvo et al., 2008). These species achieve dominance through high fire frequency which reduces the seed bank of competitors over multiple cycles. To accurately describe such a mechanism would require multispecies data on survival and reproduction after fire, and data on fire intensity and frequency. Although forest inventory data are common to many parts of the world, examples of measurements from more than one time period are much rarer. By parameterising models of dynamic processes using just two measurements it is necessary to make a space-for-time assumption. However, this ignores temporal and historical influences on currently observed forest properties. For example, episodes of stress such as disease or severe drought may influence tree performance for many years into the future (Franklin et al., 1987; Cherubini et al., 2002), meaning that poor performance observed in later years may be incorrectly assumed to be the result of current conditions. The observed forest species distribution in Spain cannot be disentangled from its very long history of human use, and forest degradation over the thousands of years of human activity in the region has dramatically changed areas of species dominance (Blondel, 2006). Recent agricultural land abandonment and new EU policy has led to an increase in forest cover since the mid 20 th century (Barbero et al., 1990; Campos et al., 2005), so many forests in the region are not in equilibrium. This increase in forested land has also changed fire regimes in the region (Viedma et al., 2006; Chauchard et al., 2007) and means that the currently observed forest structure and dynamics are likely to have been influenced by different drivers than are currently observed. Chapter 7 Discussion 118 The space for time assumption has another important implication- it means the ecotypic variation is ignored within the dataset, since it is assumed that an individual of a particular species in one location would perform in an identical manner to an individual of the same species in a different location if it were transplanted. However, many studies in the Mediterranean have shown ecotypic variation and the importance of seed provenance in tree performance (e.g. Balaguer et al., 2001; Chambel et al., 2007; Climent et al., 2008). For example, a planting experiment using seeds from multiple provenances found that the survival of Pinus canariensis seedlings in drought has high ecotypic variation, with seedlings from favourable sites (i.e. those with low or no drought) having much higher mortality rates (López et al., 2007). In the context of understanding the impacts of changing climate (particularly under increasing drought conditions) this assumption could lead a correlative model of forest dynamics to underestimate the impact on mortality and consequently incorrectly predict interspecific differences in performance. Although I have been able to show that the approach of building a forest dynamics model using component parts parameterised from inventory data is able to reproduce aspects of observed species dominance and succession, the approach taken in this thesis has been correlative. To predict and quantify the effects of climate change on Mediterranean ecosystems requires a process-based approach which in particular addresses the effects of the likely increases in drought on demographic processes (Keenan et al., 2009). The inclusion of mechanistic processes within ecosystem models will dramatically improve understanding of species distributional limits (Kearney, 2006; Keenan et al., 2011). Approaches which incorporate process-based models of the climatic dependency of demographic rates into individual-based ecological models are therefore desirable both to fully understand the role of climate and competition in driving forest dynamics and species ranges, and to make robust predictions of the future of forests. 7.4 Concluding remarks This thesis has demonstrated the flexibility of Bayesian statistical approaches for parameterising forest dynamics models with multiple drivers using large amounts of data. The flexibility of the approaches presented here, both in testing competing ecological hypotheses and in incorporating different types of data, means that they are invaluable tools for parameterising increasingly complex ecosystem models. I have also shown the wealth of information which may be extracted from national forest inventories to increase understanding of the relative importance and large-scale impacts of competitive and climatic drivers of forest structure. The procedures demonstrated in this thesis are easily applicable to other regions and future research will focus on translating these methods to other forest systems and testing their ability to reproduce species composition and successional dynamics. Chapter 7 Discussion 119 The question of the amount of model detail necessary to accurately reproduce ecosystem function and predict dynamics is one which warrants further research. For example, in my climate- driven forest dynamics model (Chapter Six) I found that climatic dependence of growth and mortality may drive rank changes in species dominance, but allometric scaling appears to only have a secondary role. Simplifying assumptions can dramatically reduce the amount of data required to parameterise forest dynamics models (e.g. Purves et al., 2008), meaning that models can be parameterised from inventory data for many more regions of the world. Models of forest dynamics need only contain as much detail as is sufficient to accurately simulate the properties of interest. Rigorous comparison of ecological hypotheses should help to determine the level of detail required for a model and an understanding of the impacts of simplifying assumptions within a forest model will dramatically improve not only its predictions but also the ease with which it may be transferred to other ecosystems and scaled to larger geographical scales. The work presented in this thesis is an important step towards understanding how the dynamics and distribution of Mediterranean tree species is controlled by climate, and also represents a framework for specifying multi-species, multi-dependency forest models which can be translated to other regions of the world. 120 Appendix A US Mortality 121 Appendix A: Eastern US mortality Table A.1. Table comparing model fits using AIC and BIC. 36 models were run within which the four types of model predictor in equation 2.4 (constant, size, environment, basal area) were left out or included with forest-level (FL) or species specific (SS) effects. Total number of parameters, AIC and BIC scores and rankings are reported. Models without size and species effects were rejected strongly, and the addition of environmental and competition variables increased model fit significantly. The best-fit model, number 26, showed a very significant improvement on the next best using both AIC and BIC. Model Constant Size Environment Basal Area # parameters AIC score AIC rank BIC score BIC rank 1 FL FL - - 3 285046.9 37 -142540.1 37 2 FL SS - - 47 250075.5 26 -125298.2 23 3 SS FL - - 25 250214.4 27 -125245.7 21 4 SS SS - - 69 249993.7 24 -125379.2 24 5 FL FL FL - 13 271511.5 35 -135827.8 34 6 FL SS FL - 57 243591.3 12 -122111.5 8 7 SS FL FL - 35 248196.5 18 -124292.2 17 8 SS SS FL - 79 243336.2 11 -122105.8 7 9 FL FL SS - 233 242284.9 9 -122433.5 9 10 FL SS SS - 277 238863.2 4 -120966.4 4 11 SS FL SS - 255 241014.1 6 -121919.9 6 12 SS SS SS - 299 237455.7 3 -120384.6 2 13 FL FL - FL 5 282448.5 36 -141252.0 36 14 FL SS - FL 49 250312.1 28 -125427.6 25 15 SS FL - FL 27 250028.6 25 -125163.9 20 16 SS SS - FL 71 249753.9 22 -125270.3 22 17 FL FL - SS 49 253033.1 30 -126788.1 30 18 FL SS - SS 93 248935.0 20 -124982.8 19 19 SS FL - SS 71 251300.0 29 -126043.4 29 20 SS SS - SS 115 247667.0 17 -124470.7 18 21 FL FL FL FL 15 270559.9 34 -135363.0 33 22 FL SS FL FL 59 245060.7 14 -122857.3 13 23 SS FL FL FL 37 247087.4 16 -123748.7 16 24 SS SS FL FL 81 245586.3 15 -123242.0 14 25 FL FL SS FL 235 248624.1 19 -125614.1 28 26 FL SS SS FL 279 242284.7 8 -122688.2 11 27 SS FL SS FL 257 244051.9 13 -123449.9 15 28 SS SS SS FL 301 241561.7 7 -122448.6 10 29 FL FL FL SS 59 254674.3 32 -127664.1 32 30 FL SS FL SS 103 249889.8 23 -125515.6 26 31 SS FL FL SS 81 253912.6 31 -127405.1 31 32 SS SS FL SS 125 249701.3 21 -125543.3 27 33 FL FL SS SS 279 242400.0 10 -122745.9 12 34 FL SS SS SS 323 237318.7 2 -120449.0 3 35 SS FL SS SS 301 239960.4 5 -121648.0 5 36 SS SS SS SS 345 236303.6 1 -120063.3 1 Appendix A US Mortality 122 Table A.2 Table of maximum likelihood estimators (MLEs), Bayesian means and 2.5% and 97.5% confidence levels calculated from the posterior distributions for each of the 15 parameters of Eqn (9) for each of the 21 common species parameterised by the adaptive MCMC algorithm. The burn-in for the algorithm was 750,000 iterations and the sampling was 250,000 iterations. Parameter Species (from equation 2.9) Acer rubrum Pinus taeda Quercus alba Liquidambar styraciflua Populus tremuloides Acer saccarum Quercus rubrum Pinus echinata Liriodendron tulipifera Quercus velutina j Mean -3.187 -3.258 -3.256 -3.510 -5.675 -5.374 -3.538 -2.151 -2.994 -3.370 MLE -3.290 -3.261 -3.251 -3.857 -5.580 -5.012 -3.568 -2.164 -3.352 -3.389 2.5% -3.613 -3.540 -3.631 -4.067 -5.960 -5.830 -3.932 -2.647 -3.896 -3.742 97.5% -2.849 -2.953 -2.952 -2.907 -5.470 -4.887 -3.198 -1.732 -2.237 -3.014 1j Mean -0.279 -0.362 -0.308 -0.254 -0.131 -0.319 -0.214 -0.344 -0.290 -0.207 MLE -0.284 -0.366 -0.305 -0.265 -0.135 -0.320 -0.214 -0.347 -0.275 -0.215 2.5% -0.296 -0.376 -0.324 -0.271 -0.146 -0.346 -0.233 -0.368 -0.319 -0.223 97.5% -0.261 -0.349 -0.293 -0.238 -0.116 -0.293 -0.193 -0.319 -0.264 -0.190 2j Mean -0.045 -0.030 -0.028 -0.032 -0.047 -0.039 -0.032 -0.029 -0.026 -0.030 MLE -0.045 -0.030 -0.028 -0.032 -0.047 -0.040 -0.033 -0.029 -0.027 -0.030 2.5% -0.047 -0.031 -0.029 -0.033 -0.050 -0.041 -0.034 -0.031 -0.027 -0.032 97.5% -0.043 -0.029 -0.027 -0.031 -0.045 -0.037 -0.031 -0.028 -0.024 -0.029 3j Mean 0.437 -1.146 0.395 -0.104 0.359 0.320 0.551 -0.185 -0.085 0.939 MLE 0.429 -1.152 0.416 -0.092 0.378 0.325 0.536 -0.139 -0.185 0.879 2.5% 0.378 -1.232 0.280 -0.205 0.257 0.174 0.441 -0.380 -0.279 0.819 97.5% 0.491 -1.054 0.502 -0.009 0.458 0.456 0.660 0.020 0.124 1.046 4j Mean -0.163 0.367 -0.257 0.041 0.351 -0.583 -0.255 -0.147 -0.083 -0.279 MLE -0.143 0.372 -0.242 0.056 0.388 -0.557 -0.264 -0.162 -0.109 -0.295 2.5% -0.203 0.316 -0.310 -0.008 0.227 -0.693 -0.308 -0.220 -0.228 -0.333 97.5% -0.124 0.419 -0.200 0.090 0.470 -0.466 -0.203 -0.081 0.050 -0.229 5j Mean -0.340 0.015 -0.139 0.124 -1.916 -0.501 -0.169 -0.340 0.439 -0.137 MLE -0.344 -0.061 -0.108 0.065 -2.026 -0.468 -0.189 -0.388 0.349 -0.118 2.5% -0.415 -0.161 -0.275 -0.096 -2.185 -0.704 -0.291 -0.590 0.040 -0.279 97.5% -0.270 0.173 0.004 0.338 -1.648 -0.302 -0.040 -0.089 0.804 -0.002 6j Mean 0.087 -0.295 0.060 -0.112 -0.695 -0.265 0.266 0.222 -0.466 0.062 MLE 0.101 -0.354 0.046 -0.047 -0.640 -0.230 0.260 0.142 -0.396 0.073 2.5% 0.045 -0.392 -0.048 -0.233 -0.799 -0.356 0.200 0.100 -0.686 -0.037 97.5% 0.129 -0.192 0.162 0.006 -0.599 -0.179 0.328 0.343 -0.216 0.160 Appendix A US Mortality 123 Parameter Species (from equation 2.9) Acer rubrum Pinus taeda Quercus alba Liquidambar styraciflua Populus tremuloides Acer saccarum Quercus rubrum Pinus echinata Liriodendron tulipifera Quercus velutina 7j Mean 0.625 2.915 -0.134 0.706 -2.849 1.424 0.426 -0.146 2.117 -0.722 MLE 0.635 2.886 -0.019 0.679 -2.878 1.407 0.417 -0.194 2.148 -0.757 2.5% 0.554 2.762 -0.346 0.290 -2.983 1.179 0.202 -0.856 1.610 -0.940 97.5% 0.700 2.994 0.073 1.136 -2.661 1.695 0.673 0.426 2.589 -0.498 8j Mean 0.240 -1.237 -0.066 -0.391 -1.064 1.126 -0.114 -0.142 -1.039 -0.343 MLE 0.249 -1.246 -0.061 -0.503 -1.065 1.154 -0.154 -0.178 -1.128 -0.379 2.5% 0.195 -1.328 -0.209 -0.648 -1.133 0.965 -0.282 -0.522 -1.399 -0.524 97.5% 0.284 -1.132 0.082 -0.150 -0.983 1.272 0.061 0.313 -0.672 -0.166 9j Mean -0.264 -0.276 -0.144 0.013 0.086 0.057 -0.072 -0.108 0.304 -0.699 MLE -0.282 -0.217 -0.141 -0.088 0.076 0.024 -0.053 -0.054 0.111 -0.701 2.5% -0.307 -0.362 -0.261 -0.079 0.028 -0.064 -0.187 -0.261 0.130 -0.817 97.5% -0.219 -0.188 -0.019 0.101 0.138 0.167 0.044 0.040 0.472 -0.587 10j Mean 0.100 -0.417 0.033 0.022 0.003 0.058 0.064 -0.085 0.262 0.190 MLE 0.104 -0.407 0.046 -0.019 0.004 0.048 0.060 -0.082 0.277 0.171 2.5% 0.053 -0.490 -0.016 -0.041 -0.028 -0.002 0.020 -0.156 0.153 0.143 97.5% 0.143 -0.344 0.080 0.086 0.034 0.117 0.107 -0.012 0.377 0.243 11j Mean 0.389 0.336 0.239 -0.209 0.295 -1.650 0.491 0.556 0.159 0.313 MLE 0.291 0.381 0.165 -0.262 0.157 -1.701 0.547 0.535 0.153 0.332 2.5% 0.159 0.188 -0.088 -0.538 0.111 -2.056 0.270 0.216 -0.727 0.071 97.5% 0.588 0.539 0.488 0.227 0.458 -1.335 0.710 0.918 0.818 0.703 12j Mean 0.129 0.092 0.108 0.040 0.104 -0.294 0.128 0.231 0.117 0.048 MLE 0.110 0.098 0.113 0.035 0.075 -0.134 0.147 0.250 0.068 0.109 2.5% 0.088 0.060 0.017 -0.016 0.071 -0.378 0.062 0.110 -0.044 -0.042 97.5% 0.165 0.133 0.178 0.119 0.135 -0.216 0.193 0.337 0.237 0.177 13j Mean 0.006 0.073 0.093 0.013 0.014 0.010 0.101 0.113 0.024 0.124 MLE 0.004 0.071 0.035 0.017 0.015 0.022 0.102 0.112 -0.015 0.106 2.5% -0.003 0.062 0.068 0.001 0.006 -0.006 0.082 0.089 0.003 0.106 97.5% 0.018 0.085 0.115 0.025 0.022 0.030 0.123 0.137 0.055 0.145 14j Mean -0.0002 -0.0007 -0.0016 -0.0001 -0.0002 -0.0003 -0.0021 -0.0015 -0.0006 -0.0022 MLE 0.0000 -0.0007 -0.0010 0.0000 -0.0002 -0.0002 -0.0021 -0.0015 -0.0006 -0.0019 2.5% -0.0004 -0.0009 -0.0019 -0.0003 -0.0004 -0.0006 -0.0025 -0.0019 -0.0011 -0.0026 97.5% -0.0001 -0.0005 -0.0012 0.0001 -0.0001 -0.0001 -0.0017 -0.0011 -0.0003 -0.0019 Appendix A US Mortality 124 Table A.2 continued Parameter Species (from equation 2.9) Quercus prinus Quercus stellata Carya spp Thuja occidentalis Nyssa slyvatica Quercus nigra Betula papyferia Fagus grandifolia Pinus virginiana Fraxinus americana N.sylv. (biflora) j Mean -2.838 -3.810 -2.731 -2.313 -1.502 -2.570 -6.793 -3.634 -2.948 -4.487 -2.881 MLE -3.166 -3.410 -1.994 -2.257 -1.636 -2.496 -6.313 -2.650 -2.865 -4.509 -2.730 2.5% -3.447 -4.162 -3.185 -3.108 -2.502 -3.279 -7.546 -5.030 -3.700 -5.073 -3.474 97.5% -2.114 -3.402 -2.211 -1.626 -0.621 -1.805 -6.233 -2.139 -2.245 -3.858 -2.204 1j Mean -0.390 -0.333 -0.316 -0.226 -0.265 -0.227 -0.103 -0.363 -0.502 -0.251 -0.276 MLE -0.388 -0.335 -0.304 -0.266 -0.247 -0.231 -0.125 -0.359 -0.497 -0.235 -0.263 2.5% -0.421 -0.354 -0.343 -0.277 -0.308 -0.248 -0.138 -0.408 -0.548 -0.279 -0.314 97.5% -0.359 -0.313 -0.291 -0.173 -0.224 -0.206 -0.070 -0.315 -0.454 -0.223 -0.238 2j Mean -0.030 -0.033 -0.036 -0.055 -0.035 -0.033 -0.053 -0.040 -0.040 -0.030 -0.037 MLE -0.029 -0.033 -0.036 -0.056 -0.035 -0.033 -0.056 -0.041 -0.039 -0.030 -0.038 2.5% -0.031 -0.034 -0.038 -0.060 -0.037 -0.034 -0.061 -0.043 -0.042 -0.032 -0.040 97.5% -0.029 -0.032 -0.034 -0.050 -0.032 -0.031 -0.046 -0.038 -0.038 -0.027 -0.034 3j Mean 0.885 -0.285 0.100 0.158 -0.403 -0.545 0.539 0.510 -0.163 0.899 0.180 MLE 0.844 -0.316 0.084 0.162 -0.377 -0.505 0.472 0.444 -0.223 0.975 0.145 2.5% 0.496 -0.468 -0.055 -0.307 -0.727 -0.689 0.363 0.355 -0.473 0.742 -0.063 97.5% 1.262 -0.101 0.279 0.624 -0.079 -0.407 0.722 0.667 0.141 1.068 0.425 4j Mean 0.369 -0.143 0.274 0.103 0.135 0.216 0.149 -0.230 0.458 -0.463 -0.183 MLE 0.410 -0.130 0.315 0.100 0.137 0.220 0.078 -0.218 0.328 -0.436 -0.155 2.5% 0.232 -0.218 0.201 -0.302 -0.144 0.150 0.016 -0.331 0.298 -0.553 -0.288 97.5% 0.503 -0.070 0.347 0.504 0.375 0.285 0.280 -0.137 0.623 -0.374 -0.082 5j Mean 0.158 -0.171 0.986 -0.471 -0.156 -0.469 -2.117 0.439 -0.081 0.479 0.133 MLE 0.097 -0.117 0.967 -0.192 -0.056 -0.365 -1.365 0.481 -0.024 0.452 0.125 2.5% -0.167 -0.421 0.684 -1.040 -0.635 -0.881 -2.557 0.233 -0.336 0.249 -0.253 97.5% 0.501 0.095 1.292 0.111 0.257 -0.136 -1.755 0.658 0.193 0.721 0.534 6j Mean -0.303 -0.191 -0.671 -0.197 0.033 -0.067 -0.895 -0.320 -0.208 -0.009 -0.090 MLE -0.247 -0.142 -0.691 -0.171 0.067 -0.047 -0.808 -0.308 -0.224 0.037 0.064 2.5% -0.560 -0.356 -0.874 -0.462 -0.192 -0.237 -1.101 -0.482 -0.387 -0.156 -0.288 97.5% -0.075 -0.031 -0.456 0.066 0.291 0.136 -0.755 -0.171 -0.035 0.121 0.115 7j Mean 0.243 0.048 1.563 2.108 -1.433 -0.018 -2.116 -0.239 2.431 -0.133 1.531 MLE -0.181 -0.446 1.520 2.642 -0.654 0.129 -0.116 -0.219 2.615 -0.034 1.496 2.5% -0.450 -0.439 1.061 1.261 -2.269 -1.028 -2.862 -0.474 1.880 -0.398 0.910 97.5% 0.952 0.535 1.964 2.831 -0.673 0.949 -1.153 -0.001 2.895 0.138 2.224 Appendix A US Mortality 125 Parameter Species (from equation 2.9) Quercus prinus Quercus stellata Carya spp Thuja occidentalis Nyssa slyvatica Quercus nigra Betula papyferia Fagus grandifolia Pinus virginiana Fraxinus americana N.sylv. (biflora) 8j Mean -1.636 -0.167 -1.411 0.668 0.565 -0.054 -0.537 0.656 -2.411 0.111 -1.094 MLE -1.805 -0.016 -1.450 0.769 0.548 0.039 -0.348 0.639 -2.332 0.100 -1.041 2.5% -2.217 -0.476 -1.720 0.414 0.207 -0.545 -0.813 0.522 -2.896 -0.071 -1.527 97.5% -1.080 0.143 -1.072 0.938 0.945 0.440 -0.145 0.794 -1.779 0.287 -0.701 9j Mean -0.261 -0.037 0.040 0.311 -0.277 -0.242 0.389 -0.008 0.297 0.195 -0.027 MLE -0.332 -0.017 0.013 0.314 -0.109 -0.258 0.415 -0.021 0.227 0.343 -0.161 2.5% -0.546 -0.176 -0.169 0.121 -0.472 -0.402 0.291 -0.136 0.078 0.016 -0.223 97.5% 0.033 0.116 0.217 0.493 -0.087 -0.086 0.492 0.126 0.529 0.375 0.167 10j Mean 0.719 0.035 0.179 0.013 -0.107 -0.058 0.107 0.210 0.961 0.148 -0.008 MLE 0.655 0.003 0.190 0.016 -0.087 -0.075 0.060 0.161 1.006 0.137 -0.005 2.5% 0.507 -0.026 0.074 -0.138 -0.243 -0.154 0.054 0.087 0.779 0.075 -0.108 97.5% 0.927 0.095 0.280 0.164 0.028 0.035 0.156 0.328 1.151 0.218 0.094 11j Mean 1.105 -0.527 0.866 -0.190 0.036 0.206 -0.246 -0.615 0.131 -1.220 0.419 MLE 1.142 -0.318 1.067 -0.014 -0.079 0.345 -0.109 0.000 -0.082 -0.557 0.009 2.5% 0.681 -0.983 0.583 -0.416 -0.633 -0.142 -0.450 -1.691 -0.579 -1.627 0.072 97.5% 1.418 -0.239 1.105 0.060 0.489 0.610 -0.038 0.694 0.675 -0.680 0.730 12j Mean 0.388 -0.088 0.111 -0.012 0.003 0.080 0.002 -0.133 -0.006 -0.175 0.124 MLE 0.535 -0.007 0.157 -0.028 0.000 0.085 0.014 -0.035 -0.024 -0.109 0.053 2.5% 0.201 -0.228 0.007 -0.051 -0.110 0.014 -0.036 -0.366 -0.295 -0.271 0.049 97.5% 0.529 0.007 0.202 0.029 0.081 0.156 0.041 0.153 0.232 -0.058 0.191 13j Mean 0.111 0.142 0.100 -0.025 -0.012 0.047 0.012 -0.018 0.131 -0.002 0.017 MLE 0.029 0.130 0.031 -0.020 -0.006 0.032 0.007 -0.008 0.120 0.001 -0.004 2.5% 0.075 0.120 0.071 -0.037 -0.025 0.017 -0.001 -0.030 0.092 -0.014 -0.010 97.5% 0.143 0.168 0.125 -0.007 0.002 0.067 0.026 -0.004 0.176 0.012 0.050 14j Mean -0.0019 -0.0021 -0.0018 0.0002 0.0001 -0.0005 -0.0003 0.0000 -0.0021 -0.0001 -0.0005 MLE -0.0011 -0.0007 -0.0017 0.0002 0.0001 -0.0004 -0.0001 0.0001 -0.0018 0.0000 -0.0004 2.5% -0.0024 -0.0026 -0.0023 0.0000 -0.0001 -0.0009 -0.0005 -0.0002 -0.0028 -0.0003 -0.0010 97.5% -0.0014 -0.0017 -0.0013 0.0004 0.0002 0.0000 -0.0001 0.0001 -0.0015 0.0000 -0.0001 Appendix A US Mortality 126 Figure A.1 Log annual mortality rates observed for the whole forest including rare species (orange) and the 21 common species (green), and the model predictions for the 21 species combined (purple) and each species individually (grey), plotted against maximum wind speed (m/sec). Species' error bars (grey) show parameter uncertainty, forest error bars (purple, orange and green) show the 95% confidence interval for the mortality rates predicted from the model-created and real datasets. Figure A.2 Log annual mortality rates plotted against soil type for the predicted forest-level mortality rate for all 21 species (purple), the real forest-level mortality rates for the 21 species (green) and the whole forest including rare species (orange). Error bars (purple, orange and green) show the 95% confidence interval for the mortality rates predicted from the model-created and real datasets. Appendix A US Mortality 127 Figure A.3 Log annual mortality rates observed for the whole forest including rare species (orange) and the 21 common species (green), and the model predictions for the 21 species combined (purple) and each species individually (grey), plotted against plot basal area (m 2 /hectare). Species' error bars (grey) show parameter uncertainty, forest error bars (purple, orange and green) show the 95% confidence interval for the mortality rates predicted from the model-created and real datasets. Figure A.4 Predicted versus observed plot-averaged annual mortality rate for all plots with at least 10 stems, showing the high correlation (r 2 =0.9). Appendix A US Mortality 128 Figure A.5 Patterns of mortality due to regional variation in stand structure and environment alone. Maps of estimated annual forest-level mortality across the Eastern United States illustrating the contributions of variation in stand structure (stem size and plot basal area) and environment, modelled across the range of Pinus taeda to control for the effects of species composition. (a) Variation in forest structure alone (stem size and plot basal area), illustrated by removing environmental effects and modelling just the most common species (P. taeda). (b) The effect of variation in environment alone, illustrated by modelling P. taeda without stand structure variation (i.e. modelling a 20 cm DBH tree) across the region. (a) (b) Annual mortality (% year-1) Appendix A US Mortality 129 Figure A.6 Map of absolute difference between predicted and observed forest level mortality across the Eastern United States. Annual mortality (% year-1) 130 Appendix B Aboveground allometry 131 Appendix B: Aboveground allometry Materials and Methods Species-level traits We found data for green wood density and modulus of elasticity (Young's modulus) in the literature (for density: Gutiérrez & Plaza 1967; Dietz 1975; Zanne et al. 2009; and for modulus of elasticity: U.S. Department of Agriculture Forest Service; Summitt & Sliker 1980; Gibson et al., 1981; Wessolly & Erb 1998; Borghetti et al. 2004; Alméras et al., 2005). When more than one estimate was given for European data we took the average. Species' shade tolerance was represented using Ellenberg scores (Ellenberg, 1988), which range from 1 (plants typically found in deep shade) to 9 (mostly found in full light). Species' values were available for 28 species, accounting for 45% of the data, whilst for a further 24 species scores were calculated from average scores of other species of the same genus. In total 52 species, accounting for 94% of the data, were assigned shade-tolerance scores. We removed species which were non-native, and for those species which are frequently coppiced and/or pollarded we used the sub-dataset of high-quality, uncut stems. For height-diameter we parameterised the model for the 15 species for which we had information on wood density and modulus of elasticity, and which had at least 1000 samples within the dataset: in total over 550,000 stems. For one of these species we did not have a shade-tolerance score, but as we were primarily interested in the variables predicted by the elastic similarity model to have an effect, we preferred to include this species. For crown diameter- DBH there were 14 suitable species (over 175,000 stems in total) for which we had wood density and modulus of elasticity data, for which we had at least 1000 samples and which were native. Choice of line-fitting methods In this analysis we chose a method of line fitting which suited our aim to separate out the effects of several environmental variables on allometric scaling. These variables were partially correlated because, for example, denser forests are found in wetter places; which meant that the parameters associated with their effects were also likely to be partially correlated (Kruschke, 2010a). Bayesian methods provide the natural solution to such a problem, since instead of providing an estimate of the mean and variance of each parameter separately from all others, they provide parameter estimates in sets, each dependent on the others. These joint parameter estimates, drawn from the joint distribution of the model parameters, may therefore be used to make model predictions in a way that robustly accounts for correlation within a dataset. Although frequentist approaches such as generalised linear modelling allow the variance and covariance of parameters to be estimated, the approach is less Appendix B Aboveground allometry 132 intuitive and flexible, and requires specification of parameter distributions, which are not easily selected well. Reduced major axis (RMA) line fitting (also known as standardised major axis (SMA) line fitting) has become the most commonly used method for fitting allometric relationships and is routinely chosen over ordinary least squares regression (OLS) without specific justification, although in many cases OLS would be more appropriate (Smith, 2009). The main reason for this is that RMA line fitting assumes error in X as well as Y, often a reasonable assumption in allometry data. However, other factors ought to be included, and there are many excellent reviews in the literature which compare the assumptions and implications of these methods (e.g. Martin & Barbour 1989; Warton et al. 2006; Smith 2009). For example, error in the measurement of X should be considered relative to the error in Y. Another major difference is that RMA is a symmetric method (it produces the same line of best fit regardless of which variable is the predictor and which the response) whereas OLS is asymmetric. Choice of one or the other may therefore depend on whether finding the relationship between two variables or predicting one from the other is the aim. Our Bayesian approach is more akin to OLS than it is to RMA, and although our primary aim was to choose a method which could accurately describe the response of height to a variety of co- varying predictors, RMA was also inappropriate because the diameter measurements were taken to the nearest 1 mm and H to the nearest 0.5 m. In these data, for a typical 15 cm DBH tree with a height of 10 m, this corresponds to measurement errors of 0.67% and 5% respectively, so the X variable was measured more accurately than the Y variable. We recognise that if we had used RMA methods then it is likely that the estimated slopes of our allometric relationships would have been greater, altering our comparisons of slope estimates with theoretical values. The hierarchical Gibbs sampler Hierarchical or multilevel modelling allows us to fit both individual species' allometric relationships and a 'universal' allometric relationship for all species together simultaneously. Hierarchical Gibbs sampling provides a natural way to constrain the allometry of less common species to be more similar to the population average (Gelfand et al., 1990; Lee, 1997; Gelman and Hill, 2007; Dietze et al., 2008). Rather than fit a universal 'species-average' allometric relationship we chose to use species co- variates (Ellenberg score, wood density, modulus of elasticity) to constrain species with closer covariates to have closer allometries. This was done within the hierarchical Gibbs sampler by regressing the species-level parameters on the covariates at the across-species level (Gelman et al., 2004; Gelman and Hill, 2007; Dietze et al., 2008). By assuming normal distributions on parameter values and standard inverse-Gamma variance structures it is possible to fit the models using Gibbs samplers to find fully described posterior distributions, since prior and conditional distributions are conjugate (Lee, 1997). Appendix B Aboveground allometry 133 We constructed a hierarchical normal Gibbs sampler to fit parameters to traditional power- law allometric equations, = b0xij b1 , to describe the relationship between tree diameter ( ) and tree height and crown diameter ( ) following the methods used by Dietze et al. (2008). However, we decided not to fit the two models simultaneously as substantially more data were available to fit the height-DBH than crown diameter-DBH model. The models take the form ijijijij bxby   ijjxβ)log()log( 01 (Eqn B.1) where, using the notation of Dietze et al. (2008), either =log(height) or log(crown diameter) (depending on the model), xij ={1,log(DBH)}, βj={log(b0), b1} and j is the species. We therefore assume that yij ~N(βjxij,σ 2 ), where σ2~IG(m,n) represents the inverse-gamma distributed variance. Moreover, we assume that all species' parameters have a common prior:, βj~N(B,Φ) where B=(B0,B1) and Ф are parameters to be fitted. This assumption makes the model hierarchical, which is an advantage because rare species' allometries will be constrained to be more similar to the universal allometry. In the fitting of the Gibbs sampler we used the standard conjugate hyperpriors, B~N(B0,V0) and Ф~Inv-Wishart(w, wΨ). For the log(height)-log(diameter) model we used the MST prediction of 2/3 as a prior for the coefficient (b1 in equation B.1) and a prior of 3 for the intercept (log(b0)). For the log(crown diameter)-log(diameter) model a prior of 3 for the intercept and 0 for the coefficient. For all parameters for the species level covariates we used a prior of 0. The prior V0 was the identity matrix, w was 1 and Ψ was the identity matrix. To test the hypotheses based on species' differences in wood density, modulus of elasticity and shade tolerance, we fitted the Gibbs sampler with each included as a regression variable at the species level, that is we fitted  ,~ ZBβj N (Eqn B.2) where Z={z1,z2,z3,z4} is a matrix of the covariates (as in Dietze et al., 2008). We parameterised the model using a wide variety of different starting values for parameters, but none affected the significance of the fitted parameters or our conclusions. The model was fitted using an algorithm written in C with code for sampling from standard distributions from Press et al. (1992), Lee (1997) and Kennedy & Gentle (2000) and model description from Gelfand et al. (1990), Gelman & Hill (2007) and Dietze et al. (2008). Specifications of the MCMC sampler We modelled DBH-H and DBH-CD relationships using an MCMC sampler. The models took the forms   2102 ,~ 3 ijbijij xbbxbNy  (Eqn B.3) Appendix B Aboveground allometry 134   21032 ,~ ijijij xbbxbbNy  (Eqn B.4) where (in equation B.3) yij is the H of each tree i of species j of DBH , and (in equation B.4) is the CD of each tree i of species j of DBH . In our final model b2 and b3 of equations B.3 and B.4 were linear functions of the environmental predictor variables: b2 = a0 + a1(competition index) + a2(annual precipitation) + a3(drought length) + a4(average temperature) b3 = c0 + c1(competition index) + c2(annual precipitation) + c3(drought length) + c4(average temperature) (Eqn B.5) We set wide and uninformative priors on all parameters of the MCMC sampler (Table B.4). Convergence for the final models for DBH-H and DBH-CD was checked using the Gelman- Rubin statistic R (Gelman and Rubin, 1992) which assesses if a model has converged by comparing within and between chain variance for multiple chains. We fitted six chains using randomly chosen starting values for each parameter across a wide range (see Table B.4). We used a burn-in of 500,000 iterations and a sampling period of 500,000 for each chain. We applied the Gelman-Rubin statistic to each parameter and found that for most species convergence was achieved (√R<1.2) within this number of iterations. However for the DBH-H model, a longer run was required to achieve convergence for seven species (Abies alba, Arbutus glutinosa, Betula spp., Juniperus thurifera, Pinus pinea, P. uncinata and Sorbus spp.). For these we run six chains with a burn-in of 2,500,000 iterations and a sampling period of 1,500,000, after which convergence was achieved. For the DBH-CD models a burn-in of 500,000 and a sampling period of 500,000 for each of six chains was sufficient to achieve convergence for all species. Additional Results Hierarchical allometry The fitted parameters for the across-species hierarchical allometry with species-level covariates are given in table S1 (parameters in equation B.2, height and DBH in cm), with 95% credible intervals on the given Bayesian means, where the coefficients of the covariates are for the normalised values. Overall, height for a given DBH increased with wood density, modulus of elasticity and Ellenberg score (i.e. decreased with shade tolerance). The effect of the Ellenberg score on the allometry was strongest and modulus of elasticity the weakest. However for all three covariates the central 95% of all parameters' posterior distributions contained 0, indicating that the effect is either not strong or not the same for all species, and therefore not well converged when species were combined. We found the Appendix B Aboveground allometry 135 same result for crown diameter: none of the three covariates showed a clear positive or negative effect when tested in the hierarchical framework. Tables and figures Table B.1 Fitted intercept and gradient parameters for the across-species height/stem diameter and crown-diameter/stem diameter hierarchical Gibbs regression using species-level covariates (equation D.2), where height and diameter are both expressed in cm, and the 95% confidence interval of each parameter. All covariate regression parameters were non-significant. (constant) Ellenberg score Wood density Modulus of elasticity Height intercept 5.263 (4.911, 5.615) -0.084 (-0.368, 0.201) -0.001 (-0.274, 0.271) -0.002 (-0.284, 0.280) Height gradient 0.480 (0.396, 0.564) 0.027 (-0.055, 0.109) -0.054 (-0.135,0.028) 0.022 (-0.061, 0.104) Crown diameter intercept 3.590 (3.065,4.116) 0.039 (-0.372, 0.451) -0.114 (-0.548, 0.320) 0.231 (-0.138, 0.600) Crown diameter gradient 0.712 (0.602, 0.822) 0.022 (-0.095, 0.139) 0.083 (-0.033, 0.198) -0.034 (-0.146, 0.078) Appendix B Aboveground allometry 136 Table B.2 Fitted coefficient and exponent parameters, with 95% credible intervals, for all 26 species fitted together and for each species' height-diameter relationship (equation 2.3), where height and diameter are both expressed in cm for species with over 1000 individuals in the dataset. Neither the across-species model nor any individual species showed support for the 2/3 exponent, although the exponents of two species (Pinus nigra and P. sylvestris) were very close. Species Number Coefficient b2 Exponent b3 Across-species model 696582 164.69 (163.48,165.91) 0.609 (0.606,0.611) Abies alba 3961 198.10 (190.33,208.70) 0.62 (0.61,0.63) Acer campestre 2194 287.98 (252.60,316.81) 0.44 (0.41,0.49) Alnus glutinosa 3825 384.18 (362.56,404.69) 0.43 (0.42,0.45) Arbutus unedo 2418 213.25 (196.76,233.55) 0.42 (0.38,0.45) Betula spp. 7238 381.23 (357.38,399.61) 0.40 (0.38,0.42) Castanea sativa 1162 682.24 (602.34,780.59) 0.16 (0.13,0.20) Corylus avellana 1558 374.38 (344.95,400.12) 0.31 (0.28,0.35) Fagus sylvatica 51479 350.88 (347.78,357.17) 0.46 (0.45,0.46) Ilex aquifolium 1612 249.60 (215.62,281.44) 0.37 (0.32,0.42) Juniperus communis 1465 154.74 (142.39,168.61) 0.42 (0.39,0.45) Juniperus thurifera 7980 172.81 (168.12,177.53) 0.39 (0.39,0.40) Olea europaea 2759 249.44 (232.14,272.62) 0.27 (0.24,0.29) Pinus halepensis 88437 154.90 (153.73,156.04) 0.58 (0.57,0.58) Pinus nigra 72548 128.13 (127.18,129.03) 0.67 (0.67,0.68) Pinus pinaster 178622 163.06 (161.93,164.03) 0.63 (0.62,0.63) Pinus pinea 25256 111.50 (108.87,114.18) 0.64 (0.63,0.65) Pinus sylvestris 145446 142.44 (142.02,142.97) 0.66 (0.66,0.67) Pinus uncinata 15239 169.00 (165.90,172.11) 0.58 (0.57,0.59) Quercus faginea 3171 158.32 (146.91,171.77) 0.51 (0.48,0.54) Quercus ilex 21551 202.44 (197.16,207.36) 0.34 (0.33,0.35) Quercus petraea 13809 269.77 (256.35,280.01) 0.48 (0.47,0.50) Quercus pyrenaica 4304 198.13 (183.32,213.11) 0.49 (0.47,0.52) Quercus robur 26550 305.18 (297.30,310.49) 0.44 (0.44,0.45) Quercus suber 3450 164.31 (147.53,179.68) 0.44 (0.42,0.48) Salix spp. 2281 350.42 (323.84,383.45) 0.36 (0.32,0.39) Sorbus spp. 1046 286.32 (256.75,323.22) 0.42 (0.37,0.45) Appendix B Aboveground allometry 137 Table B.3 Spearman's rank correlation p-values for across-species predicted heights and crown diameters at given DBH and species' mean environmental conditions. Positive associations are indicated with (+), and negative relationships with (-). Height DBH Basal area of larger trees Annual precipitation Drought length Average annual temperature 10 0.011 (+) 0.00001 (+) 0.0007 (-) 0.25 (-) 15 0.007 (+) 0.00001 (+) 0.002 (-) 0.156 (-) 20 0.0016 (+) 0.0001 (+) 0.0008(-) 0.097 (-) Crown Diameter DBH Basal area of larger trees Annual precipitation Drought length Average annual temperature 10 0.197 (+) 0.048 (+) 0.064 (-) 0.240 (+) 15 0.573(+) 0.108 (+) 0.191(-) 0.584 (+) 20 0.682 (-) 0.112 (+) 0.186 (-) 0.627 (+) Crown Diameter (angiosperms only) DBH Basal area of larger trees Annual precipitation Drought length Average annual temperature 15 0.011 (+) 0.022 (+) 0.005 (-) 0.132 (-) Table B.4.Priors and initial value sampling ranges for each parameter for the final DBH-H and DBH-CD models (the same ranges were used for both). Gelman-Rubin convergence statistics were calculated using six chains, with initial parameter values were sampled randomly and uniformly across the given range for each parameter. Parameter names refer to equations D.3-D.5. Parameter b0 b1 a0 c0 a1 - a4 c1 - c4 Prior U(-1000,1000) U(-1000,1000) U(-1000,1000) U(-1000,1000) U(-50,50) U(-10,10) Initial sampling distribution U(100,300) U(5,15) U(100,800) U(0,1) U(-5,5) U(-0.5,0.5) Appendix B Aboveground allometry 138 Figure B.1 Height-diameter data for the 14 native species with over 1000 stems in the dataset for which Hcrit (blue line) could be calculated, plotted with the MCMC-fitted allometry (green line), and the data. For most species the safety factor increased with size, and for many species, some specimens approached or even exceeded the hypothetical limit, particularly at small sizes. Appendix B Aboveground allometry 139 Figure B.2 As Fig. 4.2 (page 63), with species' names included. Predicted changes in height of 15- cm-DBH stems of 26 species in relation to (a) basal area of larger trees, (b) precipitation, (c) drought length, and (d) mean annual temperature. The symbols show height at the mean value of the environmental variables for the species - angiosperms (filled circles) and gymnosperms (open triangles) - whilst predicted variation in height is shown for ± 1 standard deviation in the environmental variable around the species' distribution mean (grey arrows). Estimates used in the predictions were produced using posterior estimates of the joint distribution of the parameters. Species' names are as follows: P.sy=Pinus sylvestris, P.un=Pinus uncinata, P.pa=Pinus pinea, P.ha=Pinus halepensis, P.ni=Pinus nigra, P.pr=Pinus pinaster, A.al=Abies alba, J.co=Juniperus communis, J.th=Juniperus thurifera, Q.ro=Quercus robur, Q.pe=Quercus petraea, Q.py=Quercus pyrenaica*, Q.fa=Quercus faginea*, Q.il=Quercus ilex*, Q.su=Quercus suber*, A.gl=Alnus glutinosa, Sa.spp=Salix spp., I.aq=Ilex aquifolium, O.eu=Olea europaea, A.un=Arbutus unedo, F.sy=Fagus sylvatica, C.sa=Castanea sativa*, B.spp=Betula spp., C.av=Corylus avellana, A.ca=Acer campestre, So.spp=Sorbus spp. (*commonly coppiced/pollarded species so data was taken from a smaller database of trees with no signs of cutting at the time the inventory was taken). Appendix B Aboveground allometry 140 Figure B.3 As Fig. 4.3 (page 64), with species' names included. Predicted changes in crown diameter of 15-cm-DBH stems of 14 species in relation to (a) basal area of larger trees, (b) precipitation, (c) dry season length, and (d) mean annual temperature. The symbols show crown diameter at the mean value of the environmental variables for the species - angiosperms (filled circles) and gymnosperms (open triangles)- whilst predicted variation in crown diameter is shown for ± 1 standard deviation in the environmental variable around the mean (grey arrows). Estimates used in the predictions were produced using posterior estimates of the joint distribution of the parameters. Species' names are as in Fig. B.2. Appendix C ABC recruitment model 141 Appendix C: ABC recruitment model MCMC algorithm We estimated parameters and credible intervals (CIs) of models of crown diameter, individual tree growth and annual mortality (described below) using an adaptive MCMC Metropolis algorithm (Lee, 1997; Gelman et al., 1999). We fitted several different functional forms for each model and compared them using the Akaike information criterion (Akaike, 1974). The MCMC algorithm compares parameter values using the log-likelihood of the data given the model. At each iteration the algorithm selects a parameter to alter and recalculates the likelihood. If the new parameter improves the likelihood then it is accepted by the algorithm. If not, it is accepted with probability of the ratio of the new and old likelihoods. In this way it returns not only a best-fit value for each parameter given the data but also estimates its distribution. The algorithm has two periods: burn-in and sampling. During the burn-in period (100,000-250,000 iterations of the algorithm) the algorithm alters the search range ("jumping distance") of each parameter value to achieve an optimal acceptance ratio of 25% (Gelman et al., 1999). After the burn-in period, the jumping distance is fixed (separately for each parameter). During sampling (100,000-250,000) iterations, parameter values are recorded every 100 iterations and the resulting parameter samples are taken as samples from the posterior distribution of each parameter. The resulting samples are then used to calculate mean and 95% confidence intervals for each parameter. We used non-informative uniform priors on all parameters so the MCMC algorithm needed to refer to the log-likelihood only. We used normalised mean annual temperature and mean annual precipitation values (taken from Gonzalo Jiménez, 2008). All models were fitted using an adaptive Metropolis algorithm written in C. Construction of crown metric CAI We parameterised models of crown diameter (CD) as a function of stem size (DBH) and climate for each species in order to calculate the crown area of adults in each plot, both in total and of each species individually. We used a subset of the IFN database in which two measurements of crown diameter were recorded for around four trees of particular silvicultural interest in each plot. The number of measurements for each species is shown in Table C.1. We parameterised DBH-CD equations using adaptive MCMC for the 30 species with more than 50 trees measurements in the data (in total >200,000 measurements), which accounted for >90% of the data. We tested a set of models Appendix C ABC recruitment model 142 (see Table C.2 for functional forms tested) for crown diameter as a function of stem size and climate and selected the best model as the best for the most species and data (model 10, see Table C.2). For each tree we used these functions to calculate, as a proportion of plot area, the total crown area of all taller trees in each plot, CAIh, and the crown area of all conspecifics, CAIsp in the plot. We also calculated the crown area of all trees in each plot, CAIpl. Observed and predicted crown diameters are shown for each of the 30 fitted species in Fig. C.1. For species lacking allometric data we estimated the crown diameter-stem diameter relationship by either using the allometric equation of the single most closely related species or by averaging the allometric parameters of all the most closely related species if there was more than one at the closest distance (determined according to a phylogenetic tree created using the software Phylomatic, Webb & Donoghue 2005, see Table C.3). Construction of priors for growth and mortality functions To construct priors for the growth and mortality functions within the ABC algorithm we fitted models to data of small trees from the second and third Spanish Forest Inventories (MMA, 1996, 2007). We selected plots that had been measured in both the second (IFN2) and third (IFN3) inventories and fitted models to trees that had stem diameter (DBH) < 10 cm in the IFN2, excluding individuals whose mortality was human induced. We fitted models to 16 species with >100 individual stems for both growth and mortality. We compared the influence of three different predictors, initial stem size (DBH1) and competition measured as crown area of all taller trees, CAIh, and basal area of taller trees (BAL) in the plot. All models were fitted with all parameters species-specific. We modelled the stem diameter measured in the IFN3 (DBH2) as a function of the initial stem diameter measured in the IFN2 (DBH1) and the growth rate using: (Eqn C.1) where GR is the predicted annual growth rate and t is the time interval (average 9 years). We modelled the annual probability of mortality using a logistic function: mortality (Eqn C.2) with corresponding likelihood: likelihood of data given model mortality if tree survived mortality if tree died We compared a set of models with different functional forms and selected the best fit model according to AIC (see Tables C.4 and C.5, for model forms fitted for growth and mortality respectively). Appendix C ABC recruitment model 143 Results of growth and mortality models We compared seven models for annual growth and four for annual mortality rate (Tables C.4 and C.5). We calculated AIC values to compare models for each species individually. The best fit growth model for most species was a three parameter function with CAIh as a predictor (model 3 in Table C.4), so we used this functional form for growth in the recruitment model. Similarly, the best fit model for mortality for most species was a two function with CAIh as a predictor (model 3 in Table C.5). Individual species' parameter values and their corresponding 95% CIs for these two models are shown in Table C.6 and C.7. Predicted and observed values for DBH2, fitted using model 3 in Table C.4, are shown in Fig. C.2. Predicted and observed values for annual mortality rate, fitted using model 3 in Table C.5, are shown in Fig. C.3. Predicted growth and mortality rates for each species across the range of values of CAIh in which it is found are shown in Fig. 5.2. Appendix C ABC recruitment model 144 Table C.1 Amount of field data for each species used to estimate DBH-crown diameter allometric equations. Species Name Count Abies alba 631 Abies pinsapo 63 Castanea sativa 4659 Chamaecyparis lawsoniana 177 Eucalyptus camaldulensis 1972 Eucalyptus globules 7127 Eucalyptus nitens 143 Fagus sylvatica 10292 Larix spp. 409 Picea abies 59 Pinus halepensis 30046 Pinus nigra 18455 Pinus pinaster 38086 Pinus pinea 8970 Pinus radiata 6609 Pinus sylvestris 28093 Pinus uncinata 2720 Platanus spp. 115 Populus alba 97 Populus nigra 1817 Pseudotsuga menziesii 172 Quercus canariensis 417 Quercus faginea 7845 Quercus ilex 36945 Quercus petraea 3660 Quercus pyrenaica 11832 Quercus robur 7958 Quercus rubra 304 Quercus suber 8693 Robinia pseudoacacia 214 Appendix C ABC recruitment model 145 Table C.2 Tested models of crown diameter (CD) as a function of stem size (DBH), drought length (DL), average annual temperature (AvT) and annual precipitation (PA), and the number of parameters in each model. Parameters fitted are denoted p0-p6. Average temperature and annual precipitation were normalised to aid convergence (using annual precipitation mean = 862, standard deviation = 378, average temperature mean = 12, standard deviation = 3). Model # Description # parameters AIC AIC rank #species' best model % data best model 0 CD ~ N(p1+p2DBH, p0) 3 5593348 11 1 0.07 1 CD ~ N(p2+p3DBH, p0+p1DBH) 4 5481178 7 5 16.92 2 CD ~ N(p1+p2DBH+p3DL, p0) 4 5584746 8 0 0.00 3 CD~N(p2+p3DBH+p4DL,p0+p1DBH) 5 5472071 3 0 0.00 4 CD~N(p1+p2DBH+p3AvT,p0) 4 5588356 9 0 0.00 5 CD~N(p2+p3DBH+p4AvT,p0+p1DBH) 5 5474664 5 2 1.98 6 CD~N(p1+p2DBH +p3PA,p0) 4 5590359 10 0 0.00 7 CD~N(p2+p3DBH+p4PA, p0+p1DBH) 5 5478742 6 4 3.34 8 CD~N(p2+p3DBH+p4DL+p5AvT, p0+p1DBH) 6 5466517 2 2 2.90 9 CD~N(p2+p3DBH+p4PA+p5AvT, p0+p1DBH) 6 5472122 4 5 19.92 10 CD ~ N(p2+p3DBH+p4PA+p5AvT+p6DL, p0+p1DBH) 7 5464760 1 12 54.87 Appendix C ABC recruitment model 146 Table C.3 IFN species code, species genus and family, the number of plots the species was found in, and the code of the species’ crown diameter allometric equations used to calculate crown area for the species (in bold if the species had its own equation), assigned using nearest phylogenetic neighbour or neighbours, if there was more than one at the closest distance. If more than one species’ code is listed then the average of those species’ parameters was used. For 93% of the data we were able to use crown diameter equations fitted to the individual species’ crown measurements. IFN code Species Family #Plots IFN code(s) of species’ allometric equation used to fit crown area. 31 Abies alba Pinaceae 293 31 32 Abies pinsapo Pinaceae 42 32 7 Acacia spp. Mimosaceae 37 92 76 Acer campestre Aceraceae 902 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72, 79,92 54 Alnus glutinosa Betulaceae 618 41,42,43,44,45,46,47,48,71,72 88 Apollonias barbujana Lauraceae 4 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,79,92 68 Arbutus unedo Ericaceae 743 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,92 73 Betula spp. Betulaceae 1424 41,42,43,44,45,46,47,48,71,72 91 Buxus sempervirens Buxaceae 29 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,92 98 Carpinus betulus Coryloideae 5 41,42,43,44,45,46,47,48,71,72 72 Castanea sativa Fagaceae 2396 72 17 Cedrus atlantica Pinaceae 17 21,22,23,24,25,26,28,31,32,33,34, 35 13 Celtis australis Ulmaceae 18 41,42,43,44,45,46,47,48,71,72 67 Ceratonia siliqua Fabaceae 218 92 18 Chamaecyparis lawsoniana Cupressaceae 76 18 9 Cornus sanguinea Cornaceae 1 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,92 74 Corylus avellana Betulaceae 433 41,42,43,44,45,46,47,48,71,72 15 Crataegus spp. Rosaceae 328 41,42,43,44,45,46,47,48,71,72 36 Cupressus sempervirens Cupressaceae 71 18 83 Erica arborea Ericaceae 183 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,92 62 Eucalyptus camaldulensis Myrtaceae 691 62 61 Eucalyptus globulus Myrtaceae 3006 61 64 Eucalyptus nitens Myrtaceae 69 64 5 Euonymus Celastraceae 1 51,58 Appendix C ABC recruitment model 147 europaeus 71 Fagus sylvatica Fagaceae 3549 71 3 Frangula alnus Rhamnaceae 7 41,42,43,44,45,46,47,48,71,72 55 Fraxinus angustifolia Oleaceae 761 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,92 1 Heberdenia bahamensis Myrsinaceae 2 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,79,92 65 Ilex aquifolium Aquifoliaceae 446 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,92 82 Ilex canariensis Aquifoliaceae 114 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,92 75 Juglans regia Juglandaceae 98 41,42,43,44,45,46,47,48,71,72 37 Juniperus communis Cupressaceae 832 18 39 Juniperus phoenicea Cupressaceae 203 18 38 Juniperus thurifera Cupressaceae 1588 18 35 Larix spp. Pinaceae 173 35 94 Laurus nobilis Lauraceae 139 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,79,92 12 Malus sylvestris Rosaceae 32 41,42,43,44,45,46,47,48,71,72 81 Myrica faya Myricaceae 202 41,42,43,44,45,46,47,48,71,72 87 Ocotea phoetens Lauraceae 2 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,79,92 66 Olea europaea Oleaceae 743 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,92 63 Other/unknown eucalyptus species Myrtaceae 1 61,62,64 89 Other/unknown laurel species Lauraceae 6 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,79,92 29 Other/unknown pine species Pinaceae 7 21,22,23,24,25,26,28 59 Other/unknown riparian species Unknown (Angiosperm Average) 6 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72, 79,92 90 Other/unknown small trees Unknown (Angiosperm Average) 1 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,79,92 99 Other/unknown species Unknown (Angiosperm Average) 252 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,79,92 84 Persea indica Lauraceae 43 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72, 79,92 8 Phillyrea latifolia Oleaceae 96 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,92 69 Phoenix spp. Arecaceae 12 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72, 79,92 86 Picconia excelsa Oleaceae 16 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,92 33 Picea abies Pinaceae 34 33 Appendix C ABC recruitment model 148 27 Pinus canariensis Pinaceae 1448 23,24,26 24 Pinus halepensis Pinaceae 10893 24 25 Pinus nigra Pinaceae 6988 25 26 Pinus pinaster Pinaceae 12372 26 23 Pinus pinea Pinaceae 3288 23 28 Pinus radiata Pinaceae 2368 28 21 Pinus sylvestris Pinaceae 9221 21 22 Pinus uncinata Pinaceae 929 22 93 Pistacia terebinthus Anacardiaceae 39 61,62,64 79 Platanus hispanica Platanaceae 72 79 51 Populus alba Salicaceae 51 51 58 Populus nigra Salicaceae 658 58 52 Populus tremula Salicaceae 158 51,58 95 Prunus spp. Rosaceae 324 41,42,43,44,45,46,47,48,71,72 34 Pseudotsuga menziesii Pinaceae 80 34 16 Pyrus spp. Rosaceae 30 41,42,43,44,45,46,47,48,71,72 47 Quercus canariensis Fagaceae 220 47 44 Quercus faginea Fagaceae 4373 44 45 Quercus ilex Fagaceae 15714 45 42 Quercus petraea Fagaceae 1695 42 43 Quercus pyrenaica Fagaceae 4596 43 41 Quercus robur Fagaceae 3821 41 48 Quercus rubra Fagaceae 154 48 46 Quercus suber Fagaceae 3537 46 4 Rhamnus alaternus Rhamnaceae 11 41,42,43,44,45,46,47,48,71,72 96 Rhus coriaria Anacardiaceae 4 61,62,64 92 Robinia pseudoacacia Fabaceae 145 92 57 Salix spp. Salicaceae 702 51,58 97 Sambucus nigra Adoxaceae 47 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72, 92 78 Sorbus spp. Rosaceae 492 41,42,43,44,45,46,47,48,71,72 53 Tamarix spp. Tamaricaceae 7 41,42,43,44,45,46,47,48,51,58,61, 62,64,71,72,92 14 Taxus baccata Taxaceae 49 18 77 Tilia spp. Malvaceae 123 61,62,64 56 Ulmus minor Ulmaceae 246 41,42,43,44,45,46,47,48,71,72 Appendix C ABC recruitment model 149 Table C.4 Set of species-specific growth models tested with corresponding maximum log-likelihoods and AICs, and the number of species for which each model was the best fit (according to AIC) out of the fourteen in the analysis. Model 3 (shown in bold) provided the best fit for the largest number of species, and was therefore chosen. Model number Annual growth (GR in equation C.1) Max log likelihood # of parameters AIC # of species' best model 0 GR=ω1 -20151.7 2 40359.4 0 1 GR= ω1DBH -20190.1 2 40436.2 0 2 GR=ω1 /(1+ ω2BAL) -19024.1 3 38132.2 4 3 GR=ω1 /(1+ ω2CAIh) -19134.9 3 38353.7 6 4 GR=ω1DBH ω2 -20134.5 3 40353 0 5 GR=ω1 DBH ω2 /(1+ ω3BAL) -18986.5 4 38084.9 3 6 GR=ω1 DBH ω2 /(1+ ω3CAIh) -19125.7 4 38363.3 1 Table C.5 Set of species-specific mortality models tested, with corresponding maximum log- likelihoods and AICs, and the number of species for which each model was the best fit (according to AIC) out of the fourteen in the analysis. Model 3 (shown in bold) provided the best fit for the largest number of species, and was therefore chosen. Model number Annual probability of mortality P(mortality)=1/(1+exp(-k)) (equation C.2) Max log likelihood # of parameters AIC # of species' best model 0 k=τ0 -3550.79 1 7129.6 2 1 k=τ0 +τ1DBH -3541.05 2 7138.1 0 2 k=τ0 +τ1BAL -3425.55 2 6907.6 5 3 k=τ0 +τ1 CAIh -3414.51 2 6885 7 Appendix C ABC recruitment model 150 Table C.6 Parameter values and 95% confidence intervals for the chosen models for growth (equation C.1) for each of the fourteen species in the analysis (model 3 in table C.4). Parameters ω1 and ω2 formed prior mean values for parameters p2 and p3 in equation 5.5 in the ABC-SMC algorithm. Species ω0 ω1 ω2 Castanea sativa 1.990 (1.972,2.008) 0.386 (0.334,0.437) 1.043 (0.662,1.424) Fagus sylvatica 1.436 (1.381,1.491) 0.412 (0.374,0.451) 1.895 (1.557,2.233) Juniperus thurifera 1.314 (1.235,1.393) 0.186 (0.165,0.208) 2.980 (1.524,4.435) Pinus halepensis 1.994 (1.983,2.005) 0.342 (0.327,0.357) 1.855 (1.585,2.125) Pinus nigra 1.907 (1.863,1.950) 0.499 (0.479,0.519) 3.167 (2.870,3.464) Pinus pinaster 1.999 (1.998,2.001) 0.843 (0.812,0.874) 4.440 (4.080,4.800) Pinus pinea 1.991 (1.975,2.007) 0.583 (0.535,0.632) 2.690 (2.073,3.306) Pinus sylvestris 1.998 (1.995,2.002) 0.572 (0.553,0.591) 2.325 (2.146,2.505) Pinus uncinata 1.840 (1.726,1.953) 0.486 (0.424,0.549) 3.445 (2.514,4.376) Quercus faginea 1.096 (1.065,1.126) 0.177 (0.170,0.185) 0.872 (0.703,1.040) Quercus ilex 1.499 (1.479,1.519) 0.161 (0.156,0.165) 0.459 (0.373,0.545) Quercus petraea 1.524 (1.415,1.633) 0.327 (0.259,0.395) 1.964 (1.064,2.864) Quercus pyrenaica 1.409 (1.377,1.441) 0.236 (0.226,0.246) 1.372 (1.191,1.553) Quercus suber 1.572 (1.473,1.671) 0.311 (0.254,0.368) 2.329 (1.299,3.359) Appendix C ABC recruitment model 151 Table C.7 Parameter values and 95% confidence intervals for the chosen models for mortality (equation C.2) for each of the fourteen species in the analysis (model 3 in table C.5). Parameters τ 0 and τ 1 formed prior mean values for parameters p4 and p5 in equation 5.5 in the ABC-SMC algorithm. Species τ0 τ1 Castanea sativa -3.422 (-3.744,-3.100) 0.283 (0.004,0.563) Fagus sylvatica -5.818 (-6.229,-5.406) 0.834 (0.603,1.066) Juniperus thurifera -6.143 (-6.695,-5.591) 2.309 (1.011,3.607) Pinus halepensis -4.791 (-4.941,-4.641) 1.596 (1.330,1.862) Pinus nigra -4.782 (-4.929,-4.635) 0.923 (0.739,1.106) Pinus pinaster -4.011 (-4.144,-3.877) 1.687 (1.493,1.882) Pinus pinea -4.042 (-4.352,-3.732) 0.665 (0.157,1.174) Pinus sylvestris -5.381 (-5.543,-5.219) 1.554 (1.406,1.702) Pinus uncinata -5.966 (-6.569,-5.363) 2.549 (1.779,3.320) Quercus faginea -5.325 (-5.543,-5.107) 0.858 (0.538,1.177) Quercus ilex -5.242 (-5.341,-5.142) 0.788 (0.620,0.956) Quercus petraea -6.562 (-7.281,-5.843) 1.988 (1.391,2.586) Quercus pyrenaica -4.640 (-4.771,-4.508) 1.051 (0.898,1.204) Quercus suber -5.124 (-5.601,-4.647) 1.434 (0.772,2.097) Appendix C ABC recruitment model 152 Figure C.1 Predicted vs observed crown diameters fitted using the chosen crown diameter model (model 10 in table C.2) for each of the 30 species for which we had >50 measurements in the dataset.. The one to one relationship is shown by the red line. Appendix C ABC recruitment model 153 Appendix C ABC recruitment model 154 Figure C.2 Predicted and observed diameters fitted using the chosen growth model (model 3 in table C.4). Growth was predicted separately for each species using initial stem size (DBH1) and CAIh, and final observed diameter (DBH2) is shown against predicted final diameter (pDBH2). The one to one relationship is shown by the red line. Appendix C ABC recruitment model 155 Figure C.3 Predicted and observed annual mortality fitted using the chosen mortality model (model 3 in table C.5). Mortality was predicted separately for each species using CAIh, and average rates for each species are shown with their 95% credible intervals. The one to one relationship is shown by the red line. 156 Appendix D Simulation model 157 Appendix D: Simulation model Data and species selection The PPA modelling framework uses individual-based models of tree growth, mortality and allometry. We used data from the second and third Spanish forest inventories (IFN2 (1986-1996) and IFN3 (1997-2007), (MMA, 1996, 2007)) to derive growth and mortality functions, and allometry data from IFN2 only. We selected species for which we had at least 1000 points with which to parameterise the mortality model and at least 100 points to parameterise the growth and allometry models. We used a higher limit for the mortality model as it is a rare event with a discrete response (alive/dead) and requires more data to accurately parameterise the model. There were 14 species which satisfied this requirement (see Table D.1). Data on height and stem diameter were taken from IFN2. Spanish forests have a long history of human management, so for five species that are commonly coppiced/pollarded we used a smaller database of trees that showed no signs of cutting at the time the inventory was taken (these were Quercus pyrenaica, Q. faginea, Q. ilex, Q. suber and Castanea sativa). For a subset of the database (around 150,000 stems of 14 species of silvicultural interest), two measurements of crown diameter were recorded for around four trees per plot. For both stem diameter and crown diameter we used the average of the two measurements taken in the inventory in the model. The time between the two surveys ranged from 6-13 years, with an average of 11 years. In order to compare between IFN2 and IFN3 to derive growth and mortality rates we used only plots whose locations had been positively identified in IFN3. We also removed all plots not classified as thinning (stand density decreasing and mean stem size increasing over the time period of the two surveys), leaving 11,057 plots for the growth models. For mortality analyses we also removed plots which had evidence of management (recorded in IFN3), leaving 8,783 plots. Submodel fitting methods: Growth, mortality and allometry To construct the models and estimate mean and 95% credible intervals for their parameters we used an adaptive MCMC Metropolis algorithm to fit different functional forms and compared them using the Akaike information criterion (AIC, Akaike, 1974) (the MCMC algorithm is described in full in Chapters Two and Four). We used a burn-in period of between 250,000 and 500,000 (with a longer burn-in for models with more parameters) and a sampling period of 250,000 iterations, retaining every 100 th sample parameter set. Details of the priors used are given in each section. All models were fitted using an adaptive Metropolis algorithm written in C (complied using MS Visual Studio 2008). We took climate data from Gonzalo Jiménez (2008), the same data used in Chapter Four, to test as Appendix D Simulation model 158 possible predictors for each model. We also used canopy area of taller trees, CAIh (Caspersen et al., 2011; Coomes et al., 2012), as derived for the Spanish data in Chapter Five, as a measure of the competition experienced by each tree, and tested this as a possible predictor for each model. Growth To model growth we used a simple power function dependent on stem diameter (DBH): (Eqn D.1) where and are parameters to be estimated, with priors set as U(-10,10). This integrates, for initial DBH = DBH0 and final predicted DBH (after t years of growth) = pDBHt, to give: (Eqn D.2) In order to incorporate the size-dependent growth decline we also used a simpler form of the power law, which gives (Eqn D.3) We used a normal distribution with standard deviation increasing linearly with initial size (DBH0), so that to compare real and predicted DBHt we used: (Eqn D.4) where ρ0 and ρ1 are parameters to be estimated (with priors U(0,10) and U(-10,10) respectively). This model had corresponding likelihood within the MCMC run (with data X and model M dependent on parameters θ): π σ σ (Eqn D.5) We tested different functional forms for growth by incorporating the effects of competition and climate into the coefficient . We modelled the effect of competition on growth using the area of taller trees (CAIh) as a predictor of competition, applied in a two-parameter functional form proposed by Coomes and Allen (2007) assuming that the assimilation rate of the plant depends non-linearly on the light available, as a multiple of the coefficient of the form: (Eqn D.6) where γ and δ are parameters to be estimated, with priors set as U(-10,10). To model the effect of temperature on growth we used Boltzmann-Arrhenius function which describes the temperature-dependency of metabolic rates (e.g. Enquist et al., 2003; Coomes and Allen, 2007), as a multiple of the coefficient of the form: (Eqn D.7) where k is the Boltzmann constant (8.62 × 10 -5 eV K -1 ), T is the absolute temperature in Kelvin and E is the average activation energy (characterising the effect of temperature on biochemical reaction rates, Allen et al., 2005). The value of E varies between different organisms but has been found to be Appendix D Simulation model 159 around 0.62 eV for plants (Gillooly et al., 2001). We therefore fit the parameter E during the MCMC algorithm, but with a constrained prior of U(0.55,0.7) to reflect this. To model the effect of annual precipitation levels and drought length on growth we fit simple linear functions as multiples of the coefficient . We also tested an alternative relationship between average annual temperature and growth, using a simple linear function of the same form: (Eqn D.8) where η, λ and ω were parameters to be estimated, with priors set as U(-10,10). We normalised values for annual precipitation (mean = 862 mm, standard deviation = 378 mm) and average annual temperature (mean = 12°C, standard deviation = 3°C) to avoid parameter convergence problems. We tested different combinations of these models as coefficients of the parameter (equation D.1), initially just with one predictor at a time and subsequently with combinations of predictors to find the best fit according to AIC (see Table D.1 for model forms). All parameters were species- specific. Mortality We modelled annual probability of mortality for each individual tree i, as P(mortality, i). Since P(mortality, i) must lie between 0 and 1, we used a logistic transformation mortality, (Eqn D.9) where ki (which can vary from ± ) is a function of the predictor variables. This had corresponding likelihood mortality, if tree survived mortality, if tree died (Eqn D.10) We used the method describe in Chapter Two to fit these models. Priors for all parameters were set as U[-10,10]. We tested different combinations of predictors within the parameter ki (equation D.9) to choose the best fit model according to AIC (see Table D.3 for model forms), and normalised annual precipitation values and average annual temperature values as before. All parameters were species- specific. Crown diameter and height allometry For the PPA model we required models for height and crown diameter allometry. For the crown diameter allometry we used the equations derived in Chapter 5. The best fit model was a seven parameter model with size dependent standard deviation which predicted crown diameter (CD) as: (Eqn D.11) where ν0- ν6 are species-specific fitted parameters. For the height allometry we used a simplified version of the model in Chapter Four. We fit a power function to describe the relationship between height (H) and stem diameter with size dependent standard deviation: Appendix D Simulation model 160 (Eqn D.12) where ϕ0- ϕ7 are species-specific parameters to be fitted. For both equations we used normalised annual precipitation values and average annual temperature values as before Results We found that for both growth and mortality the best fit model had size, competition and climatic dependencies, and these high complexity models were the best fit to the majority of species representing the majority of data (Tables D.2 and D.3). The temperature dependency of growth was not well represented by the Boltzmann-Arrhenius function (equation D.7), as the AIC was worse than a simple linear function (Table D.2). We therefore discarded this function and used a linear equation to describe the relationship between growth and temperature. Parameter values and 95% credible intervals for the best fit growth and mortality functions are given in Tables D.4 and D.5, and the parameter values and intervals for the height-diameter relationship (equation D.12) are shown in Table D.6. We found that both the size and competition dependency of growth rates was strong (Fig. D.1). All species had increasing growth with size, and most showed a strong decline in growth rates in larger sizes. All species showed a substantial decrease in growth rates in more competitive environments (with higher CAIh). Responses of growth to changes in annual precipitation and drought rate were less consistent, but there was also no clear pattern of observed changes in growth along these gradients (Fig. D.1). However we found that, taking all data together, growth increased with temperature in cooler areas and decreased with temperature in warmer areas, and the predictions of the model were able to capture this pattern (compare black and blue lines, Fig. D.1). Mortality rates were most strongly dependent on size and competition (Fig D.2). All species had high mortality rates in small stems, and lower mortality rates in larger stems, with the rates appearing to reach a minimum around 30-40 cm DBH. However, unlike the US mortality rates in Chapter Two, we did not find evidence of a U-shaped size dependency of mortality for most species (with higher mortality in larger stems). The model predicted for all species that mortality increases with temperature, drought and precipitation, but these did not match the observed patterns as well (Fig. D.2), possibly due to covariance with other factors (e.g. plots with higher competition in wetter areas). The relationship between height and size, competition and climate are shown in Fig D.3 and discussed further in Chapter Four. Appendix D Simulation model 161 Table D.1 Details of the data used for each model for the 14 species included in the analysis. Table shows the amount of data used for each species for each model and mean values of all predictors for each species (DBH= diameter breast height (mm), CAIh= crown area of taller trees (ha ha -1 ), DL= drought length (months), AVT = average annual temperature (°C) and PA= annual precipitation (mm year 1 )). Details of the crown width allometry models are given in Chapter Five, "N/A" indicates a species for which we did not have crown data and so assigned a mean crown allometry based on nearest phylogenetic neighbour. Species name Growth data Mortality data Height data Crown width data Mean DBH Mean CAIh Mean DL Mean AVT Mean PA Pinus sylvestris 35272 30684 54457 2039 226.2 0.5 0.7 8.9 974.9 Pinus uncinata 2954 2596 3873 150 232.2 0.3 0.0 5.7 1218. 3 Pinus pinea 6971 5896 11210 1080 245.0 0.3 3.2 14.3 613.7 Pinus halepensis 14875 12196 24191 2973 186.5 0.3 2.8 13.9 549.3 Pinus nigra 14998 12187 22672 1289 199.2 0.4 1.4 10.8 810.7 Pinus pinaster 40363 32620 73836 3533 234.5 0.3 2.2 12.1 838.6 Quercus robur 1181 1952 5685 289 277.5 0.6 0.6 11.9 1335. 3 Quercus petraea 3613 1056 2020 107 239.3 0.6 0.1 9.3 1090. 2 Quercus pyrenaica 1354 7160 877 866 196.8 0.5 1.4 10.8 963.8 Quercus faginea 7714 2233 478 471 180.5 0.4 1.2 11.4 825.6 Quercus ilex 3298 9598 3581 3570 213.2 0.3 2.2 13.3 749.6 Quercus suber 11248 3673 762 756 286.2 0.3 2.6 15.4 880.8 Fagus sylvatica 6354 5423 8697 303 266.9 0.7 0.1 9.0 1164. 2 Castanea sativa 2042 1618 170 168 337.3 0.6 0.9 12.4 1213. 9 Appendix D Simulation model 162 Table D.2 Comparison of different models for the coefficient of the annual growth model (parameter α in equation D.1), showing the different functional forms tested. DBH is stem diameter (cm), CAIh is the crown area of taller trees (ha ha -1 ), DL is drought length in months, AVT is average annual temperature (in °C, and nAVT is normalised as nAVT=(AVT-12)/3) and PA is annual precipitation (in mm year -1 , rescaled as (precipitation– 862)/378). are parameters that were estimated by the MCMC algorithm. Models are compared using the Akaike Information Criterion (AIC), for which the total for all species is shown. As models were fitted separately for each species we calculated the AIC for each species' models and compared these. The number of species for whom each model was the best (according to AIC) is also shown, as well as the total percentage of the data these species represent. Modles 0-11 were fitted using the integrated growth function (equation D.2), whilst model 12 was fitted using the multiplicative growth function (equation D.3). Model # Annual growth model coefficient (α in equation D.1) Parameters AIC AIC Rank # species best model % data best model 0 4 727574 .6 12 0 0.0 1 6 716597 .7 7 0 0.0 2 5 725831 .1 9 0 0.0 3 5 726928 .8 11 0 0.0 4 5 725880 .01 10 0 0.0 5 5 727603 13 0 0.0 6 7 714393 .3 4 0 0.0 7 7 716481 .6 6 1 2.6 8 7 714955 .68 5 2 6.9 9 7 716626 8 0 0.0 10 9 714359 3 0 0.0 11 9 712268 1 3 8.1 12 10 712724 2 8 82.4 Appendix D Simulation model 163 Table D.3 Comparison of different models for annual probability of mortality, showing the different functional forms tested. DBH is stem diameter (mm), CAIh is the crown area of taller trees (ha ha -1 ), DL is drought length in months, AVT is average annual temperature (in °C, rescaled as (temperature – 12)/3) and PA is annual precipitation (in mm year-1, rescaled as (precipitation– 862)/378). τ0 – τ6 are parameters that were estimated by the MCMC algorithm. Models are compared using the Akaike Information Criterion (AIC), for which the total for all species is shown. As models were fitted separately for each species we calculated the AIC for each species' models and compared these. The number of species for whom each model was the best (according to AIC) is also shown, as well as the total percentage of the data these species represent. Model # Logit (Annual probability of mortality) (ki in equation D.9) Parameters AIC AIC Rank # species best model % data best model 0 1 112408 .4 7 0 0.0 1 3 109604 .3 6 0 0.0 2 4 107969 .4 2 1 1.5 3 4 109136 .1 3 0 0.0 4 4 109279 .4 5 1 2.8 5 4 109212 .8 4 0 0.0 6 7 107092 1 12 95.6 Appendix D Simulation model 164 Table D.4 Parameter values for the chosen fit model (number 12 in Table D.2) for annual growth (cm year -1 ) for each species showing Bayesian mean and 95% credible interval (CI). Species Annual growth model parameters Castanea sativa Mean 1.972 0.047 2.779 0.482 0.138 0.358 -0.105 -0.067 0.248 0.012 95% CI (1.972, 1.972) (0.047, 0.047) (2.779, 2.779) (0.482, 0.482) (0.138, 0.138) (0.358, 0.358) (-0.105, -0.105) (-0.067, -0.067) (0.248, 0.248) (0.012, 0.012) Fagus sylvatica Mean 1.107 0.039 0.337 0.233 0.995 1.131 0.310 -0.149 -0.004 0.002 95% CI (1.036, 1.179) (0.036, 0.042) (0.243, 0.520) (0.157, 0.336) (0.549, 1.788) (0.841, 1.503) (0.290, 0.328) (-0.222, -0.074) (-0.031, 0.023) (0.000, 0.005) Pinus halepensis Mean 1.840 0.019 2.107 0.147 0.551 0.997 0.020 -0.066 0.205 0.008 95% CI (1.775, 1.905) (0.016, 0.022) (1.025, 2.944) (0.032, 0.243) (0.413, 0.740) (0.887, 1.148) (-0.009, 0.048) (-0.076, -0.057) (0.175, 0.233) (0.003, 0.013) Pinus nigra Mean 1.890 0.005 1.541 0.386 0.281 0.872 -0.125 -0.021 0.243 0.031 95% CI (1.835, 1.943) (0.003, 0.008) (0.568, 2.885) (0.284, 0.459) (0.227, 0.345) (0.775, 1.033) (-0.154, -0.096) (-0.032, -0.010) (0.225, 0.260) (0.026, 0.035) Pinus pinaster Mean 2.530 0.021 0.840 0.195 0.669 1.356 -0.001 0.026 0.165 0.011 95% CI (2.474, 2.584) (0.019, 0.023) (0.634, 1.143) (0.142, 0.251) (0.547, 0.795) (1.197, 1.506) (-0.015, 0.014) (0.016, 0.037) (0.154, 0.176) (0.009, 0.014) Pinus pinea Mean 2.401 0.001 1.554 0.065 0.544 0.832 -0.105 0.135 0.242 0.003 95% CI (2.316, 2.491) (-0.003, 0.004) (0.615, 2.880) (0.007, 0.162) (0.365, 0.857) (0.645, 1.164) (-0.127, -0.083) (0.106, 0.165) (0.208, 0.273) (0.000, 0.006) Pinus sylvestris Mean 2.258 0.000 2.152 0.202 0.388 0.700 -0.004 0.089 0.146 0.023 95% CI (2.223, 2.296) (-0.002, 0.001) (1.138, 2.925) (0.151, 0.289) (0.327, 0.440) (0.660, 0.759) (-0.021, 0.012) (0.078, 0.100) (0.132, 0.161) (0.021, 0.027) Pinus uncinata Mean 1.644 0.005 1.840 0.039 0.553 0.987 0.136 0.845 0.006 0.013 95% CI (1.530, 1.751) (0.001, 0.010) (0.817, 2.920) (0.002, 0.126) (0.399, 0.873) (0.762, 1.318) (0.097, 0.170) (0.637, 1.076) (-0.055, 0.072) (0.009, 0.018) Quercus faginea Mean 1.040 0.025 0.355 0.345 0.385 1.296 0.272 -0.039 0.179 0.008 95% CI (0.963, 1.117) (0.021, 0.029) (0.106, 1.218) (0.185, 0.486) (0.140, 0.947) (0.696, 2.080) (0.230, 0.312) (-0.061, -0.015) (0.128, 0.230) (0.001, 0.014) Quercus ilex Mean 1.854 0.007 1.668 0.040 0.313 0.761 0.174 -0.014 0.141 0.007 95% CI (1.813, 1.895) (0.006, 0.009) (0.466, 2.918) (0.003, 0.100) (0.245, 0.545) (0.630, 1.084) (0.136, 0.211) (-0.032, 0.006) (0.098, 0.182) (0.005, 0.010) Quercus robur Mean 1.495 0.082 0.915 0.039 1.836 1.407 0.384 -0.091 -0.052 0.004 95% CI (1.349, 1.645) (0.076, 0.088) (0.555, 2.006) (0.001, 0.133) (0.776, 4.160) (0.893, 2.090) (0.299, 0.466) (-0.130, -0.051) (-0.090, -0.008) (0.001, 0.008) Quercus petraea Mean 1.387 0.017 1.494 0.332 0.179 0.620 0.177 0.131 0.116 0.006 95% CI (1.262, 1.513) (0.012, 0.022) (0.198, 2.920) (0.171, 0.523) (0.103, 0.377) (0.458, 1.040) (0.132, 0.220) (0.029, 0.240) (0.030, 0.209) (0.001, 0.013) Quercus pyrenaica Mean 1.196 0.042 0.597 0.162 0.567 1.251 0.227 -0.052 0.044 0.003 95% CI (1.140, 1.253) (0.039, 0.045) (0.287, 2.288) (0.092, 0.245) (0.222, 0.941) (0.737, 1.687) (0.193, 0.258) (-0.070, -0.033) (0.017, 0.073) (0.000, 0.007) Quercus robur Mean 1.495 0.082 0.915 0.039 1.836 1.407 0.384 -0.091 -0.052 0.004 95% CI (1.349, 1.645) (0.076, 0.088) (0.555, 2.006) (0.001, 0.133) (0.776, 4.160) (0.893, 2.090) (0.299, 0.466) (-0.130, -0.051) (-0.090, -0.008) (0.001, 0.008) Quercus suber Mean 1.181 0.054 1.604 0.022 0.172 0.603 0.046 0.415 0.369 0.003 95% CI (1.066, 1.291) (0.050, 0.059) (0.437, 2.900) (0.001, 0.072) (0.139, 0.235) (0.432, 0.812) (-0.059, 0.169) (0.326, 0.515) (0.277, 0.456) (0.000, 0.006) Appendix D Simulation model 165 Table D.5 Parameter values for the best fit model (number 6 in Table D.3) for annual mortality rate (stems stem -1 year -1 ) for each species showing Bayesian mean and 95% credible interval (CI). Species Annual mortality model parameters Castanea sativa Mean -3.458 -0.005 -0.001 0.587 0.263 0.023 0.012 95% CI (-3.920, -3.015) (-0.008, -0.003) (-0.001, 0.000) (0.394, 0.792) (0.145, 0.387) (-0.274, 0.328) (-0.157, 0.182) Fagus sylvatica Mean -4.017 -0.017 -0.004 0.780 0.524 0.099 -0.201 95% CI (-4.814, -3.302) (-0.024, -0.009) (-0.005, -0.003) (0.614, 0.942) (0.183, 0.855) (-0.118, 0.304) (-0.345, -0.065) Pinus halepensis Mean -3.521 -0.014 -0.004 1.312 -0.071 0.076 0.060 95% CI (-4.019, -2.976) (-0.022, -0.008) (-0.005, -0.003) (1.124, 1.499) (-0.134, -0.003) (-0.064, 0.212) (-0.118, 0.241) Pinus nigra Mean -3.969 -0.005 0.000 0.908 0.027 0.197 0.218 95% CI (-4.263, -3.612) (-0.009, -0.003) (-0.001, 0.001) (0.770, 1.037) (-0.035, 0.089) (0.065, 0.335) (0.137, 0.302) Pinus pinaster Mean -1.758 -0.023 -0.003 0.610 0.089 -0.096 0.237 95% CI (-2.000, -1.503) (-0.025, -0.020) (-0.003, -0.003) (0.528, 0.688) (0.051, 0.127) (-0.146, -0.040) (0.198, 0.274) Pinus pinea Mean -2.060 -0.019 -0.003 0.395 0.193 -0.727 0.223 95% CI (-2.580, -1.453) (-0.024, -0.013) (-0.003, -0.002) (0.160, 0.634) (0.093, 0.295) (-0.889, -0.572) (-0.002, 0.446) Pinus sylvestris Mean -3.725 -0.010 -0.002 0.891 0.265 0.067 0.189 95% CI (-3.981, -3.456) (-0.013, -0.007) (-0.003, -0.002) (0.824, 0.955) (0.223, 0.306) (-0.010, 0.146) (0.133, 0.247) Pinus uncinata Mean -5.296 0.013 -1.397 0.502 2.504 -0.154 0.003 95% CI (-5.964, -4.622) (-0.935, 0.932) (-2.850, -0.114) (0.051, 0.963) (2.063, 2.906) (-0.463, 0.187) (-0.262, 0.268) Quercus faginea Mean -2.813 -0.028 -0.003 0.627 0.192 0.284 0.412 95% CI (-3.906, -1.902) (-0.039, -0.015) (-0.004, -0.002) (0.291, 0.940) (0.000, 0.392) (-0.046, 0.616) (0.079, 0.731) Quercus ilex Mean -2.418 -0.020 -0.003 0.037 -0.048 0.327 0.232 95% CI (-2.781, -2.051) (-0.024, -0.016) (-0.003, -0.002) (-0.146, 0.214) (-0.109, 0.009) (0.224, 0.428) (0.088, 0.366) Quercus robur Mean -3.400 -0.013 -0.002 0.652 -0.118 0.401 0.181 95% CI (-4.555, -2.438) (-0.022, -0.003) (-0.003, -0.001) (0.377, 0.932) (-0.339, 0.098) (0.011, 0.784) (-0.021, 0.388) Quercus petraea Mean -5.625 0.028 -1.375 1.999 -0.084 0.377 -0.574 95% CI (-6.100, -5.139) (-0.905, 0.926) (-2.877, -0.057) (1.609, 2.368) (-0.718, 0.501) (0.005, 0.724) (-1.082, -0.086) Quercus pyrenaica Mean -2.973 -0.013 -0.002 0.795 0.094 0.143 -0.146 95% CI (-3.382, -2.603) (-0.017, -0.009) (-0.003, -0.002) (0.662, 0.925) (0.016, 0.174) (0.001, 0.291) (-0.247, -0.048) Quercus suber Mean -3.235 -0.025 -0.004 0.318 -0.019 0.630 -0.133 95% CI (-3.943, -2.591) (-0.032, -0.018) (-0.004, -0.003) (-0.056, 0.695) (-0.144, 0.111) (0.299, 0.947) (-0.357, 0.093) Appendix D Simulation model 166 Table D.6 Parameter values for the height-diameter allometric model (equation D.11, fit for both height and DBH in cm) for each species showing Bayesian mean and 95% credible interval (CI). Species Height-diameter model parameters (equation D.12) Castanea sativa Mean 369.743 -1.142 615.467 0.216 33.035 23.476 -43.610 -21.590 95% CI (297.406, 437.934) (-2.408, 0.750) (498.539, 753.552) (0.157, 0.275) (3.870, 48.916) (-20.301, 48.483) (-49.744, -31.050) (-47.930, 28.712) Fagus sylvatica Mean 236.400 6.091 530.878 0.398 -15.437 49.639 -5.155 -47.601 95% CI (223.716, 248.372) (5.626, 6.625) (515.242, 545.224) (0.389, 0.408) (-18.611, -12.177) (48.733, 49.990) (-16.257, 5.556) (-49.855, -43.692) Pinus halepensis Mean 112.928 5.469 172.257 0.576 12.758 26.670 -9.646 21.214 95% CI (112.928, 112.928) (5.469, 5.469) (172.257, 172.257) (0.576, 0.576) (12.758, 12.758) (26.670, 26.670) (-9.646, -9.646) (21.214, 21.214) Pinus nigra Mean 63.340 8.263 119.976 0.718 5.597 9.465 -3.274 11.356 95% CI (58.310, 68.306) (7.976, 8.541) (117.508, 123.144) (0.710, 0.725) (5.040, 6.204) (8.604, 10.379) (-3.656, -2.873) (10.181, 12.473) Pinus pinaster Mean 163.963 3.805 173.859 0.617 15.319 19.176 -8.098 4.954 95% CI (160.362, 167.399) (3.654, 3.956) (170.958, 175.953) (0.613, 0.622) (14.816, 15.842) (18.513, 19.871) (-8.549, -7.662) (3.863, 6.013) Pinus pinea Mean 123.963 3.022 92.684 0.688 -0.818 13.146 -3.750 15.567 95% CI (117.313, 130.675) (2.747, 3.299) (87.898, 97.842) (0.673, 0.701) (-2.244, 0.573) (12.209, 14.129) (-4.318, - 3.174) (13.906, 17.111) Pinus sylvestris Mean 128.255 7.240 130.683 0.693 -0.216 0.399 -1.236 14.094 95% CI (124.204, 132.485) (7.034, 7.435) (128.295, 132.951) (0.688, 0.699) (-0.787, 0.367) (-0.254, 1.086) (-1.614, - 0.829) (13.385, 14.849) Pinus uncinata Mean 87.861 5.892 259.672 0.558 8.980 37.099 -37.428 -31.849 95% CI (87.861, 87.861) (5.892, 5.892) (259.672, 259.672) (0.558, 0.558) (8.980, 8.980) (37.099, 37.099) (-37.428, -37.428) (-31.849, -31.849) Quercus faginea Mean 69.323 6.225 189.159 0.503 4.933 8.642 -12.874 31.954 95% CI (39.816, 98.931) (4.650, 7.902) (162.908, 220.221) (0.452, 0.549) (-4.279, 14.826) (-0.197, 17.642) (-18.342, -7.720) (12.228, 47.352) Quercus ilex Mean 84.725 1.571 251.152 0.377 2.358 21.716 -26.286 -28.393 95% CI (79.055, 90.589) (1.359, 1.779) (240.666, 261.799) (0.364, 0.390) (-1.413, 6.214) (18.685, 25.039) (-28.701, -23.988) (-38.525, -17.935) Quercus robur Mean 196.219 4.935 380.613 0.416 -19.204 23.682 -7.266 -48.759 95% CI (184.461, 208.673) (4.477, 5.412) (365.490, 397.081) (0.403, 0.427) (-23.468, -15.113) (16.760, 30.922) (-11.060, -3.547) (-49.972, -45.982) Quercus petraea Mean 155.143 8.745 275.585 0.496 -4.780 2.417 -2.774 -22.563 95% CI (129.771, 180.021) (7.494, 10.072) (252.052, 299.669) (0.469, 0.524) (-11.854, 2.386) (-3.511, 8.462) (-11.781, 6.537) (-32.762, -13.754) Quercus pyrenaica Mean 113.468 5.931 306.082 0.409 28.361 19.223 -21.095 16.261 95% CI (87.341, 141.647) (4.621, 7.280) (267.439, 351.366) (0.367, 0.447) (20.053, 37.476) (8.121, 31.255) (-29.189, -13.354) (-3.981, 35.384) Quercus suber Mean 136.004 0.902 183.058 0.446 30.393 19.249 -11.918 -29.099 95% CI (116.457, 155.930) (0.344, 1.479) (157.201, 205.918) (0.410, 0.489) (23.925, 37.140) (11.236, 27.702) (-15.277, -8.625) (-45.093, -13.147) Appendix D Simulation model 167 Figure D.1 Annual growth rate (cm year -1 ) for the 14 selected species, with predictions plotted for each species (grey) and for all species together (blue), along with the observed growth rate along each gradient (black). Predicted growth was calculated along the gradient of each predictor by varying only that predictor and holding all others at the species' average (see table D.1 for average values), and then binning the data along the gradient to give average and 95% CIs. 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 DBH (cm) 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 CAIh(ha ha 1 ) 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Drought Length (months) 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Average Annual Temperature C 500 1000 1500 2000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Annual Precipitation (mm yr 1 ) All data predicted All data observed Individual speices predicted A n n u a l g ro w th r a te ( c m y r1 ) Appendix D Simulation model 168 Figure D.2 Annual mortality rate (stems stem -1 year -1 ) for the 14 selected species, with predictions plotted for each species (grey) and for all species together (blue), along with the observed mortality rate along each gradient (black). Predicted mortality was calculated along the gradient of each predictor by varying only that predictor and holding all others at the species' average (see table D.1 for average values), and then binning the data along the gradient to give average and 95% CIs. 20 40 60 80 0.001 0.002 0.005 0.010 0.020 0.050 0.100 DBH (cm) 0.0 0.5 1.0 1.5 2.0 0.001 0.002 0.005 0.010 0.020 0.050 0.100 CAIh(ha ha 1 ) 0 1 2 3 4 5 6 0.001 0.002 0.005 0.010 0.020 0.050 0.100 Drought Length (months) 5 10 15 20 0.001 0.002 0.005 0.010 0.020 0.050 0.100 Average Annual Temperature C 500 1000 1500 2000 0.001 0.002 0.005 0.010 0.020 0.050 0.100 Annual Precipitation (mm yr 1 ) All data predicted All data observed Individual speices predicted A n n u a l m o rt a li ty r a te ( s te m s s te m 1 y r 1 ) Appendix D Simulation model 169 Figure D.3 Tree height (m) for the 14 selected species, with predictions plotted for each species (grey) and for all species together (blue), along with the observed heights rate along each gradient (black). Predicted height was calculated along the gradient of each predictor by varying only that predictor and holding all others at the species' average (see table D.1 for average values), and then binning the data along the gradient to give average and 95% CIs. 20 40 60 80 100 5 10 15 20 25 30 DBH (cm) 0.0 0.5 1.0 1.5 2.0 5 10 15 20 25 30 CAIh(ha ha 1 ) 0 1 2 3 4 5 6 5 10 15 20 25 30 Drought Length (months) 5 10 15 20 5 10 15 20 25 30 Average Annual Temperature C 500 1000 1500 2000 5 10 15 20 25 30 Annual Precipitation (mm yr 1 ) All data predicted All data observed Individual speices predicted H e ig h t (m ) Appendix D Simulation model 170 Figure D.4 Range of dominance of the six pine species studied as predicted by the simulation model with early successional dominance determined as the tallest species after 30 years of growth under full light, split by drought length and plotted along gradients of average annual temperature and annual precipitation. 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