Macroscopic insights from
mechanistic ecological network
models in a data void
Yangchen Lin
8 April 2015
Sidney Sussex College, University of Cambridge
A dissertation submitted for the degree of Doctor of Philosophy

在缺乏数据的状态下以机械性生态网模型吸取宏观灼见林扬尘博士学位论文剑桥大

to Man

Contents
prefatory note iii
summary v
1 exposition 1
1.1 A complex systems approach to ecology . . . . . . . . . . . . . . 1
1.2 The network paradigm in ecology . . . . . . . . . . . . . . . . . . 2
1.3 Research direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Modeling strategy . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 The data void . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 stochastic bioenergetic network model 13
2.1 Network construction . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Multispecies dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Environmental noise . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 colour and synchrony of environmental noise 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1 Effects of non-predator-prey interactions . . . . . . . . . 47
3.5.2 Effects of noise colour and synchrony . . . . . . . . . . . 50
4 extinction and invasion do not add up 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5.1 Regression trees . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5.2 Effects of community structure . . . . . . . . . . . . . . . . 90
4.5.3 Effects of extinctor dominance . . . . . . . . . . . . . . . . 91
i
5 collective effect of interaction modifications 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.5.1 Supplementary figures . . . . . . . . . . . . . . . . . . . . . . 110
5.5.2 Plots of unpooled data . . . . . . . . . . . . . . . . . . . . . 113
6 species decline in compartmentalized networks 123
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.6.1 Regression trees . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.6.2 Plots from all treatments . . . . . . . . . . . . . . . . . . . . 143
7 coda 163
literature cited 167
On the cover: the arms of the University of Cambridge,
gules on a cross ermine between four lions passant gardant a book gules (granted 1573).
ii
prefatory note
This work attempts to serve as a dissertation for a degree qualification,
but even more importantly it has emerged symphonically from a voyage of
exploration and discovery of the world and of myself that involved not only
scientific research but also the concurrent pursuit of artistic and spiritual
fulfilment. I am indebted to my supervisor, Professor William J. Sutherland,
who accompanied and guided me on this exhilarating journey. I am also
grateful to Dr. Andrea Manica and Dr. Rufus Johnstone, who helped man
the proverbial Apollo mission control ensuring that the trajectory of my
voyage into the unknown remained within survivable constraints.
There are also others from all over the world who engaged me in
thought-provoking discussions, seeded me with important ideas, and taught
me skills that facilitated the execution of my plan: Stefano Allesina, Tatsuya
Amano, Jordi Bascompte, Francesco Battocchio, Tim Benton, Edouard Bul-
teau, Mark Burgman, Yohay Carmel, Tharsila Carranza, Charlotte Chang,
Zhi Yuan Chor, Damon Civin, David Coomes, Bo Dalsgaard, Lynn Dicks,
Marcin Dziubiński, Anders Eriksson, Lucas Faria, Laurens Geffert, Paulo
Guimarães Jr., Hsun-Yi Hsieh, Martin Hughes, Sonia Kéfi, Krzysztof Kozak,
Philip Krammer, Clinton Leach, Thomas Lewinsohn, Owen Lewis, Nick
Maclaren, Simon Martin, Jane Memmott, Mark Novak, Jose Manuel Ochoa-
Quintero, Steven Orzack, Amael Paillex, Kelvin Peh, Owen Petchey, Mathias
Pires, Philip Platts, Paulo Inácio Prado, Dan Reuman, Lasse Ruokolainen,
Jane Shevtsov, Marius Somveille, Phillip Staniczenko, Daniel Stouffer, Si
Tang, Elisa Thébault, Antonio Tironi, Kristian Trøjelsgaard, Marco van der
Leij, Weigang Yan, Justin Yeakel and Veronica Zamora. Furthermore, I am
grateful to the many anonymous peer reviewers who scrutinized my submit-
ted and resubmitted manuscripts in detail and provided invaluable feedback.
And not least my graduate tutors Jillaine Seymour, Berry Groisman, Julius
Ross and Ian Black, the college and department administrators Suzannah
Horner, Angela Parr-Burman, Stephanie Prior, Linda Wheatley and Alice
Jago, and my compatriots at the Conservation Science Laboratory, Depart-
iii
ment of Zoology and Sidney Sussex College, all of whom conspired to
create a genial and inspiring environment.
My scientific endeavours have also benefited from generous research and
travel grants from the São Paulo Research Foundation (fapesp), South Amer-
ican Institute for Fundamental Research, Nanyang Technological University,
Sidney Sussex College, nerc Centre for Ecology and Hydrology and the
British Ecological Society.
Parts of this dissertation were published [1–4] but have been brought
up to date and improved in the light of subsequent ideas and literature.
Insights from here have also been contributed to coauthored publications
[5, 6].
Declaration. This dissertation is the result of my own work and includes
nothing which is the outcome of work done in collaboration except as
declared in the preface and specified in the text. It is not substantially the
same as any that I have submitted, or is being concurrently submitted, for a
degree or diploma or other qualification at the University of Cambridge or
any other university or similar institution except as declared in the preface
and specified in the text. I further state that no substantial part of my
dissertation has already been submitted, or is being concurrently submitted,
for any such degree, diploma or other qualification at the University of
Cambridge or any other university of similar institution except as declared
in the preface and specified in the text. The dissertation does not exceed
the prescribed word limit for the relevant Degree Committee. ©
Yangchen Lin
Cambridge, spring 2015
iv
summary
Complexity science has come into the limelight in recent years as the scien-
tific community begins to grapple with higher-order natural phenomena that
cannot be fully explained via the behaviour of components at lower levels
of organization. Network modeling and analysis, being a powerful tool that
can capture the interconnections that embody complex behaviour, has there-
fore been at the forefront of complexity science. In ecology, the network
paradigm is relatively young and there remain limitations in many ecological
network studies, such as modeling only one type of species interaction at
a time, lack of realistic network structure, or non-inclusion of community
dynamics and environmental stochasticity. I introduce bioenergetic network
models that bring together for the first time many of the fundamental
structures and mechanisms of species interactions present in real ecological
communities. I then use these models to address some outstanding questions
that are relevant to understanding ecological networks at the systems level
rather than at the level of subsets of interactions. Firstly, I find that realis-
tic red-shifted environmental noise, and synchrony of species responses to
noise, are associated with increased variability in ecosystem properties, with
implications for predictive ecological modeling which usually assumes white
noise. Next, I look at simultaneous species extinction and invasion, finding
that as their individual impacts increase, their combined impact becomes
decreasingly additive. In addition, the greater the impact of extinction or
invasion, the lesser their reversibility via reintroduction or eradication of
the species in question. For modifications of pairwise species interactions
by third-party species, a phenomenon that has so far been studied one
interaction at a time, I find that the many interaction modifications that
occur concurrently in a community can collectively have systematic effects
on total biomass and species evenness. Finally, examining a higher level
of organization in the form of compartmentalized networks, I find that
the relationship between intercompartment connectivity and the impacts of
species decline depends considerably on network topology and whether the
consumer-resource functional response is prey- or ratio-dependent. Over-
v
all, the results vary considerably across model communities with different
parameterizations, underscoring the contingency and context dependence
of nature that scientists and policy makers alike should no longer ignore.
This work hopes to contribute to a growing multidisciplinary understand-
ing, appreciation and management of complex systems that is fundamentally
transforming the modern world and giving us insights on how to live more
harmoniously within our environment.
vi
1 exposition
1.1 a complex systems approach
to ecology
The twenty-first century has been marked by the advent of a new frontier
in humanity’s scientific enterprise: the so-called complexity science. It arose
from the recognition that many locally interacting agents can give rise
to a system with unexpected macroscopic properties, and that this system
furthermore adapts to changes in its external environment over time and
is often far from equilibrium, its trajectory highly dependent on conditions
early in its history. Consequently, such systems are not amenable to mathe-
matical analysis, and their future possible states are ‘not finitely prestatable’
[7]. Despite their real-world prevalence in the guise of financial markets,
ecosystems, ant colonies and many others, their interesting and potentially
beneficial or destructive behaviours are hardly understood or controllable. In
his seminal monograph in 1997, Prigogine [8] called for scientists to move
on from traditional deterministic approaches and embrace the indeterminism
of complex systems.
I think the next century will be the century of complexity.
Stephen Hawking, San Jose Mercury News (2000)
In ecology, it was realized as early as 1994 that research had (and as of
today, has) been dominated by the approach of studying at most a few species
or interactions at a time [9], inherited from the early days of observation
and description of living organisms. This has partly been wrought by the
difficulty of collecting detailed field data on more than a few selected
components of the system (see §1.3.2), and partly by the popularity of simple
mathematical models for finding analytical solutions of ecological equilibria.
While exact, tidy and easy to interpret, the reductionistic approach does not
encapsulate the complex interactions of nature [9]. Reductionism may be
beginning to take its toll on advancement in ecological research. In a 2014
meta-analysis of more than 18 000 published articles, Low-Décarie et al. [10]
1
Exposition
observed that despite an increasing number of hypotheses being tested per
article in ecology, their explanatory power seemed to be declining. One
possible cause cited was that the simple questions had mostly been answered,
leading ecologists to tackle increasingly complicated questions. Although
correspondingly sophisticated statistical procedures were used to attack such
questions, it is not unreasonable to suppose that even those procedures will
have difficulty explaining and predicting ecological behaviour as long as
only a few components of the system are analyzed for extrapolation to the
system as a whole. A more fundamental shift from reductionistic to holistic
approaches may be needed to bring about the next golden age in ecological
research [11].
1.2 the network paradigm in ecology
Network analysis is a powerful framework that has been applied to complex
nonlinear systems in many disciplines [12], being flexible to the idiosyncrasies
of each. It has been hailed as the tool for addressing the ‘big questions
of contemporary science’ [13]. The versatility of the network approach
for diverse purposes is illustrated, for example, by Schich et al. [14], who
traced the evolution of cultural history via networks of intellectual mobility
reconstructed from birth and death location data of individuals who had
significant impacts on culture.
Solé & Valverde [15] introduced a classification of the structure of many
different types of networks, from the Internet to metabolic maps to food
webs to brain networks, according to three axes: randomness, modularity
and heterogeneity. In this classification, food web structure has moderately
high randomness, low modularity and moderate heterogeneity and lies out-
side the domain of ‘scale-free-like networks’, while ecological mutualistic
networks have low randomness, moderately high modularity and moder-
ate heterogeneity and lie within the domain of ‘scale-free-like networks’.
Proulx et al. [16] reviewed the rising use of network analysis in various
subfields of ecology and evolution and suggested that the time was ripe
for advancing beyond pairwise interactions and building a ‘predictive science
of biological networks’. More specifically to ecology, Bascompte [17] advo-
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cated a network approach to the vital task of understanding how ecological
interactions affect the emergent properties of complex ecosystems under in-
creasing anthropogenic threat, while Brose [18] and Lewinsohn & Cagnolo
[19] highlighted the value of ecological networks for understanding and
predicting the consequences of species loss on ecosystems.
Real ecological networks are highly complex, containing much omnivory
and trophic looping [20]. Many studies of ‘networks’ and ‘food webs’,
however, categorize the nodes into distinct trophic levels, and the inter-
level links only connect adjacent levels. Moreover, such studies usually deal
with a small number (2–3) of trophic levels [21]. It is important to account for
vertical (trophic) diversity, as has been shown in the diversity-functioning
debate [22]; the network approach is powerful in accommodating both
horizontal and vertical (trophic) diversity [23].
Across the multidisciplinary network research landscape, ecological net-
works are becoming recognized for their contribution towards the under-
standing of general laws governing the properties of complex networks. Chiu
& Westveld [24] demonstrated the application of social network analysis to
food webs, while Brummitt et al. [25] drew parallels between power grids
and ecological networks. Even more significantly, analogies between ecolog-
ical and economic networks are getting increasing attention. Simple models
of networks of bank loans have been built to demonstrate shock propa-
gation through network structure, with conceptual comparisons to species
extinctions in ecosystems [26, 27], and the power-law (scale-free) degree1
distribution tendency of both economic and ecological networks has been
found to make them more susceptible to catastrophic collapse [28]. Stability
measures for financial networks are also being developed that have potential
applications to ecology [29].
Ecological network dynamics. Change over time is what really defines the
interesting and important characteristics of ecological communities and com-
plex systems in general, and is integral to any attempt to understand the
impact of perturbation on ecological networks or make predictions for prac-
tical management [16, 30–32]. The study of networks with temporal changes
1The number of nodes (species) to which a given node in the network is connected.
3
Exposition
in information or energy flow between nodes is, however, still an emerging
field in many areas of network science [33–37]. Ecology is no exception;
much research on ecological networks has dealt with static networks, and
there tends to be a disconnect between reductionistic dynamical studies of
trophic modules versus static whole-network studies [38]. The importance of
incorporating dynamics in whole-network studies has been underscored by
the work of Eveleigh et al. [39], who found different outcomes in diversity
cascades caused by outbreak species when realistic non-equilibrium dynamics
were accounted for.
Furthermore, most network-level dynamical ecological studies to date
have dealt exclusively with food webs. Community-level conclusions from
such studies may not hold true in real communities, where multiple non-
predator-prey interaction types co-occur and interact with one another
and with trophic interactions [40]. Forays into dynamical representations
of multiple interaction types are still in their infancy [41]. For example,
Melián et al. [42] incorporated dynamics for plant species in a community
with multiple interaction types but assumed that animal population densities
were constant. Blonder et al. [43] review the methodology for incorporating
temporal dynamics into network analysis, and its application to networks in
ecology and evolution.
1.3 research direction
1.3.1 modeling strategy
The broad aims of this work are to investigate, using theoretical mod-
els, how various interesting ecological phenomena at the mesoscale (species)
level affect community-level behaviour and response to perturbation. I try
to determine the presence or absence of general patterns that apply across
different contexts, which addresses concerns about the predominance of spe-
cific case studies in community ecology with very little general understanding
[44].
Given the long-standing dichotomy between ‘simple mathematical mod-
els’ and ‘large simulation models’ [45], it is essential to justify the simulation
approach this research adopts. Ecological dynamics have traditionally been
4
Yangchen Lin
understood via stability analysis of simple deterministic equations of a few
interacting species [46–49], although community matrices of relatively large
networks are increasingly being analyzed to elucidate the influence of net-
work structure on interaction strengths [50] and the influence of both of
these on stability [51]. Mathematical analysis continues to make valuable
contributions to ecological network science, not least when allied with tech-
niques such as model selection [52]. The so-called generalized modeling
has also been promulgated [53, 54], which examines the dynamical stabilities
of a large number of ecological network replicates with different model
structures and parameter values and therefore avoids making simplifying
assumptions. Like the other analytical approaches, however, it is largely
focused on quantifying deterministic equilibrium stability.
Simulation models, used in my study, are another powerful tool for un-
derstanding ecological network behaviour [55], and complement mathemat-
ical analysis [56], especially in nonequilibrium situations and where model
complexity precludes analytical solutions. I depart from the emphasis on
equilibrium stability, whether of traditional local equilibrium points in pop-
ulation dynamics or of the latest ‘structural stability’ of ecological networks
[57], and focus on tackling the confusion of a reality where nonequilibrium
dynamics prevail [58, 59]. Indeed, in the latest overview of the continuing
debate on model simplicity versus complexity, Evans et al. [60] reiterate
that complex, mechanistic models can be better than simple models when
seeking ecological generality, as well as when attempting specific prediction
outside the boundary of current conditions [61, 62], and that the current
preponderance of ecology on simple models may even be stifling ecological
progress [63]. This is not to say that all models should be complex. We
need to harness the complementary powers of both simple and complex
models to give us the best overall understanding [60]. For example, efforts
to isolate small trophic modules from theoretical or empirical food webs
for the prediction of equilibrium responses to perturbations [64–66] should
be matched by efforts that scale back up from modules to the larger eco-
logical networks in which they are embedded [67], in order to capture all
the feedback loops.
5
Exposition
How does the oft-invoked complexity-stability relationship colour the
dichotomy between simple and complex models? The relationship has been
of central interest in ecology since the pioneering work of May [68, 69].
Despite significant advancements [58, 70–72], the issue of when and how
ecological complexity stabilizes or destabilizes an ecosystem remains a hot
topic of research and debate [73–75]. At the same time, deeper understanding
of this relationship has become ever more relevant to scientists and policy
makers hoping to mitigate potentially debilitating effects of anthropogenic
pressures on ecosystems [74, 76]. My prognosis is that more complex ecolog-
ical networks may be more resilient in overall function due to redundancy
and degeneracy [77], but may also be more prone to unexpected endoge-
nous behaviour caused by internal feedback loops and thresholds, making
the complexity-stability relationship ill-defined. The purpose of my thesis,
however, is not to leap into the fray of the complexity-stability debate which
continues to receive ample attention. Instead, the complexity-stability issue
serves here to highlight that simple and complex systems can behave differ-
ently. Although simple models may capture some of the essential features of
the complex system that exists in reality, they may not capture the rare but
potentially important or catastrophic behaviour, or the unprestatable future
states [7, 78] discussed earlier (§1.1). Conversely, and rarely considered in
ecological research, relationships that exist within simple models might well
be nonexistent in more complex systems with large numbers of alternate
pathways.
There are two main system-level simulation approaches to ecology that
are of the level of complexity pertinent to this thesis: those of ecosystem
ecology and community ecology. The ecosystem ecology approach [79–82] is
an established system dynamics modeling methodology used mostly in marine
ecology [83], where trophic flows of energy and nutrients between functional
compartments in a particular system are specified and various indices of
energy flow are calculated. More rarely, both trophic and nontrophic flows
are included [84]. The ecosystem ecology approach has been used extensively
for the analysis of aquatic ecosystems [85–88] and applied to the assessment
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Yangchen Lin
of environmental sustainability [89] and of the effectiveness of marine
protected areas [90].
Aggregated biotic and abiotic compartments and flows, however, are
perhaps not the most optimal design for studying the effects of species-
species transactions. Moreover, the ‘positive effects’ sometimes referred to
in ecosystem ecology mainly concern negative local interactions having net
positive effects on network-level energetics [79, 91, 92], rather than facilita-
tive interspecific interactions. This is where community ecology complements
ecosystem ecology. Simulation modeling in community ecology is best repre-
sented by bioenergetic modeling of consumer-resource dynamics [41, 93–95],
which has recently acquired increased ecological realism via integration with
models of ecological network structure [96] and with facilitative interactions.
The bioenergetic approach, where species interactions are mechanistically
parameterized in terms of energy and metabolism, is equally amenable to
studying the response of complex ecosystems to environmental changes in-
cluding those brought about by climate change (e.g. Gilbert et al. [97]).
Current research in this area is moving towards a more applied flavour even
if the actual modeling remains theoretical, such as a study of how food web
diversity and modularity mediate the effects of hypothetical pollutants [98].
Olesen et al. [99] (p. 37) suggest that networks of species interactions
may be no more than arbitrary structures constructed by scientists that do
not exist in reality, but the authors do not elaborate. Insights into what
this might mean can be found in the paradigm of agent-based modeling
(abm). A strong case for abm is that ecological interactions really take place
between individual organisms rather than between the non-physical aggrega-
tions known as species. To be reasonably realistic, individual-level network
models have to incorporate at least the two main classes of interactions:
between conspecifics and between individuals of different species—perhaps
this is the true ecological network that actually exists. Most existing abm
studies, however, have dealt with only either conspecific [100] or interspe-
cific [101] interactions at the individual level. An additional challenge is that
abm can become very computationally intensive if many species are involved.
Again, the kind of species-level model used in my study treads a reasonable
7
Exposition
compromise between simplicity and complexity, while incorporating both in-
terspecific and intraspecific interactions, the latter in a phenomenological yet
realistically ‘individual’-inspired way via terms such as density dependence
and cannibalism (§2.1).
1.3.2 the data void
But there are also unknown unknowns—the ones
we don’t know we don’t know.
Donald Rumsfeld
A discourse on modeling cannot be complete without some discussion of
empirical data for model validation. Ecology continues to stand apart from
many other sciences as a particularly data-deficient discipline. It is often
impossible to collect comprehensive data on entire communities. Where this
is attempted, data may be insufficient or biased and the model is usually
restricted to one part of a system at one specific locality. Data shortage is
particularly severe in the case of ecological networks, where it is extremely
difficult to detect the presence of all the interspecific interactions, let alone
measure interaction strengths.
Many data sets of food web structure that are regarded as highly resolved
are nevertheless incomplete [102], encompassing only a subset of taxonomic
or ecological functional groups. For example, there is a ‘food web’ data
set of over 1700 species that comprises only plants and herbivores [103],
and a ‘predator-prey network’ data set of high temporal resolution over
millennial time scales that comprises only mammals [104]. Most data sets
also exclude parasites [105, 106], Preston et al. [107] being a rare exception.
The system is often simply too diverse to sample comprehensively. For
example, 156 ‘kinds of organisms’ (species, distinct life history stage of a
species, or group of closely related species) are recorded in the data set
from tropical forest at El Verde Field Station in Puerto Rico [108], but the
actual number of species in tropical rainforests is widely estimated to be
much higher. Large data sets from marine biomes are probably closer to the
8
Yangchen Lin
actual number of species of the specific system [109–111], but there remains
the aforementioned problem of imperfect observation of interactions. Wirta
et al. [112] recommended combining multiple data sources, including field
observations, lab experiments and molecular data, to gain better resolution of
ecological networks and reduce bias. This, however, may still be insufficient
for diverse systems due to the sheer number of unobserved species and an
even greater number of their interactions. Besides, the structural properties
of any one or a few comprehensively sampled food webs may not be
representative of food webs in general, especially with so much contingency
of food web structure [113] and other ecosystem characteristics [114].
There is another way in which data are incomplete: all the above
examples have only predator-prey interactions. Any given ecosystem is
an intricate web of co-occuring trophic and nontrophic interactions (see
Chapter 3 for further discussion about combining them in research), but
data on these different types of interactions are rarely collected sym-
patrically. There are abundant trophic and nontrophic interaction data,
such as the Interaction Web DataBase at the National Centre for Eco-
logical Analysis and Synthesis, University of California at Santa Barbara
(http://www.nceas.ucsb.edu/interactionweb), and calls continue for ecolo-
gists to pool their data into collective databases in order to harness the
power of big data [115]. Such data, however, are from different localities
and cannot be aggregated to represent a single system. Empirical datasets
on the network structure of multiple interaction types in the same system
are rare, notable examples being those from an agroecosystem in Britain
[116] and from Chilean rocky shores [117].
When one considers superimposing community dynamics on top of net-
work structure, there arises a further problem, that of interaction strengths.
Reliable specific predictions of complex ecological communities require
highly accurate estimates of the interaction strengths [118], but interaction
strengths are very difficult to measure accurately in the field [119], especially
in terms of rates of energy flux with standardized units of measurement
across species. For example, mutualistic interactions are often quantified as
the number of pollinator visits to flowers, but the relationship of visit fre-
9
Exposition
quency to actual nectar consumption or flower fertilization rate is unclear.
Admittedly, more work is needed to develop the empirical domain of a
research topic that has been dominated by theoretical modeling [120].
Some advances have nevertheless been made in fitting mechanistic dy-
namic models to empirical data of medium-sized networks of 24–50 nodes.
Two studies [121, 122] found that when fitted to data on food web structure,
observed biomass and body sizes, such models reproduced seasonal dynamics
and community patterns (such as size-abundance distribution) relatively well
once dynamical parameters were allometrically scaled. This can be useful
where taxonomically comprehensive time series data for dynamical param-
eterization are scarce [122]. These studies, however, involved only trophic
interactions in specific systems; fitting a limited model to data from a specific
locality may not be the most appropriate paradigm in my study searching
for general patterns in communities with multiple interaction types.
There is a popular culture in science that regards models as useless or
even misleading for the real world unless they are validated by sufficient
quantity and quality of data [123]. More complex models are naturally
more susceptible to this criticism. It is indeed the case that policy and
action informed by predictions from such a model are considered risky
[124]. But I do not think this is the whole story. Even with abundant
and quality data, a model may not make specific real-world predictions
to the required accuracy for practical use, even with sensitivity analysis.
In contrast, whether data are sufficient or not, a suite of models based
on sensible, documented assumptions and comprising a range of plausible
model structures and parameter values [125, 126] can provide valuable general
insights and anticipation about how real systems might behave under given
circumstances—as is the premise of this work. Indeed, this so-called Monte
Carlo approach has been advocated even for predictive purposes, as in the
case of complex climate models which have so far not been subject to
stochastic parameterization [127]. In any case, researchers have advocated
general insights rather than specific prediction of ecosystem responses to
disturbance [128], and suggested that the appreciation of and preparedness
for the variability of the future is as important as, if not more important
10
Yangchen Lin
than, prediction [129]. The virtues of the general approach should all the
more be embraced given that the ecosystem-level ‘data vacuum’ [124] cannot
be filled before conservation needs to happen [130, 131].
In summary, my model tries to capture the essential features of the
system by incorporating all the main classes of interaction types and dy-
namics in a succinct way, producing a model that is ‘realistic yet simple’
[132]. Furthermore, it is grounded upon empirical data of network struc-
ture and knowledge of the dynamics of ecological processes, and subject
to Monte Carlo methods. Any model should be based on the foundations
of all available knowledge and data on the system of interest [133]—it
is the synergy across multiple sources of quantitative data and qualitative
knowledge that allows a model to transcend artificial boundaries and tell
us something of which we would otherwise be ignorant. At the same time,
however, the model does not blindly adhere to these data and knowledge,
since they are not necessarily omnipotent representations of reality (recall
the discussion above about incomplete data). Nevertheless, the model serves
as a springboard to explore the endless possibilities of the state space. Pat-
ten [134] cautioned that too much of ecological research has been focused
on empirical approaches for immediate local application, with too little ef-
fort in advancing the foundational science and theory that is important
for deep understanding, especially at the systems level. My study aims to
contribute to addressing this imbalance, in the hope of facilitating future
synergy between empirical and theoretical efforts in the science of ecology
[135] and beyond.
11

2 stochastic bioenergetic
network model
This chapter is a documentation of the network simulation model used
in this dissertation; see §1.3.1 for the philosophy and motivations behind
the model. The architecture of the model involves the construction of a
network of interacting species and the specification of allometrically scaled
bioenergetic equations governing changes in species states over time [93–95].
The combination of structural models with bioenergetic dynamics has been
suggested as a way towards greater ecological realism [96]. My model is
among the first to extend the bioenergetic model to include nontrophic
interaction types alongside trophic interactions (see Kéfi et al. [41]), and
introduces various refinements for greater realism. Overall, it could be said
to apply the principles of pattern-oriented modeling [136] by incorporating
observed patterns in ecological networks that are important to the research
question.
Simulations and analyses were programmed in R [137] and C++ with
Boost libraries for random number generation. In addition, the simulations
for Chapter 6 were executed in parallel using Message Passing Interface
on the Darwin supercomputer at the University of Cambridge. Numerical
integration was performed using the fourth-order Runge-Kutta algorithm
with dt = 0:1.
2.1 network construction
A food web topology is generated using the niche model [138], which
has been found to be one of the most robust so far in emulating the
structure of real food webs using relatively simple rules [96, 139–142] and
continues to be used in the latest research [41]. The model allows for the
variety of complications observed in nature, such as trophic loops (rock-
paper-scissors), omnivory, apparent competition and intraguild predation
[143, 144], although no correction is made for the relative frequencies of
occurrence of these topological ‘motifs’ (see Bascompte & Melián [144] and
Stouffer et al. [145]). There have been various generalizations of the niche
13
Stochastic bioenergetic network model
model [139, 142, 146–149]. Here a modified version of the generalization by
Stouffer et al. [146] is combined with that by Warren et al. [148], as
explained below.
Unlike the original niche model, real food webs are not fully interval
[146]. The method of Stouffer et al. [146] introduces an adjustable degree
of diet discontiguity, with their parameter x fixed at 0:8, which closely
matches empirical data [146]. One additional modification is made: if the
upper bound of a predator’s original diet range exceeds the predator’s niche
value, the so-called ∆k (Stouffer et al. p. 19017, last paragraph) number
of species is selected from non-prey species of niche values smaller than
or equal to the aforementioned upper bound rather than the predator’s
niche value. This improvement, particular to my study, avoids the logical
problems associated with reducing the upper bound of the potential range.
More recent generalizations of the niche model [139, 142, 149] have
achieved better fit to data than Stouffer et al. These versions, however,
are less suited to the current application. It is difficult to specify the con-
nectance in the formulation of Allesina et al. [139]. The models of Williams
et al. [142, 149] do not assign parameter values using statistical distributions
that would keep the model sufficiently simple. As my purpose is to con-
struct multiple plausible food webs rather than predict the links of specific
empirical webs as accurately as possible, the method of Stouffer et al. is
adequate, even in the light of more biologically motivated models based
on optimal foraging (see Petchey et al. [150] and skepticisms as to whether
optimal foraging reflects reality [151, 152]). Contrary to Williams et al. [142],
I am not overly concerned by the assumption that species near the centre
of a predator’s diet range are not more likely to be prey than those near
the edges, since unmodeled multidimensional niche space [149, 153]) could
account for that. The one-dimensional niche axis I use does not preclude
the effects of multidimensional niche space.
The inverse niche model of Warren et al. [148] is used to optionally
add parasite species to the food web. This institutes one of the most
fundamental differences between predators and parasites, by ‘letting little
things eat big things’ [105]. The inverse niche model excludes parasite-
14
Yangchen Lin
parasite and predator-eats-parasite links, the connectance being defined
as the proportion of possible host-parasite links that are realized [154].
Although predation on parasites is not included as direct links in the
inverse niche model, it is manifested via predators consuming the hosts that
harbour the parasites [148], which has been found to be the most common
form of parasite ingestion in aquatic food webs [155]. Hyperparasitism
and competition among parasites are not incorporated; these are avenues
for future empirical and theoretical work. It has also been suggested that
parasitism may be incorporated into food webs as indirect rather than direct
interactions [156] (see §5.1).
The connectances of the predator-prey and host-parasite subwebs are
set at 0:15 [154, 157]. Connectance in real webs can vary by as much as an
order of magnitude [158]; the value used here is a reasonable starting point
close to the connectances of several large and highly resolved food webs
[158]. The niche algorithms are run repeatedly until a web is produced that
has subwebs of the desired connectances ±0:01 and that satisfies the criteria
that the web must not contain any totally unconnected species, and that no
heterotrophic species should consume only itself, since this is energetically
unsustainable.
In the qualifying web, all nonparasite species having no prey are desig-
nated as basal species [94] flexibly representing autotrophs and detritivores,
which are not distinguished. Species that consume prey could also qualify
as detritivores or autotrophs, such as carnivorous plants, but these are ex-
cluded for simplicity. Model realizations containing only one basal species
are discarded, as direct competition among basal species (see below) cannot
be implemented in these realizations.
Random networks. For comparison with the niche models (Chapter 4 on-
wards), models with random network topology and at least two basal species
were also generated. The species niche values conformed to a uniform dis-
tribution as in the niche model. The consumer-resource links were assigned
in random pairs and directions regardless of niche values, with the condition
that basal species could not be consumers. This implementation also implic-
itly incorporates more trophic loops or parasitism, because a consumer can
15
Stochastic bioenergetic network model
be of smaller body size (niche value) than its resource with a probability
equal to the converse.
Competition. Direct competition interactions, where implemented, were
assigned randomly to 25% of the total number of potential interactions
between all pairs of basal species, rounded up to the nearest whole num-
ber. This procedure changes the overall connectance, but this is permissible
because the previously published connectance values pertain only to their
respective interaction types. The random assignment of competitive interac-
tions gives rise to varying degrees of intransitivity [159], reflecting natural
variability.
Mutualism. For both niche and random networks, mutualistic interactions
are assigned randomly to the predator-prey subweb, excluding species pairs
already allocated to other interaction types, with a connectance of 0:15
of the total number of potential interactions between all pairs of nonpar-
asite species, rounded up to the nearest whole number. This procedure
allows for a wide variety of motifs including plant facilitation, predator
facilitation [160], intraguild mutualism [5, 161, 162], plant-animal mutualism
and three-way mutualism such as that among sea anemones, zooxanthellæ
and anemonefish [163]. It also does not exclude less commonly studied
mutualistic associations in ecological networks, such as herbivore-detritivore
mutualisms e.g. gut microbes. Some of the even more esoteric mutualisms,
however, are not yet implemented, such as vector-parasite mutualism [164];
these are again avenues for future inquiry.
The non-implementation of the nestedness and modularity observed in
bipartite mutualistic networks [165–167] is not considered problematic. This
is because the overall structure of mutualistic interactions in the whole
community is unlikely to be as highly nested or modular as that of many
pollination- or seed dispersal-based bipartite mutualistic subwebs that have
been observed in isolation, when species from the different subwebs interact
with one another in other kinds of mutualism not encompassed by any of
those subwebs, such as habitat provision2 in return for nutrients [163, 168].
2Where habitat provision enhances protection of the occupant from particular predators,
tripartite interaction modification (Chapter 5) is a more appropriate representation.
16
Yangchen Lin
Table 2.1. Parameters of ecosystem assembly.
Parameter Value Reference
fraction of species that are parasites (:33 [154, 173]
x (:0 [146]
connectance (trophic subweb) (:)5±(:() [157]
connectance (host-parasite subweb) (:)5±(:() [154]
connectance (competition subweb) (:25 this paper
connectance (mutualistic subweb) (:)5 this paper
Furthermore, nestedness may not be a significant predictor of community
persistence [169]. The mutualistic connectance used here is lower than the
value of approximately 0:3 measured from bipartite mutualistic networks
[170]. It makes sense to have a lower connectance for the full ecological
network because its constituent species encompass a broader range of taxa
than any one specialized bipartite network of taxa that rely heavily on
mutualism.
The overall concept of different types of binary interactions used here
was first put forward by Burkholder [171] who listed nine ‘coactions
between weak and strong organisms’ comprising different combinations of
‘+’, ‘−’ and ‘0’ denoting gain, loss, and neither gain nor loss by each of
a pair of interacting species. This concept has been further developed by
Fath & Patten [91] and Fath [92]. My model simplifies it by incorporating
the three ‘elementary particles’ that are arguably the minimum set that
represents the different types of interactions at the most basic level: +−,
−−, ++.
Finally, the complete ecological network (Fig. 2.1) is tested to ensure
that there are no isolated components by computing the number of times
0 appears as an eigenvalue of the Laplacian, which is the number of
components of the graph [172]. If isolated components have been produced,
the network is regenerated until there is none. Network assembly parameters
are summarized in Table 2.1.
Fontaine et al. [174] were among the first to propose combining struc-
tured subwebs of different interaction types; my network takes their concep-
17
Stochastic bioenergetic network model
Predator axis Parasite species
P
re
y
 o
r 
h
o
st
 a
x
is
predation
cannibalism
competition
mutualism
parasitism
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Figure 2.1. One stochastic realization of the network construction algorithm com-
bining the niche model of predator-prey interactions (columns 1–20), inverse niche
model of host-parasite interactions (columns 21–30), direct competition and mutual-
ism. The first 20 columns represent the same set of species as the 20 rows. Black
boxes below the diagonal dashed line represent trophic loops. Columns with no
predation are defined as basal species. Direct competition interactions are assigned
randomly among basal species. The number of parasite species is set at half that of
nonparasite species. Connectances: (:)5± (:() for niche and inverse niche models
and mutualism.
tual framework one step further in that the predator-prey and host-parasite
subwebs are niche models rather than the more rudimentary nested or mod-
ular structures. For computational tractability, the number of species in the
networks used in this dissertation (10–30 species) is only moderately large.
The results nevertheless remain relevant to complex ecosystems—trophic
ecological networks of less than 14 nodes and 40 links have been found
to behave similarly to larger networks due to thermodynamic and other
mechanisms constraining food chain length and trophic complexity [175].
18
Yangchen Lin
2.2 multispecies dynamics
Network dynamics are modeled using allometrically scaled bioenergetic
equations [93, 94] with various modifications. The dynamics of a given basal
species i and a given non-basal species j with energy states Ei and Ej
respectively in arbitrary units are described by the equations
dEi
dt
= ri
[
1− Ei
Ki
]
Ei − 
i(1− ϵi)fiEi −
∑
k∈consumers

k Φk;ifkEk

(2.1a)
dEj
dt
= 
j
(
−1 + ϵj +
∑
k∈resources
 Φj;k
)
fjEj −
∑
k∈consumers

k Φk;jfkEk

(2.1b)
with time t, maximum consumption rate per unit metabolic rate  [94],
assimilation efficiency  [93], predator-prey functional responses Φ (see
below), nontrophic functional response f (see below) and ϵ, the species
response to multiplicative environmental stochasticity (§2.3).  is assigned
a value of 6 which is between 4 for ectotherm vertebrates and 8 for
invertebrate predators [95].  is assigned a value of 0:65 which is between
0:45 for herbivores and 0:85 for carnivores [95, 125], since the model includes
omnivory. It is assumed that parameter values for predator-prey and host-
parasite relationships are similar.
The traditional intrinsic growth rate r and mortality rate 
 are given
versatile interpretations. In this case, r represents both somatic and repro-
ductive growth arising from any net exogenous abiotic input such as detritus,
insolation or inorganic compounds, subtracting egestion and excretion. This
takes the cue from Halnes et al. [147] to embrace detritus in the niche
model, but does it in a more species-oriented way suited to the focus of
this study. The number of parameters for multiple basal species is minimized
by drawing r⃗ from lognormal distributions whose expected values and vari-
ances are equal to ⃗−1=4, accounting for metabolic scaling where metabolism
in plants is defined as photosynthesis instead of respiration [176, 177]. The
lognormal distribution is a sensible choice because it produces positive and
19
Stochastic bioenergetic network model
Table 2.2. Parameters of multispecies dynamics.
Parameter Description Unit Constraints Reference
 species niche value none ((; )) [138]
 assimilation efficiency none [(:45; (:05] [93, 95, 125]
r net exogenous input rates time−1 ((;∞)
to basal species
K carrying capacity arbitrary ((;∞)
biomass

 net exogenous output rates time−1 ((;∞)
 scaling coefficient for none this paper
mutualism and competition
 maximum consumption rate none [93, 94]
normalized to metabolic rate
! resource preference none ((; )) [95]
h Hill exponent none [); 2] [94, 125]
‘Exogenous’ refers to all abiotic energy and matter.
mostly moderate values, with allowance for some high values representing
the occasional fast-growing species.
The parameters 
⃗ are coefficients of exogenous output in terms of both
metabolism and mortality. Being related to body size [95, 178, 179], 
⃗ are
assigned the values 0:01⃗−1=4. There is no lognormal randomization here
because doing so would give some species large mortality rates leading
to rapid extinction; unlike for r, there is no (positive) density dependence
implemented in the model to prevent this. The normalization constant [177]
0:01 is chosen to produce ecologically sensible outputs i.e. to minimize
extinctions and prevent excessive segregation into high-energy basal species
and low-energy non-basal species, while keeping species states mostly within
(0; 1) and producing a long-tailed species biomass distribution at steady
state, as observed in preliminary simulations. Environmental stochasticity is
formulated to act on 
, as 
 is a basic population parameter common to
both autotrophic and heterotrophic species.
Negative density dependence is enforced on each basal species, with its
carrying capacity K = 1=4 where 0 <  < 1. This complies with the positive
general relationship of total species biomass to body size [177]. The fact
20
Yangchen Lin
that  is not perfectly correlated with trophic position, due to dependence
on the species’ behavioural and life-history traits, is already manifested in
the randomized assignments of diet midpoints and diet ranges in the niche
model.
The predator-prey functional responses are given by the fraction of
consumer i’s maximum consumption rate allocated to resource j [94, 95],
Φi;j =
!i;jEj
h
0:5h +
∑
k∈resources
!i;kEk
h
(2.2a)
h ∼ U [1; 2], (2.2b)
where resource preference ! is kept uniform and constant as 1=(no¯ of re-
source species), the half-saturation density, i.e. resource density at which
consumption rate is half the maximal rate [93], is fixed at 0:5 [125], and
the Hill exponent h is drawn from a uniform distribution in the range
[1; 2] giving a continuum between Holling types ii and iii functional re-
sponses [94, 125]. The half-saturation density is kept constant for simplicity;
0:5 is ecologically reasonable in this model because species energies are
initialized on the scale of [0; 1]. Overall, this kind of formulation allows
interaction strength to be modulated as a function of prey preference and
half-saturation density.
Ratio dependence. Most research on community dynamics, particularly bioen-
ergetic modeling of food webs, has assumed prey dependence. Ratio de-
pendence, a form of predator-prey functional response that depends on
the abundances of both predator and prey, was introduced by Arditi &
Ginzburg [180] in 1989. It has long been a matter of controversy [181]
as to whether real communities are prey- or ratio-dependent; the ecologi-
cal community now generally accepts that reality is somewhere in between
[182]. Arditi & Ginzburg [183] review the latest empirical evidence-based
arguments for ratio dependence. Specific studies on ratio dependence have
mostly been restricted to single prey and predator species or aggregated
functional groups [184, 185]. In a theoretical discourse with multiple prey
21
Stochastic bioenergetic network model
and predator species, van den Berg [186] found that ratio dependence ex-
plained the attenuation of energy fluxes of trophic cascades in food webs.
Ratio-dependent functional responses have been highlighted as an important
mechanism to account for in models [185], and as a promising avenue for
future theoretical work [187]. The present study draws comparisons between
exclusively prey-dependent (with no predator interference term) versus ex-
clusively ratio-dependent functional responses, representing the two ends of
the continuum in order to clearly reveal any effects of predator influence.
My ratio-dependent functional response is adapted from Piana et al. [184],
Φi;j =
!i;j(Ej=Ei)
h
1h +
∑
k∈resources
!i;k(Ek=Ei)
h
(2.3a)
h ∼ U [1; 2], (2.3b)
where the half-saturation of the prey:predator ratio is given a value of 1
as an ecologically sensible first approximation. If the state of a predator is
zero, the ratio for that predator is assigned a value of 0 to avoid division
by zero. This is also ecologically sensible since it would give Φ = 0.
Nontrophic interactions. The bioenergetics of nontrophic interactions are
very rarely implemented compared to trophic interactions. Here, mutualism
and/or competition act on 
 for a given species i, according to
fi =
1 + comp
∑
g∈competitors
EiEg
1 + mut
∑
g∈mutualizers
EiEg
(2.4)
where  controls the overall strength of mutualism or competition and is
assigned a value of 1 in this study. In my model, mutualism reduces 
while competition augments 
 (Eqns. 2.1); when mutualism and competition
are absent, 
 is unchanged. The functional response is linear; this is not
an unreasonable assumption as the real shape of the functional response
of mutualism is not yet generally known [41]. Although the formulation is
more rudimentary than that suggested by Kéfi et al. [41] and assumes that
22
Yangchen Lin
a given species benefits or suffers to the same degree with every interactor,
it is a reasonable starting point given that data are unavailable.
Parameter values for community dynamics are summarized in Table 2.2.
My parameterization procedure is somewhat different from that of Brose
et al. [95], who use basal growth rate as a starting point for defining the
time scale and metabolically scaling the other parameters. Nevertheless, my
parameter values are metabolically scaled and ecologically sensible, while
slightly relaxing the metabolic scaling constraints to allow for species with
unusual or extreme natural histories that occasionally occur in real ecosys-
tems. Like Brose et al., in my study the time scale is not defined in absolute
terms; the emphasis is on producing ecologically plausible simulated trajec-
tories and on the simulation duration being long enough to capture the
patterns of density dependence, species interactions and stochasticity, such
that any general results found would be applicable to a wide variety of
ecosystems operating at different time scales.
At the start of the simulation, each species in both niche and random
networks, including basal and parasite species, is assigned an initial state
E0 = 
1=4, with the same rationale as for carrying capacity K . Basal species
are therefore at carrying capacity at the start of a simulation; their interac-
tions may nevertheless bring them to equilibrium states different from their
carrying capacities. In plausible extreme cases, we may have r(1−E=E0) < 0
or 
(−1+∑ Φ) S 0 when the population exceeds the carrying capacity or
when resources are abundant, causing mortality to exceed growth or vice
versa. No distinction is made between parasite and nonparasite species when
assigning initial states; contrary to intuition, parasites in an ecosystem can
sometimes have a total biomass equal to or greater than that of nonparasites
[188].
The model has two other simplifications. Firstly, there are real-world
cases where an interaction between two species is both trophic and mutu-
alistic, for example in pollination; considering merely the net effect may
obfuscate some aspects of the dynamics [161]. There is scope for more
research in this area [41]. Secondly, the model does not implement adap-
tive rewiring of network links [189] in a mechanistic way based on optimal
23
Stochastic bioenergetic network model
foraging theory [190–192]. The model, however, does have intrinsic adaptive
rewiring in both trophic and nontrophic interactions, via the dependence of
interaction strength on species energy, although the values of the resource
preference ! are kept fixed over time. It also accounts for the formation
of new links if one interprets the links assigned to a given species at the
beginning as including all the species it could potentially interact with under
all scenarios. This has the merit of not assuming optimal foraging behaviour.
The overall model specification strikes a compromise between simplicity and
complexity, and avoids setting arbitrary and discrete thresholds at which to
change links.
2.3 environmental noise
This section documents the implementation of environmental noise in the
models; see §3.1 for background. Code written by the author can be down-
loaded at https://github.com/linyangchen/noise. The noise term ϵ is based
on a time series c generated using the equations of Cohen et al. [193] given
by
ct =
T
2∑
i=1
√(
i
2.i
)ϱ
sin
(
2.i
i
t+ Ωi
)
(2.5a)
Ω ∼ U [0; 2.] (2.5b)
where i is the number of cycles (frequency) within the total duration of c ,
i is the length of c , ϱ is the spectral exponent i.e. noise colour, and Ω is
the phase shift. i is set to the length of the network model simulation, in
which case the period of the shortest noise cycle is 2dt and the period of
the longest cycle is equal to the simulation duration. Each sine curve can be
interpreted as representing one environmental variable; summing multiple
sine curves of different phases and periods creates the overall environment
experienced by the species.
24
Yangchen Lin
A
m
p
li
tu
d
e
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
∑
am
p
li
tu
d
es
−4
−2
0
2
4
6
Time (arbitrary units)
Figure 2.2. Generation of coloured environmental noise of spectral exponent ϱ 5 )
using the spectral synthesis method [193] of sine wave summation (Eqn. 2.5a).
The time series of noise terms ϵ is finally produced using spectral mimicry
[194], permutating a Gaussian-distributed random sequence of mean 0 and
specified variance such that it matches the spectral density of c . This gives
a Gaussian-distributed colour noise series for a fair comparison with white
noise cases [195], although it does not necessarily have to be Gaussian
in reality [196]. This mechanistic noise implementation has advantages over
the commonly used phenomenological autoregressive models [193], and does
not underestimate extinction risk in red environments like both 1=f and
autoregressive models [195]. There remains a limitation in the method; as
ϱ ≥ 3, the noise increasingly looks like a smooth sine curve, which is not
realistic (L. Ruokolainen pers. comm.).
The total variance of the noise experienced by each species is fixed at 0:1
(see Sæther et al. [197]), with ϵ constituting all, half or none of this variance.
In the second of these three different treatments, the remaining variance
experienced by each species comprises values independently drawn from
Gaussian distributions with means equal to ϵ⃗ and variance 0:05. In the third
treatment, the variance experienced by each species over time comprises
values independently drawn from Gaussian distributions with mean 0 and
variance 0:1. These procedures produce different degrees of interspecific
25
Stochastic bioenergetic network model
synchrony; partial synchrony represents the environment being either good
or bad for most species in a given time step, but to different extents for
each species.
26
a multispecies Pythagorean time series
Anton Bruckner: Symphony no¯ 8

3 colour and synchrony of
environmental noise affect
network variability
Abstract. I use stochastic bioenergetic ecological network models to explore
the effects of non-predator-prey interactions, colour of environmental noise,
and the degree of synchronization of environmental noise across species on
the coefficients of variation over time of total ecosystem energy content and
Shannon entropy. In regression trees derived by binary recursive partitioning,
the presence or absence of synchrony gave the greatest difference in the
means of data points for both system energy and Shannon entropy, followed
in turn by white versus coloured noise and pink versus red and black noise.
The effects are present despite the large variation in the Monte Carlo
simulations reflecting the variability of real ecosystems. Non-predator-prey
interactions explained relatively small proportions of the total deviance, and
each had different directionalities of effects depending on the presence of
the other interaction types and on whether energy or entropy was measured.
These results underscore the importance of modeling more realistic colours
of environmental noise in understanding and predicting the dynamics of
ecological communities.
3.1 introduction
God is noisy. Real ecosystems are ‘buffeted by a more or less continual series
of perturbations, and transient behavior may be the norm rather than the
exception in nature’ [198]. In the face of increasing environmental change,
it is important to understand the impact of different types of environmental
variation on populations and communities [199]. Sutherland et al. [200]
identified the question of how environmental stochasticity interacts with
density dependence to influence population dynamics as one of the most
important contemporary questions in ecology. Stochasticity cannot be un-
derstood adequately by linear analysis, because the stochastic system is never
in equilibrium [201]. Indeed, Ruokolainen & Fowler [202] reported that
29
Colour & synchrony of environmental noise
analytical solutions did not capture many of the features of the simulation
outputs of stochastic Lotka-Volterra models.
The biggest problem is the real world.
md at Goldman Sachs
The influence of stochasticity on population dynamics was first inves-
tigated in the 1960s and 1970s [196, 203, 204] and have attracted increasing
attention in recent years. Stochasticity has been found to magnify extinc-
tion risk and reduce invasion risk [205, 206], and has been highlighted as
an area for further research in community viability analysis [207]. With
respect to species interactions, Ripa et al. [208] first presented a theory
of the population-level effects of environmental noise in two-species ‘food
webs’. A theoretical study [209] of the effects of noise on the transient
dynamics of two, three and n interacting species with Lotka-Volterra dy-
namics found that noise can enhance stability, but that analysis used random
interaction parameters. Subsequently, Ripa & Ives [210] took an analytical
approach to understanding the effects of environmental synchrony on the
dynamics of populations in a two-species Lotka-Volterra competition model,
and showed that the effects can be large and unexpected as synchrony can
either amplify or dampen cyclic behaviour. Gravel et al. [211] investigated
population-dynamical criteria for species coexistence in a stochastic envi-
ronment, while Gjata et al. [212] used stochastic simulations to study the
effects of indirect interactions resulting from trophic interactions. Vasseur
& Fox [213], using a theoretical four-species ‘diamond’ food web, reported
that noise can stabilize food webs by synchronizing population dynamics.
Wells et al. [32] found that environmental stochasticity, on top of popula-
tion dynamics, affected the structural indices of ecological networks. Most
recently, Novak [187] highlighted nonequilibrium dynamics as one of the
promising avenues for future theoretical work.
Reddened noise is experienced by most natural populations [214–219] and
is exhibited by many other complex systems ranging from stock markets
to protein-dna binding to rhythmic synchronization in music-making [220],
30
Yangchen Lin
yet most stochastic studies have assumed white noise. The colour of noise
has been shown to have significant impacts on the dynamics and stability of
single populations [221–228] or simple systems with a few interacting species
[202, 206, 210, 229]; also see review by Ruokolainen et al. [199]. Multispecies
studies include that of Greenman & Benton [206], who reported that au-
toregressive noise caused resonance and threshold effects in a simple system
of three species on three trophic levels. Using a two-species bioenergetic
trophic model, Vasseur [229] showed that the temporal variability of each
species can respond differently to changes in environmental noise colour.
Ruokolainen et al. [230] and Ruokolainen & Fowler [202] constructed
theoretical models showing how various factors affected the extinction risk
of populations in competitive Lotka-Volterra communities exposed to dif-
ferent environmental noise colours. The question remains, however, as to
whether findings from studies of noise in single-species populations or a
few interacting species would hold for networks of many interacting species.
With multiple interacting species, the question also arises as to the
extent to which individual species’ responses to stochasticity are correlated,
or synchronized, with one another. Total interspecific asynchrony has often
been used in studies of competition and niche differentiation [231], but this
setup is unsuitable for predictive models because environmental forcing itself
is one of the synchronizers of population dynamics for many taxa [232–236],
especially among species with similar traits [233, 237]. Niche differentiation
nevertheless gives rise to asynchrony, which can make diverse communities
less variable and more stable in the face of environmental disturbances
[238, 239]; partial synchrony would therefore seem consistent with most real
communities. Very little is known about how the degree of synchrony affects
the dynamics and stability of ecological networks, existing studies being
limited to white noise and either triangular tritrophic food webs [240] or
Lotka-Volterra competitive communities [241].
Nontrophic interactions. Recent studies have investigated the properties of
networks of various non-predator-prey interaction types, one of which is
competition. For example, species coexistence has been reported to depend
on the mean and variance of interaction strengths in competitive networks
31
Colour & synchrony of environmental noise
[242]. Competitive networks constructed on game theory principles have
been found to promote diversity [243], which implies that this enables
more species to coexist in equilibrium. In networks of competing species
that cannot be ranked in a strict hierarchy of competitive ability, negative
frequency dependence can arise that promotes diversity [159]. In contrast,
Loreau & de Mazancourt [241] found in a stochastic competition model
that competition was generally destabilizing, but this may have been partly
due to the inherent instabilities of the discrete-time modelling that was
used.
Ecological theory has traditionally been dominated by research into
antagonistic interactions with, until recently, a relative neglect of facilitative
or mutualistic interactions [244]. Although the co-occurrence of negative
and positive interactions was already recognized by Burkholder [171], only
in recent years have researchers begun to more widely acknowledge the
importance of accounting for the mutualistic interactions that pervade real
ecological communities [161]. Mutualism is not only of theoretical interest
but has also been shown to enhance sustainability in systems exploited by
man [245] and has become more prevalent under increasing environmental
stress [246].
Various studies have taken the network approach towards mutualistic
interactions. For example, Suweis et al. [167] analytically and numerically
demonstrated the positive correlation between species abundances and the
nestedness of animal-plant mutualistic networks. Okuyama & Holland [247]
found that the structural attributes of dynamic mutualistic networks give
rise to positive complexity-resilience relationships, while Ramos-Jiliberto
et al. [192] reported that incorporating adaptivity of interactions increased
robustness in dynamic mutualistic networks. Discoveries about mutualistic
networks have also inspired further inquiry into food webs. For example,
Kondoh et al. [248] found that food webs had nested substructures like
in bipartite mutualistic networks, and proposed that nestedness in food
webs hinders coexistence because it increases consumer niche overlap, an
effect opposite to nestedness in mutualistic networks. A comparative study by
32
Yangchen Lin
Thébault & Fontaine [249] revealed how dynamic food webs and mutualistic
networks achieve stability via different structural mechanisms.
The aforementioned studies have greatly advanced our understanding of
complexity-stability relationships in ecosystems, but one of their limitations
is that interaction types were considered in isolation [132, 249, 250]; little is
known about the combined effect of all these interaction types. The relative
neglect of non-predator-prey interactions can have profound ramifications
for our understanding of ecosystem function [251]. For example, non-
predator-prey interactions may exacerbate human impact on ecosystems
[252]; not accounting for such interactions in fisheries models has reduced
the capacity of these models to predict stock collapse [82]. Food webs
implicitly include some non-predator-prey interaction in the form of indirect
competition, but competition among basal species and, more strikingly,
facilitation are absent. Conversely, studying non-predator-prey interactions
in isolation can also compromise our ability to make useful predictions. For
example, the dynamics of host-parasite interactions depend not only on
those interactions per se but also on predators and alternative hosts, with
implications for infectious disease control [253].
Several researchers have thus advanced the design of models containing
multiple interaction types. The analytical models of Gross [161], involving
multiple consumer species feeding on a single resource, revealed that in-
traguild mutualism could be an important ingredient for species coexistence
in otherwise competitive environments. Filotas et al. [254] simulated how
spatial processes affect the structure and stability of multiple-interaction-
type networks created with random topology and link strengths. Most
recently, the so-called multiplex network approach [255] has increasingly
been adopted, where different interaction types are partitioned into differ-
ent networks that are interlinked with one another [174, 256, 257].
Some of the most general results so far emerging from networks of
multiple interaction types come from Allesina & Tang [51], whose ana-
lytical models suggested that the addition of mutualistic and competitive
interactions reduce the probability of stability of predator-prey networks.
Stability of equilibrium points, however, is just one part of the story, as
33
Colour & synchrony of environmental noise
first highlighted in §1.3.1; the response in nature of a system to perturba-
tions constitute another important line of inquiry. Indeed, some researchers
have advocated that the maintenance of resilience, rather than avoidance of
disturbance, should be the focus for conservation efforts [258, 259].
Host-parasite interactions. The importance of incorporating parasites into
mainstream food webs [260] and potential effects of parasitism on food
web stability [154, 261] have also been highlighted recently, as most studies
have investigated host-parasite networks in isolation [262, 263]. There have
even been calls for including parasites in food webs by ‘default’ [264].
The impacts of parasites on food webs have been found to be diverse,
various interspecific and intraspecific mechanisms having been reported for
both stabilizing and destabilizing effects [105, 156, 265, 266] and for effects
on network structure mainly via changes in diversity and complexity [267].
Parasitism has also been paired with mutualism in a study that explored the
dynamics and equilibria of a theoretical ‘food web module’ comprising a
plant, pollinator and nectar robber [268].
To date, no study has investigated the effects of the colour and syn-
chrony of noise on networks of many species and multiple interaction
types. Most of this kind of work has been on the dynamics of single
populations; even when interspecific interactions are considered, often us-
ing Lotka-Volterra models of predation or competition, the focus is on
population-level rather than ecosystem-level dynamics. In this study, I use
the simulation model documented in Chapter 2 to ask how environmen-
tal stochasticity and synchrony, and non-predator-prey interactions, affect
the variability of ecological networks. The foundations of the model in-
clude empirical data from which network structures have been derived, and
well-established consumer-resource equations with known assumptions. Pre-
liminary simulations using this class of model by Kéfi et al. [41], comprising
food webs with added plant facilitation and predator interference, indi-
cated that nontrophic interactions result in higher diversity at equilibrium,
but their study did not look at environmental stochasticity. I show that
the colour and degree of synchrony of noise have striking effects on the
temporal variability of ecological networks.
34
Yangchen Lin
3.2 methods
The models are based on Chapter 2, using the prey-dependent functional
responses. Williams & Purves [149] found no consistent trend across empir-
ical food webs in the distribution of niche values within a given web and
reported that the distributions were often nonuniform. In this chapter, I
therefore modify the original niche model for greater realism by using beta-
distributed ( = 1:5;  = 5) niche values ⃗. Given that  in the niche model is
roughly proportional to body size [269, 270], my values of  and  produce
a distribution that approximates two relationships3: the proportionality of
species diversity to (body size)−1=4 [177], and the lognormal distribution of
body sizes as shown in studies of terrestrial and stream ecosystems [271].
As in the food web, I set  ∼ B(1:5; 5) in the parasite web, since species
having parasitic or parasite-like lifestyles can have macroscopic (e.g. hyenas
and avian brood parasites) as well as microscopic body sizes, and there is
higher diversity at small body sizes as for nonparasitic species.
Independent replicate 15-species models with different combinations of
interaction types were generated. I used eight configurations: f, fc, fm,
fcm, fp, fcp, fmp and fcmp, where f denotes food web (predator-prey),
c competition among basal species, m mutualism and p parasitism. In the
parasitic model configurations, five of the species were parasites; the pro-
portion of parasites takes guidance from Lafferty et al. [154] and Sukhdeo
[173]. Environmental noise was implemented as detailed in §2.3. ϱ was varied
from 0 to 2 in increments of 1, 0 being white noise, and 1 and 2 being
widely regarded as pink and red respectively. For each degree of synchrony
(none, half, full—see §2.3) in each noise colour treatment in each model
configuration, 100 independent network models were generated.
The networks were simulated using the ranges of dynamical parameter
values listed in Table 3.1. Temporal variability of the ecosystem was quantified
in terms of the coefficients of variation (cv) over time [58, 272] of two metrics:
the total system energy and the exponent of the Shannon entropy [273].
3 In later chapters, species niche values are uniform rather than beta distributed, to facilitate
comparison with random-topology models where body size hierarchy no longer applies, and
to avoid making species of lower niche values that go extinct more similar to surviving species.
35
Colour & synchrony of environmental noise
Table 3.1. Parameter values of multispecies dynamics.
Parameter Value Reference
 B():5; 5) this study
 (:25 [276]
r E(r) 5 −1=4 [177]
K E0 this study

 (:()−1=4 [177]
 ) this study
 6 [93]
! )=(no¯ of resource spp.) [95]
h U[); 2] [125]
The cv combines the amplitude and frequency of temporal variation in one
convenient index. The total system energy or biomass at each time step
was obtained by summing the species states. The exponent of the Shannon
entropy measures the directly comparable effective number of species [273],
given by exp
(
−∑Si=1 pi ln pi) where pi is the proportion of system energy
constituted by species i at a given time step and h is the total number
of species. The metrics were measured from time step 10 001 (to exclude
initial transients)4 for a duration of 25 000 divisions. Preliminary simulations
indicated that this time scale of measurement encompassed all the main
features and periodicities of the stochastic behaviour.
The effects of noise colour and synchrony on ecosystem temporal vari-
ability were analyzed using regression trees grown using binary recursive
partitioning using the tree package [275] in R. Regression trees were also
used to examine the effects of dummy variables representing the presence
or absence of each non-predator-prey interaction. The advantages of re-
gression trees over regression models are that the former is nonparametric
and does not assume unimodality and linearity of the response variable.
4 A possible question for future work is how transient dynamics modulate stochasticity and
vice versa, as has been done for single-species plant populations [274].
36
Yangchen Lin
3.3 results
The results reported here are from a rerun of the entire simulation study
using a different pseudorandom number generator seed state from Lin &
Sutherland [1] and with noise for basal species implemented in mortality
rate rather than growth rate, which represents a minor advancement in
model design since publication. While the results for non-predator-prey
interactions are slightly different (as would be expected and, as emphasized
by Lin & Sutherland [1], to be interpreted with caution given the very
small effect sizes), those for the effects of noise are qualitatively unchanged,
reaffirming them.
In a small number of simulations, numerical integration produced neg-
ative species states when dE=dt < 0 and |dE=dt| S E. These were due to
the noise terms and the small errors inherent in numerical integration; the
affected simulations were omitted from the analysis. The total sample sizes
for the various model configurations are 894 (f), 894 (fc), 893 (fcm), 897
(fcmp), 896 (fcp), 893 (fm), 895 (fmp) and 895 (fp), of 900 simulations each.
The stopping criterion for growing the regression trees was set as
the point where the within-node deviance became less than 1% of the
root node deviance. The resulting trees are shown in Fig. 3.1 and
numerically summarized in Table 3.2. The split between unsynchronized
and synchronized (half and full) species environmental responses explains
most of the deviance, with considerable differences in the mean values of
the cv of system energy (0:0381 and 0:307 respectively) and exponent of
Shannon entropy (0:0319 and 0:145 respectively). The next most important
factor is noise colour, with white noise giving lower cv than coloured
noise for system energy (0:0463 and 0:437 respectively), and white or pink
giving lower cv than red noise for entropy (0:0884 and 0:258 respectively).
The effects of synchrony were much more pronounced when the noise
was coloured (Fig. 3.3). The non-predator-prey interaction types never
explained more than 1% of the deviance when the dummy variables
representing their presence or absence were analyzed together with the
noise variables.
37
Colour & synchrony of environmental noise
envar = {0}
fpow = {0}
fpow = {1}
0:21770
0:03805 0:30720
0:04627 0:43690
0:32350 0:55040
Energy
envar = {0}
fpow = {0; 1}
fpow = {0}
0:10740
0:03186 0:14490
0:08842
0:03656 0:13990
0:25770
Entropy
Figure 3.1. Regression trees of effects of noise colour and synchrony on cv of system
energy (top) and of ]pp[Shannon entropy] (bottom). Explanatory variables: envar,
noise synchrony; fpow, colour (denoted by spectral exponent). The categorical
explanatory variable levels partitioned to the left-hand branches are indicated.
Numbers under nodes are mean values of the response variables apportioned to
the respective branches above.
Regression trees using only dummy variables of non-predator-prey
interaction types, with data points pooled across different colours and
degrees of synchrony of noise, are summarized in Fig. 3.2 and Table 3.3.
Mutualism explains most of the deviance in the cv of system energy,
while parasitism explains most of the deviance in the cv of the exponent
of Shannon entropy. Both have stabilizing effects i.e. reduce system
variability. Competition generally explains the smallest proportion of the
deviances.
38
Yangchen Lin
Table 3.2. Regression tree nodes and deviances of
effects of noise colour and synchrony on the cv
of total system energy or ]pp[Shannon entropy].
Tree Node n Deviance Mean cv
energy envar 7)57 402 (:2)0
terminal 2371 )7:3 (:(30)
fpow 4770 35( (:3(7
terminal )506 )(:3 (:(463
fpow 3)12 )70 (:437
terminal )517 34:7 (:323
terminal )515 )(2 (:55
entropy envar 7)57 )36 (:)(7
terminal 2371 5:11 (:(3)1
fpow 4770 )(1 (:)45
fpow 3)03 3):4 (:(004
terminal )506 4:34 (:(366
terminal )517 )0:6 (:)4
terminal )515 47:6 (:250
Tree and node names refer to Fig. 3.1; n, number of data
points.
Regression trees were also grown separately for each noise colour-
synchrony combination (Appendix 3.5.1). The results were similar to the
aforementioned effects pooled across noise colours and degrees of syn-
chrony. Mutualism explained most of the deviance most of the time for
the cv of both energy and entropy, often causing considerable if not
greatest increase in node purity. Where they account for most of the
deviance, mutualism and parasitism both tend to have stabilizing effects,
except in the case of unsynchronized species responses to noise, where
mutualism tends to be destabilizing. The effects of direct basal competi-
tion are again inconsistent and relatively small. The effects of mutualism
and parasitism, where they do not account for most of the deviance, are
also relatively small and inconsistent.
39
Colour & synchrony of environmental noise
M = {0}
P = {0}
C = {0} C = {0}
P = {0}
C = {0} C = {0}
0:2177
0:2342
0:2426
0:2415 0:2438
0:2259
0:2324 0:2194
0:2012
0:1967
0:1976 0:1958
0:2057
0:2070 0:2045
Energy
P = {0}
C = {0}
M = {0} M = {0}
M = {0}
C = {0} C = {0}
0:10740
0:11250
0:10990
0:11140 0:10830
0:11520
0:11640 0:11400
0:10220
0:10730
0:10960 0:10510
0:09704
0:09778 0:09630
Entropy
Figure 3.2. Regression trees of effects of non-predator-prey interactions on cv of
system energy (top) and of ]pp[Shannon entropy] (bottom), pooled across noise
colours and degrees of synchrony. Explanatory variables: C, direct basal competition;
M, mutualism; P, parasitism. The categorical explanatory variable levels partitioned
to the left-hand branches are indicated (( and ) respectively denote absence and
presence of the interaction type). Numbers under nodes are mean values of the
response variables apportioned to the respective branches above.
3.4 discussion
Models with coloured noise, but not white noise, exhibit considerably
larger and longer-term fluctuations when species environments are partially
or fully synchronized. This is most likely due to constructive interference
with density-dependent dynamics within [224] and across populations,
arising from temporal autocorrelation and synchrony of the environment.
These large fluctuations may mimic periodic outbreaks and crashes of
40
Yangchen Lin
Table 3.3. Regression tree nodes and deviances of
effects of non-predator-prey interactions on the cv
of total system energy or ]pp[Shannon entropy].
Tree Node n Deviance Mean cv
energy M 7)57 402 (:2)0
P 3571 20) (:234
C )700 )57 (:243
terminal 014 77 (:242
terminal 014 71:0 (:244
C )71) )24 (:226
terminal 015 65:2 (:232
terminal 016 50:7 (:2)1
P 3570 2(( (:2()
C )706 00:) (:)17
terminal 013 44:3 (:)10
terminal 013 43:0 (:)16
C )712 ))) (:2(6
terminal 015 57 (:2(7
terminal 017 54:3 (:2(5
entropy P 7)57 )36 (:)(7
C 3574 77 (:))3
M )707 36:) (:))
terminal 014 2(:1 (:)))
terminal 013 )5:2 (:)(0
M )707 4(:1 (:))5
terminal 014 24:3 (:))6
terminal 013 )6:6 (:))4
M 3503 50:6 (:)(2
C )71) 3):6 (:)(7
terminal 015 )6:6 (:))
terminal 016 )5 (:)(5
C )712 26:1 (:(17
terminal 015 )4:7 (:(170
terminal 017 )2:2 (:(163
Tree and node names refer to Fig. 3.2; n, number of data
points.
species in real systems,5 something that my white-noise and/or unsyn-
chronized models do not appear to represent adequately. These species
5Or even bubbles and crashes in the economy.
41
Colour & synchrony of environmental noise
0.0
0.5
1.0
1.5
Configuration FCMP
a
0.0
0.5
1.0
1.5
System energy
exp[Shannon entropy] b
0.0
0.5
1.0
1.5
0 1 2
Noise colour (spectral exponent)
c
C
V
 o
f 
ec
os
y
st
em
 m
et
ri
c 
ov
er
 t
im
e
Figure 3.3. The influence of noise colour and degree of interspecific synchrony
on variability of ecosystem energy and Shannon entropy, in terms of kernel
density distributions of outputs from multiple simulations (n ≈ 1((), for model
configuration fcmp. Plots for the other model configurations are qualitatively similar
(Appendix 3.5.2). a, unsynchronized species environments; b, half synchronized; c,
fully synchronized. Violins scaled to constant width.
may be native species, invasive pests or viruses, the latter two of which
would be of direct concern to human welfare and development. It is also
notable that with white noise, synchrony did not destabilize systems more
than the unsynchronized case. This could be because of the weakness
of reinforcing effects due to the absence of temporal autocorrelation in
white noise, whether synchronized or not. Although Lögdberg & Wen-
nergren [228] found that environmental reddening reduced rather than
increased species extinction risk, the focus of their model on the spatial
environment experienced by a single species was different from mine. My
42
Yangchen Lin
results generally concur with the findings of Greenman & Benton [224]
for single populations subject to coloured versus white noise. Stochastic
food web models that assume white noise may give the idea that ecosys-
tems are less variable, less prone to large and sudden changes and more
predictable than they really are, with implications for the reliability and
efficiency of ecological and conservation projects that do not conduct
research, monitoring, data collection or management over sufficiently long
time scales.
There was more than a two-fold difference in both measures of
ecosystem variability between simulations with synchronized compared to
unsynchronized noise. These large differences highlight the importance
of accounting for synchrony in ecological modeling and prediction. A
multispecies model that implements stochasticity independently for each
species may underestimate instability and extinction risk, especially if
those models also assume white noise. That synchrony has relatively little
influence on ecosystem variability in white-noise scenarios, but that even
partial synchrony amplifies the effect of coloured noise (or vice versa) in
increasing ecosystem variability, underscores the risks of making multiple
simplifying assumptions in nonlinear ecological modeling in general and
the need to minimize such risks by striking the best possible compromise
between simplicity and realism of models. Modeling noise as realistically as
possible is especially crucial in fields of research such as critical transitions
in complex systems (see Scheffer et al. [277]), where stochasticity plays
a key role in the dynamics before, during and after critical transitions.
I recommend pink noise [216, 218] with partial synchrony as a starting
point for a more realistic representation of environmental stochasticity in
dynamic species interaction models.
In the literature, direct competition among basal species rarely appears
outside exclusively plant-based research. Some studies [278–280] have
taken the leap by combining direct basal-species competition and plant-
herbivore interactions in bitrophic designs, but have not examined the
effects of such competition on a web of many trophic levels and multiple
interaction types, including facilitation. My full webs indicate that the
43
Colour & synchrony of environmental noise
effects of direct basal competition depend on whether parasitism or
mutualism, or both, are also present. Competition, being an antagonistic
interaction, may stabilize systems containing facilitation (mutualism) by
dampening the enhanced growth rates brought about by mutualism (see
below), and amplify the variability of systems containing parasitism—
another antagonistic interaction—by depressing species abundances further
and thereby causing larger abundance fluctuations and knock-on effects
through multispecies interactions. Continuing along this line of argument,
the very small effects of competition on systems containing both mutualism
and parasitism could be because the opposing effects of competition in
the presence of either mutualism or parasitism alone balance out each
other when both mutualism and parasitism are present. Although the
‘connectance’ of direct competition (0:25) seems higher than those of
the other interaction types, the absolute number of competition links is
relatively small because of the small number of basal species. We can
expect any effects of direct competition to become clearer with increasing
connectance. I did not, however, increase the connectance further because
it is more important in the systems-oriented approach here to have a
sensible balance between different interaction types. In any case, there
is little basis for comparing the relative effect sizes between different
interaction types.
The finding that mutualism tends to stabilize the system could be
attributed to the tendency of mutualism to dampen predator-prey cycles.
The occurrence of both negative and positive effects of mutualism in
this study, however, indicates that the influence of mutualism could
be highly context dependent. Nevertheless, the results complement the
findings of Mougi & Kondoh [281] who found that when increasing
proportions of randomly selected antagonistic interactions in a model food
web were converted6 to mutualistic interactions, intermediate amounts of
6 A potential limitation with their methodology is that the conversion of antagonistic interac-
tions to mutualistic interactions meant that food web structure could not be kept constant in
their study.
44
Yangchen Lin
such mutualistic interactions generally had stabilizing effects in terms of
equilibrium stability based on the Jacobian community matrix.
In my model, the key difference between having and not having
parasite species is the difference in the prevalence of species of smaller
body size gaining energy at the expense of species of larger body size.
This allowed me to examine the effect of this difference while keeping
the other attributes of the parasite species, such as assimilation efficiency,
distribution of niche values and metabolic scaling, statistically similar to
those of the nonparasite species. Nevertheless, the model assumptions
and parameterizations been made with nonparasite species in mind may
not necessarily apply to parasites [105]. More investigation is needed on
how parasites differ topologically and dynamically from nonparasites in
the context of ecological networks [105, 106].
Lafferty et al. [105] reviewed various mechanisms by which parasites
can destabilize or stabilize food webs, notably destabilization via the
inversion of body size structure [282] and lengthening of trophic chains
[154], and stabilization via shared pathogens. All these mechanisms are
represented in my model. My finding that parasitism tends to reduce
ecosystem variability appears to contradict part of the aforementioned
literature, but there is in fact no contradiction because both stabilizing
and destabilizing effects occur in my study depending on each of many
model realizations with contingent network topologies and dynamics. In
addition, the difference of the average result of parasitism from that of
Lin & Sutherland [1] could be partly due to the smaller number of
species used there in nonparasitic configurations, implying that additional
diversity may also have stabilizing effects on average, corroborating current
consensus on the biodiversity-stability relationship.
Overall, the finding that environmental stochasticity has a far greater
role in ecosystem variability than non-predator-prey interactions is not sur-
prising insofar as stochasticity is the source of the variability. Nevertheless,
the parameter values used for both environmental noise specification and
network topology are reasonably realistic in ecological terms, and show
that the effects of noise colour are qualitatively similar across different
45
Colour & synchrony of environmental noise
combinations of interaction types. Although one could possibly increase
the effect sizes of non-predator-prey interactions relative to those of
environmental noise by making the former interactions stronger and/or
more abundant, and/or making the environmental noise weaker, doing so
may compromise realism. These results illustrate the importance of incor-
porating stochasticity in ecosystem modeling and prediction, perhaps even
more important but even less explored than incorporating non-predator-
prey interactions in food webs. This study also shows that for either
general understanding or prediction of the dynamics of a specific ecosys-
tem, it is important to model the particular kind of stochasticity present
in that ecosystem. For example, noise in terrestrial ecosystems tends to
be whiter than that in marine ecosystems, and similarly for temperate
versus non-temperate latitudes [219]. Overall, the patterns elucidated in
this study emerged over and above the considerable variation in network
model realizations emulating real-world contingencies across ecosystems.
The implications of this study extend to practical conservation. Fo-
cusing on single charismatic species without also paying attention to
their ecological interactions, or forgetting about cryptic species with dis-
proportionate influence, may be an obvious but not necessarily optimal
approach [116]. A safer way might be to conserve ecosystem structure
and function by protecting a suite of species that represent the widest
possible range of ecological functional groups [283] and species that
serve a given function but have the widest possible range of responses
to different types of stress [284], including responses to environmental
noise. Diversity and asynchrony in species responses to noise means less
extreme overall fluctuations in ecosystem properties and function. Much
research remains to be done to understand how the interactions of a
species make it a better or worse candidate for these ends [285]. The
results so far support the notion that more systems-oriented research
strategies that accommodate stochasticity have an important role to play
in cultivating such understanding and helping us make more judicious
ecological decisions.
46
Yangchen Lin
3.5 appendix
3.5.1 effects of non-predator-prey interactions
Regression trees of effects of non-predator-prey interactions on cv of
system energy and exp[Shannon entropy], by noise colours and degrees
of synchrony. Explanatory variables: C, direct basal competition; M, mutu-
alism; P, parasitism. The categorical explanatory variable levels partitioned
to the left-hand branches are indicated (0 and 1 respectively denote
absence and presence of the interaction type). Numbers under nodes
are mean values of the response variables apportioned to the respective
branches above.
47
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&
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of
environm
ental
noise
M = {0}
P = {0}
C = {0} C = {0}
C = {0}
P = {0} P = {0}
0:04035
0:03653
0:02731
0:02379 0:03090
0:04570
0:05105 0:04041
0:04417
0:03582
0:04436 0:02736
0:05257
0:05101 0:05412
white no sync
C = {0}
M = {0}
P = {0} P = {0}
P = {0}
M = {0} M = {0}
0:04305
0:04865
0:04735
0:02983 0:06470
0:04995
0:06560 0:03446
0:03749
0:04360
0:03373 0:05327
0:03141
0:03113 0:03170
white half sync
C = {0}
M = {0}
P = {0} P = {0}
M = {0}
P = {0} P = {0}
0:04947
0:04759
0:04967
0:03993 0:05930
0:04551
0:05604 0:03508
0:05135
0:04677
0:03996 0:05364
0:05596
0:05908 0:05290
white full sync
M = {0}
P = {0}
C = {0} C = {0}
P = {0}
C = {0} C = {0}
0:03708
0:03371
0:02775
0:02738 0:02811
0:03964
0:03190 0:04738
0:04048
0:04680
0:04346 0:05011
0:03420
0:03595 0:03248
pink no sync
M = {0}
P = {0}
C = {0} C = {0}
P = {0}
C = {0} C = {0}
0:2752
0:2961
0:3089
0:3126 0:3051
0:2834
0:2819 0:2849
0:2542
0:2435
0:2491 0:2379
0:2650
0:2720 0:2581
pink half sync
M = {0}
P = {0}
C = {0} C = {0}
P = {0}
C = {0} C = {0}
0:3716
0:4065
0:4189
0:4191 0:4187
0:3940
0:4031 0:3848
0:3369
0:3098
0:3068 0:3128
0:3640
0:3652 0:3627
pink full sync
P = {0}
M = {0}
C = {0} C = {0}
M = {0}
C = {0} C = {0}
0:03672
0:03821
0:02860
0:01690 0:04029
0:04782
0:04890 0:04671
0:03524
0:04291
0:04683 0:03899
0:02768
0:03107 0:02433
red no sync
M = {0}
P = {0}
C = {0} C = {0}
P = {0}
C = {0} C = {0}
0:4595
0:5031
0:5308
0:5380 0:5235
0:4752
0:4839 0:4667
0:4159
0:3958
0:3869 0:4046
0:4358
0:4437 0:4278
red half sync
M = {0}
P = {0}
C = {0} C = {0}
P = {0}
C = {0} C = {0}
0:6411
0:7037
0:7646
0:7606 0:7686
0:6434
0:6641 0:6226
0:5789
0:5550
0:5702 0:5399
0:6028
0:6116 0:5940
red full sync
System energy.
48
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M = {0}
P = {0}
C = {0} C = {0}
P = {0}
C = {0} C = {0}
0:03221
0:02986
0:02686
0:02869 0:02500
0:03284
0:03668 0:02904
0:03456
0:04010
0:03484 0:04536
0:02905
0:02380 0:03436
white no sync
M = {0}
P = {0}
C = {0} C = {0}
P = {0}
C = {0} C = {0}
0:03454
0:03065
0:02858
0:02806 0:02909
0:03269
0:03610 0:02931
0:03842
0:04609
0:05302 0:03929
0:03076
0:02682 0:03470
white half sync
C = {0}
P = {0}
M = {0} M = {0}
P = {0}
M = {0} M = {0}
0:03857
0:03691
0:03917
0:03522 0:04312
0:03467
0:04095 0:02838
0:04024
0:03923
0:03305 0:04553
0:04126
0:04897 0:03362
white full sync
M = {0}
P = {0}
C = {0} C = {0}
C = {0}
P = {0} P = {0}
0:03245
0:02768
0:02246
0:02262 0:02230
0:03288
0:02831 0:03745
0:03726
0:03386
0:03674 0:03097
0:04062
0:04450 0:03678
pink no sync
P = {0}
M = {0}
C = {0} C = {0}
C = {0}
M = {0} M = {0}
0:1182
0:1204
0:1180
0:1167 0:1192
0:1228
0:1197 0:1260
0:1161
0:1175
0:1191 0:1159
0:1146
0:1143 0:1148
pink half sync
M = {0}
P = {0}
C = {0} C = {0}
C = {0}
P = {0} P = {0}
0:1616
0:1665
0:1595
0:1481 0:1709
0:1735
0:1786 0:1684
0:1567
0:1536
0:1482 0:1589
0:1598
0:1655 0:1541
pink full sync
M = {0}
P = {0}
C = {0} C = {0}
P = {0}
C = {0} C = {0}
0:03092
0:02807
0:02450
0:01631 0:03269
0:03167
0:03380 0:02954
0:03375
0:04035
0:04250 0:03815
0:02718
0:02721 0:02714
red no sync
M = {0}
P = {0}
C = {0} C = {0}
P = {0}
C = {0} C = {0}
0:2040
0:2195
0:2419
0:2273 0:2565
0:1969
0:1994 0:1944
0:1884
0:1958
0:1886 0:2030
0:1811
0:1838 0:1783
red half sync
P = {0}
M = {0}
C = {0} C = {0}
M = {0}
C = {0} C = {0}
0:3115
0:3386
0:3673
0:3781 0:3566
0:3102
0:3050 0:3154
0:2844
0:3021
0:3111 0:2931
0:2667
0:2816 0:2519
red full sync
Exponent of Shannon entropy.
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Colour & synchrony of environmental noise
3.5.2 effects of noise colour and synchrony
The influence of noise colour and degree of interspecific synchrony on
variability of ecosystem energy and Shannon entropy, in terms of kernel
density distributions of outputs from multiple simulations (n ≈ 900), for
model configurations other than fcmp (shown in Fig. 3.3 in the main
text). a, unsynchronized species environments; b, half synchronized; c,
fully synchronized. Violins scaled to constant width.
0.0
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0.4
0.5
Configuration F
a
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System energy
exp[Shannon entropy] b
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1.5
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Noise colour (spectral exponent)
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Configuration FC
a
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exp[Shannon entropy] b
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2.0
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Configuration FM
a
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1.0 System energy
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1.2 System energy
exp[Shannon entropy] b
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53

4 extinction and invasion
do not add up
Abstract. Species extinction and invasion concurrently affect the composition
and properties of ecological communities, yet their effects have largely
been studied separately, and with more focus on species and ecological
functional groups than on the whole-community level. I adopted a dynamic
ecological network approach to compare the effects of simultaneous single-
species primary extinction and invasion to the effects of extinction and
invasion in isolation, using a set of ecosystem metrics. I also investigated
the relationship between the impact and reversibility of extinction or
invasion through reintroduction or eradication respectively. I used Monte
Carlo simulations of bioenergetic ecological network models that combined
trophic and mutualistic interactions, contained either prey-dependent or
ratio-dependent trophic functional responses, and incorporated either white
or pink environmental stochasticity. As the separate extinction or invasion
impact increased, the simultaneous extinction-invasion impact increased
but was decreasingly additive of the two separate impacts, across all
ecosystem metrics. Greater extinction or invasion impact was associated
with lower reversibility for most model types and ecosystem metrics. There
were also systematic differences between models with prey- and ratio-
dependent functional responses. These results highlight the importance of
considering the combined effects of extinction and invasion in ecological
studies, management and restoration.
4.1 introduction
Biodiversity loss is causing changes in ecosystem structure and functioning
on a global scale [286], while biotic invasion has been listed as one of the
most important global change drivers that influence biotic interactions
[287]. Accordingly, there has been extensive research on the impacts
of extinction and invasion on communities, with the ecological network
approach being increasingly used for understanding and predicting impacts
on complex ecosystems [18]. Furthermore, there is increased recognition of
55
Extinction & invasion do not add up
the importance of incorporating community dynamics in network studies
of extinction and invasion. For example, Curtsdotter et al. [288] found
that the effects of sequential species extinctions differ between static and
dynamic food webs.
Numerous studies have dealt with various aspects of food web structure
and dynamics as causes [289–292] and consequences [125, 293, 294] of
primary and secondary extinctions at different trophic levels. Because
whole-network time-series data are scarce, most such studies are theoretical,
although the models are usually constructed and parameterized using
empirical network topologies and community dynamics. Nevertheless, they
provide community- and ecosystem-level insights that are impractical to
obtain empirically. Such studies, however, have been restricted to trophic
interactions. Some recent studies have used empirical data to investigate
the consequences of extinction on networks of nontrophic interactions
either including [116] or excluding [295] trophic interactions, but these
studies did not incorporate community dynamics.
The effects of invasive species on ecological networks also constitute
an area of increasing research (see Olesen et al. [99] p. 44 for a review).
Empirical research has progressed from early descriptive studies [296] to
recent analyses of long-term data combined with dynamical simulations
[297]. Recent theoretical modeling studies have identified predictors of
invasion success [126, 298] and which measures best predict invasion impacts
on large food webs [30]. In terms of the effects of invasion on nontrophic
networks, most research has been on bipartite mutualistic networks of
pollinators and seed dispersers [299–301] and, as with extinction studies,
largely restricted to network topology without community dynamics [302–
304]. A few studies have examined other kinds of mutualistic [305] or
parasitic networks [306] restricted to particular pairs of taxonomic groups.
Existing network research on the effects of extinction or invasion
has one or more of the following limitations: they do not incorporate
community dynamics, the models contain only one interaction type, the
models are deterministic, or the metrics of perturbation impact pertain
to population- rather than community- or ecosystem-level properties.
56
Yangchen Lin
Indeed, responses of ecosystems as a whole to invasions are much less
known than responses of populations or communities [307]. Furthermore,
the vast majority of studies of the impacts of extinction and invasion
on ecological networks have so far examined extinction and invasion
in isolation from each other, or at invasion as a cause of extinction
[308]. There is a lack of studies examining the combined effects of
simultaneous extinction and invasion driven by separate causes [309]. The
importance of examining both processes concurrently is suggested by
Forys & Allen’s [310] empirical study of functional group change caused
by extinctions and invasions, and Jackson & Sax [311] who promulgate the
notion of ‘biodiversity dynamics’ in a changing environment as being the
shifting balance between species loss and gain. Indeed, non-native species
introduced by humans may mitigate the global ‘trophic downgrading’ of
food webs [312, 313].
Reintroduction of extinct species and eradication of invasive species are
increasingly being carried out in restoration ecology; an understanding
of the reversibility of extinction and invasion is therefore crucial to
restoration [314]. Lundberg et al. [315] found in dynamic models of
competitive communities that cascading extinctions could sometimes cause
community changes that preclude reinvasion, although they did not
include trophic and other types of nontrophic interactions and treated
reinvasion as a binary rather than continuous variable. The eradication of
invasive species can also have unexpected ecosystem-level outcomes because
of species interactions (review by Zavaleta et al. [316]). For example,
alien plant removal has been found to have negative impacts on rare
native plants in pollinator networks [317]. The importance of adopting
a network perspective on ecological restoration is increasingly being
recognized [116, 300]. Existing studies are mostly empirical investigations
of specific taxa and habitats (see Kardol & Wardle [314] and references
therein); more recent research has begun to demonstrate the role of
network simulation models. For example, Raymond et al. [318] dynamically
simulated the effects of invasive species eradication in a subantarctic
island’s species interaction network using and comparing different model
57
Extinction & invasion do not add up
structures, although they used random interaction strengths and did not
metabolically scale parameter values.
As Wardle et al. [309] state about the effects of alien consumers,
‘we have yet to move from a collection of impressive examples to the
development of general principles’. In my study, ‘general principles’ are
sought for the impacts and reversibilities of extinction and invasion
in ecological networks, where extinctions and invasions are driven by
independent processes. I also attempt to address issues recently identified
as high ecological research priorities [200, 307]: the role of rare species in
ecosystem functioning, what kinds of invasive species will affect ecosystem
properties, and how changes in ecosystem properties are related to changes
in community structure.
My strategy entails ensemble simulations to seek universal patterns in
the impacts of extinction and invasion (if any) across multiple contexts.
Empirical studies have limitations such as being snapshots in time, which
do not capture the actual mechanisms of biodiversity loss, or short-term
studies capturing mostly transient effects rather than long-term impacts
[319]. There can also be biogeographical or taxonomical biases, as has
been seen in field studies of the impacts of invasive alien plants [320].
This is where simulation modeling can provide valuable insights.
More specifically, I look at the effects of network connectance and
various extinctor (species going extinct) and invader species traits on the
impacts of primary extinction and invasion, the impact of simultaneous
extinction and invasion, and the reversibility of extinction and invasion
via reintroduction and eradication respectively. The impacts and reversibil-
ities are quantified in terms of changes in several ecosystem metrics
representing community structure, biomass, diversity and periodicity. This
multidimensional approach to quantifying ecological stability gives a more
comprehensive understanding, anticipation and management of effects of
perturbation on ecosystems [321]. In addition, it may be important to ac-
count for real-world environmental fluctuations in models; for example, Joo
et al. [322] found that deterministic models of bacteriophage-mediated bac-
terial invasion did not adequately capture dynamics that involved stochastic
58
Yangchen Lin
processes. This is the first study to incorporate positive interactions and
environmental stochasticity into bioenergetic food web modeling to inves-
tigate how extinction, invasion and combined impacts propagate through
noisy dynamics of species interactions to influence properties at the ecosys-
tem level of organization. This may ultimately help determine extinction
and invasion management actions with the best chance of success [323].
4.2 methods
The stochastic model is based on that documented in Chapter 2. Models of
11-species communities were generated with all different combinations of
niche and random trophic network topologies, prey- and ratio-dependent
functional responses and white and pink noise. No parasitism or direct
basal competition was implemented, for comparability across niche and
random network topologies. Trophic and mutualistic connectances were
varied within the range [0:05; 0:3] (similar to Romanuk et al. [126]) using
a Sobol’ low-discrepancy sequence [324] across replicates within each
topology-dynamics-noise configuration. The degree of interspecific syn-
chrony of response to noise was set at half-synchyronized for all models
(see Chapter 3). Dynamical parameter values are listed in Table 3.1.
The starting sample size for each topology-dynamics-noise configura-
tion was 100. Within each of the 100 replicates, four ‘parallel universe’
simulations were set up with the same parameters, initial conditions and
pseudorandom seed states, but with different combinations of extinction
and invasion (Fig. 4.1). For each simulation within a replicate, one ran-
domly selected species was removed prior to the start of the simulations,
to subsequently invade the system in the relevant simulations. Similarly,
another species was randomly selected for future extinction.
For each extinctor and invader, several species traits were quanti-
fied. These, and the trophic and mutualistic connectances, constitute the
explanatory variables (Table 4.1) whose effects on the impacts and re-
versibilities of extinction and invasion were investigated using regression
trees using the tree package in R [275]. The effects of individual ex-
planatory variables on the absolute magnitudes of the response variables
59
Extinction & invasion do not add up
(except extinction-invasion interaction) were also analyzed separately using
Kendall correlations, or Mann-Whitney j-tests in the case of categorical
variables (extinctor and invader basality). This procedure was then repeated
with [extinction and/or invasion impacts on community structure] as an
explanatory variable, against extinction and/or invasion impacts on the
five other ecosystem metrics.
Time (arbitrary units)
a b c d e
a
d
b
e
c
control
invasion eradication
extinction reintroduction
both
0 2000 4000 6000 8000 10000 12000
Figure 4.1. Simulation experiment design. Each of the four rows symbolizes a single
stochastic simulation of multispecies dynamics. The four simulations had identical
starting model structure, invader and/or extinctor species, parameter values, initial
conditions and noise time series. The diagram as a whole represents one of 100
replicates for each noise colour within each type of dynamics (prey-dependent and
ratio-dependent) within each network topology (niche model and random). The
red segments denote the parts of the time series over which various ecosystem
metrics were calculated. The difference in each metric between pairs of time series
was calculated as indicated by a (impact of invasion), b (impact of extinction), c
(impact of simultaneous invasion and extinction), d (reversibility of invasion, by
invader eradication) and e (reversibility of extinction, by extinctor reintroduction).
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Table 4.1. Abbreviations and description of explanatory variables.
Abbreviation Description
trocon trophic connectance
mutcon mutualistic connectance
invbasalnon is invader a basal species
extbasalnon is extinctor a basal species
invniche invader niche value (trophic level)
extniche extinctor niche value (trophic level)
invsim invader mean interaction similarity
(modified from Romanuk et al. [126] p. 1746 to combine trophic and nontrophic interactions)
extsim extinctor mean interaction similarity (modified from Romanuk et al. [126])
iesim mean interaction similarity of invader and extinctor (modified from Romanuk et al. [126])
ipred no¯ of consumer species of invader
iprey no¯ of resource species of invader
imut no¯ of mutualiser species with invader
epred no¯ of consumer species of extinctor
eprey no¯ of resource species of extinctor
emut no¯ of mutualiser species with extinctor
dominance relative extinctor abundance prior to extinction, when not under extinction pressure
(mean species energy over 2( (() ≤ time t ≤ 4( ((( divided by mean total system energy over same period)
ilink total no¯ of network links of invader
elink total no¯ of network links of extinctor
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Extinction & invasion do not add up
Extinction, invasion or simultaneous extinction and invasion was en-
forced at time t = 40 001, and reintroduction or eradication at t = 80 001.
A species was made extinct by instantaneously setting its state to zero.
This implementation is the most straightforward in representing different
causes of mortality. Isolated network clusters may occasionally be created
by species removal, but this does not violate ecological laws and the
separate clusters are still resource-limited. If a cluster ended up with
no basal species, it would die out as expected in reality. The state of
an invader at the time of invasion was set at 10% of what its initial
state would be if it were present as a native species at the start of
the simulation, as invasions usually begin with a few individuals; simi-
larly, species previously made extinct were reintroduced at 10% of their
initial states. Invader eradication was similar to extinction, with species
state instantaneously set to zero. Six ecosystem metrics (Table 4.2) were
calculated from time steps 20 001 to 40 000 (before extinction or inva-
sion), 60 001 to 80 000 (after extinction or invasion) and 100 001 to 120 000
(after reintroduction or eradication) to allow for most transients to pass
and to avoid capturing any spurious rate-dependent hysteresis caused by
rapid extinction. Preliminary simulations indicated that this time scale of
measurement encompassed all the main features and periodicities of the
stochastic behaviour.
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Table 4.2. Ecosystem metrics.
Metric Description
cv energy coefficient of variation over time of total biomass of all species
cv entropy coefficient of variation over time of ]pp[Shannon entropy] of relative species biomasses [273]
average energy average over time of total biomass of all species
average entropy average over time of ]pp[Shannon entropy] of relative species biomasses [273]
community structure set of all species states at each time step
power spectrum amplitude distribution of frequencies of biomass fluctuation [225] of each species
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Extinction & invasion do not add up
Impacts of extinction, invasion and simultaneous extinction and invasion
(hereafter referred to as ‘extinction-invasion impact’) were quantified
by comparing the ecosystem metrics of the perturbed versus control
simulations. Differences between parallel simulations make better measures
of extinction and invasion impacts than measuring the same simulation
before and after perturbation, because in the latter we would not know
whether any changes in the system after extinction or invasion were
actually caused by the perturbation or not.
Differences in the means and coefficients of variation of energy
and entropy were calculated as proportionate change, with the control
simulation as the baseline. The difference in community structure was
quantified as the absolute root mean square deviation of the time series
of a given species between the impacted system and the same time
steps in the control simulation (i.e. as if nothing happened throughout),
calculated species by species, and then summed across all species and
divided by the number of time steps, given byvuuut S∑s=1∑t {(Es;tpost;start − Es;tpre;start)2; : : : ; (Es;tpost;end − Es;tpre;end)2}
tpre;end − tpre;start + 1 (4.1)
where the subscripts pre and post indicate pre- and post-decline periods,
and start and end refer to the first and last time steps at which the
ecosystem metrics were measured.
The difference in the log power spectrum was quantified as for
community structure. The power spectrum, calculated as one of the
ecosystem metrics using default settings in the spectrum function in R,
is informative about the ‘internal dynamics’ of the system [214, 225, 325].
By revealing the detailed relationship between power and frequency, this
measure paints a picture of stability and periodicity at different time
scales—the system can be stable at one scale and unstable at another [326].
It can be indicative of changes in the interaction between stochasticity and
deterministic density-dependent cycles. The power spectrum was calculated
over 10 000 uniform frequency intervals from 0:000 05 to 0:5 (the Nyquist
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Yangchen Lin
frequency) cycles per time step. An alternative way to measure change in
power spectra is by means of spectral entropy, or Shannon entropy of the
power spectrum [327], but this method loses frequency information—a
spectrum that is a mirror image of another will have identical entropy.
In addition to measuring the extinction-invasion impact per se, I also
developed a measure of the interaction between extinction and invasion,
hereafter referred to as the extinction-invasion interaction. For each
ecosystem metric, the extinction-invasion interaction was calculated as
(extinction-invasion impact)−
[(extinction impact) + (invasion impact)]
(4.2)
from the parallel simulations. The expected impact of simultaneous extinc-
tion and invasion if they were purely additive is [(impact of extinction)+
(impact of invasion)]. If the interaction has a value of 0, extinction and
invasion do not interact (i.e. have purely additive effects). In the case
of changes in community structure and power spectra, which are always
positive, a negative interaction value means that extinction and invasion
mitigate each other’s impact while a positive value means that they re-
inforce each other. In the case of changes in average or coefficient of
variation of energy and entropy, which can take positive or negative
values depending on whether the extinction and/or invasion causes them
to increase or decrease, a negative interaction value simply means that
the interaction between co-occurring extinction and invasion is less than
purely additive, while a positive value indicates the opposite. In the case of
coefficient of variation of energy or entropy, positive may be detrimental
because it indicates that the variablility has increased i.e. stability is lower;
in the case of average energy or entropy, positive may be beneficial as
it indicates that the overall biomass or diversity has increased.
Reversibility of extinction or invasion depends on two factors: the
magnitude of the impact of extinction or invasion, and the extent to which
the system can return to its pre-extinction or pre-invasion characteristics
following reintroduction or eradication respectively. The impact was quan-
65
Extinction & invasion do not add up
tified as above, while the difference of the post-reintroduction/eradication
ecosystem from the control simulation where no disturbance ever happened
(this difference henceforth referred to as ‘reintroduction or eradication
effect’) was quantified by calculating the differences in the above metrics
after reintroduction/eradication compared to the same time steps in the
control simulation. The ‘relative reversibility’ of extinction or invasion—
relative to the extinction or invasion impact—was then calculated as extinction or invasion impact(reintroduction or eradication effect) + 10−100
. (4.3)
Thus the perturbation is relatively more reversible if after a larger
perturbation the system reverts to its unaffected condition. The addition
of 10−100 avoided division by 0; the value was chosen to be several
orders of magnitude smaller than the smallest non-zero data point. As
reversibility is a non-negative concept, what matters here is the magnitude
but not the sign of the impacts, so the absolute value is taken. To
see how the reintroduction or eradication effect itself is related to the
impact of extinction or invasion, ‘absolute reversibility’ was calculated as
1 minus the sample-normalized absolute values of the reintroduction or
eradication effect, where the smaller the effect, the greater the absolute
reversibility:
1− |reintroduction or eradication effect|
max|reintroduction or eradication effect| . (4.4)
Finally, the overall distributions of the response variables were com-
pared across topology-dynamics-noise configurations, to probe the relative
extents to which model type matters to model outputs. For the energy-
and entropy-related response variables, which could take on either posi-
tive or negative values, the distributions would also indicate the relative
directionality of the effects in addition to their magnitudes as described
above.
66
Yangchen Lin
4.3 results
Replicates were discarded in which noise terms and numerical integration
errors resulted in negative species states. The final sample size of each
topology-dynamics-noise configuration ranged from 72 to 98. Relatively
extreme values of response variables occurred in a small proportion of
replicates because some of the response variables, being proportionate
changes in some quantity, are divided by the original value of the quantity
which sometimes takes on very small fractional values, when populations
are very small. This is ecologically plausible and not abnormal, especially
as species states are in arbitrary units.
The regression trees showed no consistent interaction structure among
the explanatory variables of web connectances and species traits. The
explanatory variable explaining most of the deviance for each topology-
dynamics-noise configuration is listed in Appendix 4.5.1. For extinction
impact and extinction-invasion impact in most topology-dynamics-noise
configurations, extinctor dominance often accounts for most of the de-
viance in changes in power spectra and average entropy. Invader niche
value usually explains most of the deviance of invasion reversibility in
ratio-dependent niche models, across noise colours and ecosystem metrics.
An explanatory variable that explains most of the deviance in a regression
tree is not necessarily highly correlated with the response variable.
Tables of effects of individual explanatory variables on magnitudes of
response variables are found in §3.5 of the online supplement to Lin &
Sutherland [2] at http://dx.doi.org/10.1016/j.baae.2014.07.008. Extinctor
dominance is moderately strongly correlated to extinction impact for
many model types and ecosystem metrics across both white and pink
noise, in addition to explaining most of the deviance in many of the
regression trees. The correlations are mostly positive for all ecosystem
metrics except power spectrum, where the correlations are all negative.
The number of invader predators is usually moderately to moderately
strongly negatively correlated to invasion reversibility for all model types
except prey-dependent niche models, despite not often explaining most
67
Extinction & invasion do not add up
of the deviance in regression trees. Relationships for other explanatory
and response variables are mostly insignificant in comparison.
Relationships between impacts on community structure and impacts
on the other five community metrics are shown in Fig. 4.2 (tabulated in
Appendix 4.5.2). There are mostly moderately strong positive correlations
between community structure and energy- and entropy-related response
variables, across model types and noise colours (sample sizes: prey-
dependent models, 387; ratio-dependent models, 319). Correlations of
community structure with power spectra are generally not as strong, but
there is a notable difference between the positive correlations for prey-
dependent models and negative correlations for ratio-dependent models
in terms of invasion impact.
68
Y
angchen
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Figure 4.2. Relationships between impacts on community structure and absolute values of impacts on the other ecosystem
metrics, pooled across network topologies and noise colours within each of prey-dependent (black triangles) and ratio-
dependent (red squares) model types. Apparent bimodality of data points is an artifact of arctangent transformation
which facilitates visual comparisons but compresses large values into a small range. Appendix 4.5.2 gives correlations of
untransformed data.
69
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Community structure
Average energy
CV energy
Power spectrum
Average entropy
CV entropy
E
x
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Figure 4.3. Relationships of separate extinction and invasion impacts to (a) extinction-invasion impact and (b) extinction-
invasion interaction, pooled across model types and noise colours (n 5 7(6), with fitted generalized additive model surface.
Axes: E, extinction impact; I, invasion impact; all arctangent transformed for visual interpretability. Data points coloured
according to relative extinction-invasion impact, for clarity. Occasional bimodality of data points is an artifact of arctangent
transformation that does not affect the statistical analysis. In b, impacts of extinction and invasion are purely additive
when extinction-invasion interaction 5 (.
70
Y
angchen
L
in
Table 4.3. Kendall correlations of separate impacts of extinction and invasion to extinction-
invasion impact and to extinction-invasion interaction, pooled across model types and noise
colours.
Extinction-invasion… Ecosystem metric ext pext inv pinv
impact community structure (:62) ( (:6(5 (
cv energy (:404 ( (:301 (
cv entropy (:42 ( (:300 (
average energy (:5(1 ( (:43) (
average entropy (:5)5 ( (:344 (
power spectrum (:547 ( (:550 (
interaction community structure −(:400 2:75× )(−84 −(:552 3:27× )(−107
cv energy −(:257 ):70× )(−24 −(:)7 ):)4× )(−11
cv entropy −(:235 0:5)× )(−21 −(:202 2:42× )(−29
average energy −(:(7(5 (:((5(5 −(:)23 1:66× )(−7
average entropy −(:(45) (:(726 −(:2(4 5:32× )(−16
power spectrum −(:47 3:13× )(−78 −(:47) ):01× )(−78
Subscripts: ext, extinction; inv, invasion. Sample size n 5 7(6, significance level  5 (:((033 after Bonferroni
correction.
71
Extinction & invasion do not add up
Kendall correlations were calculated for impact of extinction against
extinction-invasion impact and against the extinction-invasion interaction,
and for impact of invasion against those same two variables (n = 706). The
analysis was done on data pooled across all model types and noise colours,
since there was little qualitative difference between them. Increasing impact
of extinction or invasion is strongly associated with increasing extinction-
invasion impact (Fig. 4.3a, Table 4.3). The extinction-invasion interaction,
however, becomes less additive and, as separate positive extinction and
invasion impacts increase, smaller than what it would be if the impacts
of extinction and invasion were purely additive (Fig. 4.3b, Table 4.3).
The relationship between extinction impact and absolute reversibility,
and between invasion impact and absolute reversibility, are shown in Fig.
4.4. Niche and random network topologies and white and pink noise
colours were pooled within each of prey-dependent and ratio-dependent
model types (sample sizes: prey-dependent models, 387; ratio-dependent
models, 319), since there were no qualitative differences in the data between
the pooled categories. All correlations are negative and often weak to
moderately strong. Impacts of extinction and invasion are usually more
strongly correlated with their respective reversibilities for prey-dependent
models than for ratio-dependent models.
Across all topology-dynamics-noise configurations, differences in the
outputs of prey- and ratio-dependent models are generally greater than
those between different network topologies and noise colours, except
for power spectra where niche and random network topologies make
the greatest difference (Fig. 4.5; sample sizes: prey-dependent niche, 194;
prey-dependent random, 193; ratio-dependent niche, 172; ratio-dependent
random, 147). Ratio-dependent models tend to produce less variable
impacts, reversibilities and extinction-invasion interactions, except for
power spectra. As expected, extinction tends to reduce average energy
and entropy while invasion tends to increase them (Fig. 4.5a), and
extinction and invasion tend to cancel out each other’s effects (in a
statistical-distribution context) when combined (comparing Figs. 4.5a and
4.5b). The directionality of energy- and entropy-related impacts, however,
72
Yangchen Lin
is less symmetric, the upper tails of the distributions tending to be
longer than the lower tails for both extinction and invasion (upper rows
of Figs. 4.5a and 4.5b). Interestingly, the shapes of distributions are
more similar between extinction impact and invasion impact for a given
model type than they are across model types within either extinction
or invasion (Fig. 4.5a). The extinction-invasion interaction is mostly near
zero (i.e. almost purely additive impacts of extinction and invasion) for
energy and entropy-related ecosystem metrics, but tends to negative and
highly negative for community structure and power spectra respectively,
but this is because of the abundance of high extinction-invasion impacts
for those metrics.
73
E
xtinction
&
invasion
do
not
add
up
Figure 4.4 (facing page). Relationship between (a) extinction
impact and absolute reversibility and (b) invasion impact and
absolute reversibility, pooled across network topologies and
noise colours within each of prey-dependent (upper row) and
ratio-dependent (lower row) model types. Kendall  signifi-
cance level  5 (:((033 after Bonferroni correction. Legend:
triangles/circles, niche/random network topologies; black/red
data points, white/pink noise.
74
Y
angchen
L
in
Community struct.
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τ = −0.545
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a CV energy
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| normalized extinction impact |
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Community struct.
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75
E
xtinction
&
invasion
do
not
add
up
Figure 4.5 (facing page). Effects of model type on (a) the im-
pacts (upper row of a) and relative reversibilities (lower row
of a) of extinction (dashed lines) and invasion (solid lines)
and (b) the extinction-invasion impact (upper row of b) and
extinction-invasion interaction (lower row of b). Black half-
violins, white noise; red half-violins, pink noise. Model types:
PDn, Prey-Dependent niche model; RDr, Ratio-Dependent
random-network model. Violins scaled to constant width and
connected at medians. The arctangent transformation produces
the artifact of bimodality in some violins, but facilitates visual
comparisons. Statistical analyses used untransformed data.
76
Y
angchen
L
in
0.0
0.5
1.0
1.5
Community structure
−0.5
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−0.5
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CV entropy
−0.5
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1.5
Average energy
−0.5
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1.0
Average entropy
1.44
1.46
1.48
1.50
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1.54
1.56
Power spectrum
a
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PDn RDnPDr RDr
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PDn RDnPDr RDr
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77
Extinction & invasion do not add up
4.4 discussion
That more dominant species tend to have greater impacts after extinction
is not surprising, since one would expect the loss of a dominant species
constituting a large proportion of the abundance or biomass to bring
a greater change to community structure, system energy and entropy.
For extinction-invasion impact, the effect of dominance may have been
weakened by the impact of invasion. Interestingly, consistent negative
rather than positive correlations of extinctor dominance with changes in
power spectra were observed. This may be because dominant extinctors
tend to have fewer predators (Appendix 4.5.3a); the extinction of a sparsely
linked dominant species may cause less disruption to the overall profile of
interacting population and noise cycles. The relationship between extinctor
dominance and number of predators was not substantially due to niche
model structure and metabolic scaling (Appendix 4.5.3b). Question 55 of
Sutherland et al. [200] asks, ‘How important are rare species in the
functioning of ecological communities?’ These results suggest that rare
species may not necessarily be more important than common (dominant)
species in the functioning of ecological communities. Rare species may,
however, be more prone to total extinction owing to their rarity, and there
may be cumulative impacts if multiple rare species are lost concurrently.
Where historical data are lacking, it may also be difficult to distinguish
between species that are naturally rare and species that are normally
common but are rare at sampling time because they are in the process
of going extinct.
Strayer [307] suggested that there does not seem to be a general
answer to the question of which invasions will change ecosystem function;
my results support this view that the impact of invasion is highly
context-dependent. Similarly, insights into Question 50 of Sutherland
et al. [200]—‘How relevant are assembly rules in a world of biological
invasion?’—can be gained from my results in that assembly rules, insofar
as the rules relate to the degree of niche overlap of the invasive and
native species, may have limited predictive power in invasion given the
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Yangchen Lin
multitude of counteracting factors. For example, for invasion success per
se, it is widely recognized that greater niche overlap may make it harder
to invade, but a smaller overlap may mean that the invader is less well-
adapted to the ecosystem than the natives and therefore less successful
in invasion [328]. This tradeoff may carry over to the invasion impact
on the ecosystem as a whole, and be further tempered by other factors.
My results support Simberloff’s [323] prognosis that invasive species
management decisions should be considered on a case-by-case basis, and
underscore the importance of multi-pronged strategies synthesizing other
lines of inquiry as disparate as invasive species distribution modeling [329]
and risk and decision analysis [330, 331]. Community ecology and network
dynamics are essential components of any such strategy, as might be
illustrated by a case of unexpected community consequences of a species
introduced for biological control [332].
For both extinction and invasion (and the two combined), larger
changes in community structure can potentially be accompanied by larger
changes in energy- and entropy-related ecosystem metrics, as expected,
although large changes in community structure are more likely to accom-
pany small than large changes in the energy- and entropy-related metrics
given the distribution of points in Fig. 4.2. Nevertheless, the results
imply that if large changes in the variability or magnitude of biomass
or overall species evenness are observed in an ecosystem of interest, it
may be generally expected that there has been a considerable change in
community structure that could potentially translate to changes in various
aspects of ecosystem function and ecosystem services.
The extinction, invasion and extinction-invasion impacts on energy-
and entropy-related ecosystem metrics are similarly likely to be negative
or positive in most cases (Fig. 4.5), suggesting that extinction and invasion
can be both beneficial and detrimental in terms of ecosystem function,
insofar as high carbon storage (system energy) and high diversity (Shannon
entropy) are widely recognized to be of ecological benefit. This outcome
supports shifting attitudes towards invasive species partly brought about
by a more system-oriented world view, with increasing debate on both
79
Extinction & invasion do not add up
positive and negative impacts at the ecosystem and socioeconomic levels
[333–335]. The upper tails of the distributions of those impacts tend to
be longer than the lower tails; this is attributed not to any ecological
phenomenon but to the measurement of impacts as proportionate change,
which sometimes generates large positive values as explained in §4.3.
In most of the models, the magnitude of the impacts on energy-
and entropy-related metrics is relatively small, partly due to just a single
extinctor or invader. The impacts on community structure and power
spectra, however, take on a wider range of magnitudes, probably because
these metrics deal with more mechanistic aspects of the ecosystem’s
behaviour, which can be affected in many qualitatively different ways,
rather than the total energy and diversity. In cases where separate
impacts of extinction and invasion are high, the extinction-invasion impact
increases but the extinction-invasion interaction becomes increasingly less
than purely additive. This might be attributable to a kind of system-
level ‘density dependence’ arising from the intrinsic resistance of the
system to perturbation, due to reproduction- and mortality-mediated
negative feedback loops acting within and across interacting species.
Such homeostasis, however, may break down with multiple invasions and
extinctions including secondary extinctions, and should not therefore be
taken for granted in the maintenance of ecosystem function.
The difficulty of reversing extinctions or invasions with large impacts
may be discouraging, but is not unexpected news. In the case of extinction,
this may be due to secondary extinction cascades that cannot be reversed
by simply reintroducing one species. In the case of invasion, it may
be due to invasive species having self-reinforcing effects on ecosystem
processes over time [336]. The results suggest that reversal efforts may
sometimes not be worth the difficulty, because of either small extinction
or invasion impact or low reversibility. One, however, needs to consider
reintroduction or eradication on a case-by-case basis, as the desired end
result need not be identical to the original state, and there have been
spectacular success stories in wolf reintroduction [337] and rat eradication
[338]. Further research in simultaneous extinction and invasion and their
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Yangchen Lin
reversibilities will also help in reconciling the sometimes conflicting goals
of endangered species management and invasive species eradication [339].
It has also been reported that complex interactions among multiple
invading species can affect invasion impact and reversibility on native
species [340]; this kind of approach could be extended to account for
network structure and dynamics.
Differences in behaviour between prey- and ratio-dependent models
tend to be more pronounced than differences in topology or even whether
extinction or invasion is involved. If niche models are considered as much
more appropriate than random networks, then perhaps more attention
should correspondingly be paid to whether prey or ratio dependence is
more ecologically appropriate. The proportionate fluctuations of predators
and prey that characterize ratio dependence may be responsible for the
differences from prey-dependent models observed in this study, such as
smaller variability in model behaviour across models of the same type of
dependence. This recalls the long-debated question as to whether prey-
or ratio-dependent models better represent biomass distributions in the
real world, and underscores the need to ‘get it right’ in predictive models
of specific systems.
The median extinction, invasion and extinction-invasion impacts on
power spectra are considerably higher for models with random topology
than those with niche topology (Fig. 4.5). This is possibly because
random networks are less hierarchical and less nested than niche-structured
networks; when one perturbs a niche model, it is more likely that there
are surviving species more similar to the extinctor and/or invader to
maintain the periodicity structure than when one perturbs a random
model. Random networks also translate to more trophic complementarity
(sensu Poisot et al. [341]) in the network overall; this implies that trophic
complementarity is potentially a predictor of the structure of community
periodicity, in addition to the other ecosystem functions studied by Poisot
et al. [341]. Nevertheless, looking at the rest of the ecosystem metrics
other than power spectra, it appears that communities with interaction
topologies that do not conform to the traditional niche model will not
81
Extinction & invasion do not add up
behave in radically different ways from those that obey niche rules;
indeed, Solé & Valverde [15] regarded food webs in general as having
relatively high randomness.
At the same time as general relationships were found across contingen-
cies of model structure and parameter values,7 the considerable variation
in model behaviour reflects real-world ecological variation and context
dependence of extinction and invasion impacts [307, 342], potentially across
both aquatic and terrestrial ecosystems. Future work along the lines of
this study will provide further insights into how we might influence
ecological network dynamics through selection of species (nodes) to eradi-
cate or reintroduce for increased recovery and resilience sensu Cornelius
et al. [343], perhaps using techniques inspired by control theory [344]
and appreciating that overall network topology may be more important
than individual node degree as a determinant of resilience [345]. Also
worthy of further inquiry is the relatively little known phenomenon of
degeneracy [77], in which species differing in traits can perform similar
functions, potentially mediating the impacts of extinction or invasion in
unexpected ways.
Finally, it may also be mutually beneficial for different disciplines
to understand the similarities and differences between ecological and
other kinds of networks in their responses to perturbations [346] and
to control measures [347–350]. At the same time, it is important to bear
in mind that the time-irreversibility of complex dynamical processes [8]
precludes perfect reversal of disturbances. As ecology matures into a
predictive discipline alongside the other sciences, the systems approach
offers rich prospects for exciting new fundamental research and more
effective environmental management.
7The similar outcome across noise colours does not contradict Chapter 3, because the concern
here is with the impacts of invasion and extinction and not with the ‘day-to-day’ behaviour
of the system per se. Modeling noise correctly for any given system is still important if more
specific predictions are desired.
82
Yangchen Lin
4.5 appendix
4.5.1 regression trees
Root predictors that explain the largest proportions of deviances in
extinction and invasion impacts, extinction-invasion interaction and relative
reversibilities, for different model types, noise colours and ecosystem
metrics. See Table 4.1 in main text for key to predictor abbreviations.
83
Extinction & invasion do not add up
Predictors of extinction impact.
Ecosystem metric Model type Noise Root predictor
community structure prey-dependent niche white dominance
pink extsim
prey-dependent random white extbasalnon
pink dominance
ratio-dependent niche white epred
pink extsim
ratio-dependent random white dominance
pink dominance
cv energy prey-dependent niche white eprey
pink dominance
prey-dependent random white elink
pink extniche
ratio-dependent niche white dominance
pink extbasalnon
ratio-dependent random white elink
pink dominance
cv entropy prey-dependent niche white eprey
pink eprey
prey-dependent random white extniche
pink extniche
ratio-dependent niche white extsim
pink emut
ratio-dependent random white dominance
pink extniche
average energy prey-dependent niche white eprey
pink extniche
prey-dependent random white elink
pink extsim
ratio-dependent niche white extbasalnon
pink dominance
ratio-dependent random white dominance
pink extbasalnon
average entropy prey-dependent niche white dominance
pink dominance
prey-dependent random white dominance
pink dominance
ratio-dependent niche white eprey
pink dominance
ratio-dependent random white dominance
pink dominance
power spectrum prey-dependent niche white dominance
pink dominance
prey-dependent random white dominance
pink dominance
ratio-dependent niche white extniche
pink dominance
ratio-dependent random white dominance
pink dominance
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Yangchen Lin
Predictors of invasion impact.
Ecosystem metric Model type Noise Root predictor
community structure prey-dependent niche white invsim
pink ilink
prey-dependent random white invbasalnon
pink imut
ratio-dependent niche white ipred
pink ipred
ratio-dependent random white ipred
pink invbasalnon
cv energy prey-dependent niche white invniche
pink invsim
prey-dependent random white invniche
pink ilink
ratio-dependent niche white invsim
pink iprey
ratio-dependent random white invsim
pink imut
cv entropy prey-dependent niche white invniche
pink invsim
prey-dependent random white invniche
pink invniche
ratio-dependent niche white ipred
pink iprey
ratio-dependent random white invniche
pink imut
average energy prey-dependent niche white invniche
pink imut
prey-dependent random white imut
pink ipred
ratio-dependent niche white iprey
pink invbasalnon
ratio-dependent random white invbasalnon
pink invbasalnon
average entropy prey-dependent niche white invniche
pink invsim
prey-dependent random white iprey
pink invniche
ratio-dependent niche white invniche
pink iprey
ratio-dependent random white invbasalnon
pink invbasalnon
power spectrum prey-dependent niche white invniche
pink invbasalnon
prey-dependent random white ipred
pink invsim
ratio-dependent niche white ipred
pink invniche
ratio-dependent random white ipred
pink ipred
85
Extinction & invasion do not add up
Predictors of extinction reversibility.
Ecosystem metric Model type Noise Root predictor
community structure prey-dependent niche white elink
pink eprey
prey-dependent random white dominance
pink extsim
ratio-dependent niche white dominance
pink dominance
ratio-dependent random white dominance
pink extsim
cv energy prey-dependent niche white elink
pink eprey
prey-dependent random white dominance
pink extsim
ratio-dependent niche white extsim
pink elink
ratio-dependent random white dominance
pink extniche
cv entropy prey-dependent niche white elink
pink eprey
prey-dependent random white dominance
pink extsim
ratio-dependent niche white extsim
pink elink
ratio-dependent random white dominance
pink epred
average energy prey-dependent niche white elink
pink eprey
prey-dependent random white elink
pink extsim
ratio-dependent niche white dominance
pink elink
ratio-dependent random white dominance
pink dominance
average entropy prey-dependent niche white elink
pink eprey
prey-dependent random white dominance
pink extsim
ratio-dependent niche white dominance
pink dominance
ratio-dependent random white dominance
pink epred
power spectrum prey-dependent niche white elink
pink eprey
prey-dependent random white elink
pink extsim
ratio-dependent niche white dominance
pink dominance
ratio-dependent random white dominance
pink extsim
86
Yangchen Lin
Predictors of invasion reversibility.
Ecosystem metric Model type Noise Root predictor
community structure prey-dependent niche white invsim
pink iprey
prey-dependent random white ilink
pink invsim
ratio-dependent niche white invniche
pink invsim
ratio-dependent random white iprey
pink ipred
cv energy prey-dependent niche white invsim
pink iprey
prey-dependent random white ilink
pink invsim
ratio-dependent niche white invniche
pink invniche
ratio-dependent random white ipred
pink invniche
cv entropy prey-dependent niche white invniche
pink iprey
prey-dependent random white ilink
pink invsim
ratio-dependent niche white invniche
pink invniche
ratio-dependent random white imut
pink iprey
average energy prey-dependent niche white invniche
pink iprey
prey-dependent random white ilink
pink invniche
ratio-dependent niche white invniche
pink invniche
ratio-dependent random white ipred
pink invniche
average entropy prey-dependent niche white invsim
pink iprey
prey-dependent random white ilink
pink invniche
ratio-dependent niche white invniche
pink invniche
ratio-dependent random white ipred
pink invniche
power spectrum prey-dependent niche white invniche
pink iprey
prey-dependent random white ilink
pink imut
ratio-dependent niche white invniche
pink invsim
ratio-dependent random white ipred
pink ilink
87
Extinction & invasion do not add up
Predictors of extinction-invasion impact.
Ecosystem metric Model type Noise Root predictor
community structure prey-dependent niche white extsim
pink trocon
prey-dependent random white iesim
pink dominance
ratio-dependent niche white trocon
pink ipred
ratio-dependent random white trocon
pink trocon
cv energy prey-dependent niche white eprey
pink extsim
prey-dependent random white trocon
pink extsim
ratio-dependent niche white invsim
pink dominance
ratio-dependent random white invsim
pink imut
cv entropy prey-dependent niche white eprey
pink eprey
prey-dependent random white invsim
pink extsim
ratio-dependent niche white extniche
pink dominance
ratio-dependent random white invsim
pink epred
average energy prey-dependent niche white eprey
pink imut
prey-dependent random white iesim
pink trocon
ratio-dependent niche white extbasalnon
pink dominance
ratio-dependent random white invbasalnon
pink dominance
average entropy prey-dependent niche white dominance
pink dominance
prey-dependent random white dominance
pink dominance
ratio-dependent niche white dominance
pink dominance
ratio-dependent random white dominance
pink dominance
power spectrum prey-dependent niche white dominance
pink dominance
prey-dependent random white dominance
pink dominance
ratio-dependent niche white extniche
pink invniche
ratio-dependent random white dominance
pink dominance
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Yangchen Lin
Predictors of extinction-invasion interaction.
Ecosystem metric Model type Noise Root predictor
community structure prey-dependent niche white ipred
pink extsim
prey-dependent random white trocon
pink extniche
ratio-dependent niche white trocon
pink ipred
ratio-dependent random white dominance
pink epred
cv energy prey-dependent niche white dominance
pink eprey
prey-dependent random white invniche
pink extniche
ratio-dependent niche white ipred
pink dominance
ratio-dependent random white dominance
pink imut
cv entropy prey-dependent niche white dominance
pink eprey
prey-dependent random white invniche
pink extniche
ratio-dependent niche white ipred
pink dominance
ratio-dependent random white dominance
pink invsim
average energy prey-dependent niche white emut
pink epred
prey-dependent random white ilink
pink extsim
ratio-dependent niche white invniche
pink extsim
ratio-dependent random white mutcon
pink invniche
average entropy prey-dependent niche white dominance
pink dominance
prey-dependent random white imut
pink trocon
ratio-dependent niche white imut
pink invniche
ratio-dependent random white mutcon
pink dominance
power spectrum prey-dependent niche white dominance
pink dominance
prey-dependent random white dominance
pink dominance
ratio-dependent niche white emut
pink invniche
ratio-dependent random white dominance
pink dominance
89
Extinction & invasion do not add up
4.5.2 effects of community structure
Kendall  correlations between impacts on community structure and
impacts on the other ecosystem metrics. Response variables were converted
to absolute values before statistical testing. Asterisked parentheses *( )*
indicate p <  where  = 0:01 after Bonferroni correction for five
ecosystem metrics.
EXTINCTION IMPACT
Model type Noise cv . energy cv . entropy av . energy av . entropy powspec
prey - dep . niche white *(0.412)* *(0.425)* *(0.538)* *(0.422)* -0.0653
pink *(0.401)* *(0.304)* *(0.489)* *(0.32)* -0.0781
prey -dep . random white *(0.517)* *(0.469)* *(0.607)* *(0.419)* *( -0.191)*
pink *(0.439)* *(0.378)* *(0.564)* *(0.359)* -0.149
ratio - dep . niche white *(0.35)* *(0.26)* *(0.399)* 0.0812 -0.176
pink *(0.271)* 0.111 *(0.447)* 0.171 *( -0.206)*
ratio - dep . random white *(0.585)* *(0.367)* *(0.53)* *(0.347)* *( -0.355)*
pink *(0.247)* 0.159 *(0.5)* 0.182 *( -0.295)*
INVASION IMPACT
Model type Noise cv . energy cv . entropy av . energy av . entropy powspec
prey - dep . niche white *(0.437)* *(0.379)* *(0.424)* *(0.234)* 0.0855
pink *(0.409)* *(0.298)* *(0.453)* *(0.18)* 0.107
prey -dep . random white *(0.533)* *(0.489)* *(0.644)* *(0.495)* 0.0421
pink *(0.461)* *(0.454)* *(0.54)* *(0.408)* 0.0481
ratio - dep . niche white *(0.297)* *(0.189)* *(0.484)* 0.117 *( -0.309)*
pink *(0.211)* 0.127 *(0.367)* 0.028 *( -0.233)*
ratio - dep . random white *(0.569)* *(0.498)* *(0.659)* *(0.423)* *( -0.342)*
pink *(0.495)* *(0.357)* *(0.701)* *(0.55)* *( -0.225)*
EXTINCTION-INVASION IMPACT
Model type Noise cv . energy cv . entropy av . energy av . entropy powspec
prey - dep . niche white *(0.396)* *(0.429)* *(0.462)* *(0.383)* 0.07
pink *(0.309)* *(0.204)* *(0.35)* *(0.226)* 0.0667
prey -dep . random white *(0.389)* *(0.291)* *(0.474)* *(0.272)* 0.00789
pink *(0.305)* *(0.372)* *(0.463)* *(0.364)* -0.105
ratio - dep . niche white *(0.279)* *(0.257)* *(0.22)* 0.145 -0.113
pink 0.133 -0.00728 0.137 0.121 -0.188
ratio - dep . random white *(0.387)* *(0.246)* *(0.336)* 0.0931 -0.129
pink *(0.226)* 0.0544 *(0.341)* 0.0739 -0.0595
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4.5.3 effects of extinctor dominance
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Relationships between extinctor dominance and (a) number of predators and (b)
niche value. Pooled across all models (a); pooled across niche-topology models (b).
91

5 collective effect of
interaction modifications
Abstract. Interaction modification (im), where one species modifies the
strength of the density-mediated direct interaction between two other
species, is an important ecological process, but little is known about
the collective effect of multiple im on overall community dynamics. I
use stochastic bioenergetic modeling of ecological networks with differ-
ent network topologies, functional responses and parameter values, to
investigate the effects of im connectance and im strength on ecosystem
properties including the evenness of species abundances and variability of
system biomass. It was found that the maximum system biomass observed
across the model systems increased with im connectance and strength
when the models had nonrandom topology and prey-dependent functional
responses as opposed to random topology and ratio-dependent responses.
The maximum observed species evenness increased with im strength but
decreased with increasing im connectance, when all modifications were
negative. These findings underscore the importance of accounting for
multiple im across the community for understanding complex community
dynamics.
5.1 introduction
Indirect effects are widely recognized as important drivers of the dynamics
of ecological communities, and are a subject of active empirical and
theoretical research [40, 351, 352]. There are two main types of indirect
effects, acting via interaction chains and interaction modifications [353].
In an interaction chain, one species can indirectly affect the population
density of a target species via density-mediated interactions with one
or more intermediary species, such as in exploitative competition and
trophic cascades [354], rather than direct interaction. On the other hand,
an interaction modification [355], also known by the jargonologies of ‘trait-
mediated indirect interaction’ [356], ‘non-consumptive effect’ [357] and
‘rheagogy’ [358], refers fundamentally to one species affecting the target
93
Collective effect of interaction modifications
species by modulating the latter’s interactions with other species. Both
types of indirect effects have been found in both empirical and theoretical
studies to vary widely in their impacts on the stability and relative species
abundances of ecological communities, depending on the relative strengths
of direct and indirect effects and the quantitative balance between positive
and negative feedback loops in ecological networks [354, 359]. This chapter
focuses on interaction modification (im), which is also a useful implicit
representation of behaviour [352].
The prevalence of im in natural communities has been recognized
in recent years. For example, predator modification of prey grazing has
been shown to be a mechanism accounting for much of the observed
variation in the strength of trophic cascades [360], and im has been
suggested as an alternative way to look at host-parasite interactions,
where the parasite affects the feeding efficiency of the host [156]. Some
mutualisms commonly formalized as pairwise interactions are actually part
of tripartite im, such as bipartite networks of ant-plant mutualisms where
ants mediate plant susceptibility to herbivory, or fish protected from
predation while living amidst and contributing nutrients to sessile marine
organisms [163, 168]. Werner & Peacor [356] reviewed the mechanistic
basis and empirical evidence for the presence of im in both aquatic and
terrestrial ecosystems and found that im were widespread in ecosystems
and often as strong as or stronger than density-mediated effects.
The importance of embracing im in understanding the response of
real-world communities to global change has recently been highlighted
[342], especially since im may play a role in engendering alternative stable
states [355]. Bolker et al. [361] found that most of the early theoretical
work on the effects of im on community dynamics pertained mostly to
the mathematical stability of simple communities and that actual dynamics
depended on details that were rarely measured empirically, such as func-
tional responses. An ecosystem-level understanding also requires a grasp
of the collective effects of the multiple im present in a given community,
but most research has involved analysis of one im in a particular com-
munity (e.g. Hsieh & Perfecto [351]). There is little knowledge on either
94
Yangchen Lin
the overall topology and distribution of strengths of im in the entire
community, or how multiple im collectively affect ecosystem properties.
Multiple im present in a community may interact antagonistically with
one another to reduce the total im effect on a given direct interaction,
but little is known about the detailed relationships [40]. There is also
evidence for cascading im, where a species modifies the feeding activities
of another species that, in turn, modifies the density-mediated direct
interaction between yet another two species [362], but its implications at
the ecosystem level are unknown.
Theoretical studies have begun to show that incorporating im can
have a range of ecosystem effects. In models with four trophic levels, the
presence of positive im gave rise to increased efficiency of abiotic nutrient
use [358]. Goudard & Loreau [251] showed that greater connectance or
magnitude of im in interaction web models with three distinct trophic
levels tended to reduce species richness, biomass and production. More
recently, Kamran-Disfani & Golubski [352] found that different kinds
of im-related adaptive behaviour can either facilitate or inhibit the
propagation of disturbances through food webs with three or four distinct
trophic levels and that the effects also depended on food web structure.
Although they examined each behaviour separately, they proposed that
future models should incorporate multiple types of im (representing
both foraging and defence) simultaneously. The incorporation of im in
models of entire ecological communities is still in its infancy; most
bioenergetic models of food webs, which represent the state of the art in
dynamic community modeling, do not incorporate im. Models that have
incorporated im are often simplified in terms of network topology, such
as having a small number of trophic levels with no omnivory or looping
[251, 352].
I investigate the collective effects of different degrees of im on various
ecosystem-level properties of bioenergetic ecological network models with
realistic network structure and metabolically scaled community dynamics.
I compare the responses of models with random and nonrandom network
topologies and also with prey- and ratio-dependent functional responses,
95
Collective effect of interaction modifications
while incorporating realistic partial synchrony of species responses to pink
environmental noise. A wide range of ecologically plausible parameter
values are explored. This approach aims to clarify how im may influence
the macroscopic properties of complex ecological communities in noisy
real-world environments, with potential implications for their predictive
modeling and management.
5.2 methods
Models were based on the specifications in Chapter 2 and categorized into
four possible configurations of trophic network topology and functional
response: (1) topology based on the niche model of food webs combined
with prey-dependent consumer-resource functional responses, (2) random
topology with prey dependence, (3) niche topology with ratio dependence
and (4) random topology with ratio dependence. All networks comprised
25 species. Parasitism and direct basal competition were omitted for
comparability with random-topology models. Pink environmental noise
with partial synchrony of species responses (§2.3) was used, as this was
considered the most realistic.
The prey-dependent consumer-resource functional response was mod-
ified from Eqn. 2.2a following the im implementation of Goudard &
Loreau [251]) and is given by
Φi;j =
Mi;j!i;jEj
h
0:5h +
∑
k∈resources
Mi;k!i;kEk
h
(5.1)
with resource preference ! and Hill exponent h as before. The additional
term M is defined as
Mi;j =
S∏
k=1
(1 + Ek)
i;j;k , (5.2)
being the total effect on the interaction between species i and j caused
by all species that modify that interaction,  being the strength of
interaction modification by a given species k [251]. This formulation of
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Yangchen Lin
im satisfies basic ecological requirements [251], such as increasing with
the modifier species’ biomass, having no effect when the biomass is zero,
and giving the same direction of energy flow regardless of the sign of
the modification. The corresponding ratio-dependent functional response
is given by
Φi;j =
Mi;j!i;j(Ej=Ei)
h
1h +
∑
k∈resources
Mi;k!i;k(Ek=Ei)
h
. (5.3)
Although im, as implemented by Goudard & Loreau [251], can be both
antagonistic and facilitative, it is still important to retain in the model
facilitative interactions that depend directly on the states of the interacting
species, as this is fundamentally different from im [353]. Kéfi et al. [41]
recommended having both of them in dynamical models of ecological
networks.
There is little empirical knowledge of real topologies of im in eco-
logical networks. The topology of im across the network was therefore
assigned randomly; any type of species could influence the interaction
between one or more species pairs of any type, representing all pos-
sible types of adaptive behaviour—foraging, defence or otherwise (see
Dambacher & Ramos-Jiliberto [355] p. 231). Furthermore, a modifier species
can concurrently interact directly with one or both of the species whose
mutual interaction it modifies (see de Roode et al. [363] for an empirical
example). Cannibalistic links were excluded from im.
im connectance is defined in this study as Goudard & Loreau’s ‘non-
trophic connectance’ [251]: the number of realized interaction modifications
divided by the total number of possible interaction modifications, where
number of possible modifications is the total number of species multiplied
by the number of trophic links, minus two per link to account for the
fact that the two species interacting via a given link cannot modify their
own interaction. The im connectance was varied from 0 to 1 [251] in
increments of 0:1, and the calculated number of im links was rounded
to the nearest integer. I examined the effects of magnitude and spread
97
Collective effect of interaction modifications
of im strengths within each level of im connectance. Two separate sim-
ulation experiments were conducted, one looking at different range sizes
(hereafter ‘spread’) of im strengths around mean 0, the other looking at
different magnitudes of negative im strengths. In the first experiment, 
was drawn from a uniform distribution of mean 0 and range size from
[−0:1; 0:1] to [−0:5; 0:5] in increments of 0:1 on each side of the mean
(Goudard & Loreau [251] used [−0:2; 0:2]). In the second experiment, 
was drawn from a uniform distribution of range [−0:5;−0:4] to [−0:1; 0:0]
in increments of 0:1. Positive magnitudes were not used because they
were often found to give ecologically unrealistic behaviour, with species
undergoing ‘runaway’ population growth to extreme values. For both
experiments, a set of 100 control simulations was done with no im.
For each combination of model configuration, im connectance and
im strength, 100 independent model realizations and simulations were
executed. The models were simulated for 20 000 time steps, and time
steps 10 001 to 20 000 were extracted for analysis, having allowed initial
transients to pass and all species to either reach steady state or be close to
steady state. Pilot simulations indicated that the extracted time window
encompassed all the main features and periodicities of the stochastic
behaviour.
Four ecosystem metrics were measured from the community time series
as response variables: the means and standard deviations (sd) of system
energy and entropy over time. Although a measure of the effective
number of species [273], the entropy reduces here to a measure of
species evenness, since the total number of species is constant in this
study. Unlike previous chapters, the coefficients of variation were not
extracted here, because I wished to distinguish between systems with high
mean and variability and those with low mean and low variability, both
of which would result in similar coefficients of variation. In addition to
looking at the energy and entropy metrics individually, I also examined
via the evenness-variability (mean entropy-sd energy) relationship, which
is one way of looking at the diversity-stability relationship. Higher
variability is interpreted as lower stability.
98
Yangchen Lin
5.3 results
Simulations were discarded where extreme species behaviour resulted in
computational infinity values caused by high positive im strength (also
responsible for a few very large data points discussed below), and
occasionally where negative species states were produced due to the noise
terms and intrinsic numerical integration errors. The sample size of each
treatment ranged from 97 to 100. Overall, there was little difference
between the control simulations with no im and those with the lowest
connectance and lowest spread and magnitude of im, for all of the
response variables; the control experiment is therefore omitted from the
analysis.
99
Collective
effect
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interaction
m
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0.2 0.4 0.6 0.8 1.0
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Figure 5.1. Effect of im connectance and im spread on maximum observed means (upper row) and sd (lower row) of system
energy. The surface vertex at each connectance-spread setting denotes the mean response of the models with the highest
5% of response values among about 100 independent replicate models with the given connectance and spread. The four
main model configurations are segregated by column.
100
Y
angchen
L
in
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Figure 5.2. Effect of im connectance and mean im magnitude on maximum observed means (upper row) and sd (lower row)
of system energy. The surface vertex at each im connectance-mean setting denotes the mean response of the models with
the highest 5% of response values among about 100 independent replicate models with the given im connectance and im
mean. The four main model configurations are segregated by column.
101
Collective effect of interaction modifications
In the ‘spread’ experiment where both positive and negative im are
present, the maximum mean energy over time, as observed across all
models, increases dramatically with both im connectance and strength
for niche-topology prey-dependent models (Fig. 5.1, note log-transformed
data), while the other three model configurations exhibit much less change,
if at all. All model configurations show limited influence by all-negative
im (Fig. 5.2). In both experiments, the minimum observed energy changes
much more slowly, if at all (Figs. 5.5, 5.6 in Appendix 5.5.1). The sd of
system energy over time also increases with im connectance and strength
for niche-topology prey-dependent models (Figs. 5.1, 5.2). Ratio-dependent
models also have lower maximum observed mean and sd energy, and are
appreciably less variable in these maxima, across im connectances and
strengths compared to prey-dependent models (Figs. 5.1, 5.2). The mean
and sd of system energy are strongly positively correlated across the
board (Fig. 5.7 in Appendix 5.5.1) as both are caused by high production
rates leading to greater tendencies of cycles of population overshoot and
collapse.
102
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angchen
L
in
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Niche topo, ratio-dep. Random topo, ratio-dep.
Figure 5.3. Effect of im connectance and mean im magnitude on maximum (a) and minimum (b) observed mean entropies,
for different model configurations of topology and functional response. The surface vertex at each im connectance-mean
setting denotes the mean response of the models with the highest (a) or lowest (b) 5% of response values among about
100 independent replicate models with the given im connectance and im mean.
103
Collective effect of interaction modifications
The maximum observed mean species evenness decreases when all im
in a given system are increasingly negative and of increasing connectance
(Fig. 5.3a). Niche-topology ratio-dependent models have on average the
highest maximum observed mean evenness for any given im connectance
and magnitude. The minimum observed mean evenness shows little or
no systematic change, with niche-topology ratio-dependent models having
on average considerably higher minimum evenness than the other model
configurations (Fig. 5.3b). Models with random topology tend to have lower
evenness than models with niche topology, for the same im connectance
and strength (Fig. 5.3a). The sd of evenness shows little or no change
with increasing im negativity and connectance; this is also the case when
both negative and positive ims are present (all Pearson |/| and Kendall
| | < 0:1).
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0 5 10 15 20 25
a
b
Species evenness (mean exp[Shannon entropy])
V
ar
ia
b
il
it
y
 (
S
D
 s
y
st
em
 e
n
er
gy
)
Figure 5.4. Relationships between species evenness and system variability, by model
configuration (columns). Row a, both positive and negative im co-occurring; row b,
only negative im present. Data pooled across im connectances and strength levels.
PD, Prey-Dependent; RD, Ratio-Dependent.
104
Yangchen Lin
Evenness-variability relationship. Prey-dependent models exhibit two clus-
ters of data points with opposite relationships, while ratio-dependent
models have single clusters (weak to moderately strong positive Kendall  ,
most p < 0:05) resembling the lower variability-higher evenness clusters of
the prey-dependent models with the respective niche or random network
topologies (Fig. 5.4). Inspection of simulated species trajectories of prey-
dependent models showed that systems with both very low evenness and
very high variability tended to be characterized by one or two species
having considerably higher energy states than the other species and
continuing to increase monotonically in energy within the measurement
window for the response variables while the other species had reached
steady state. These simulations constituted the minority of the sample and
were not omitted because their behaviour merges gradually with those in
which all species reached steady state, with no thresholds of evenness or
variability. The data clusters with opposite relationships occur regardless
of im settings and therefore appear to be unrelated to im (see Appendix
5.5.2 for plots of unpooled data).
5.4 discussion
The implementation of im by Goudard & Loreau [251] used in this study,
with randomization of interaction modification strengths, is considered
by Golubski & Abrams [40] as an ‘extreme simplification’. The overall
formulation, however, is sufficient for studying the fundamental effects
of im compared with no modification, across contingencies represented
by the randomization of the magnitudes and directions of im strengths.
Olff et al. [82] suggest further types of indirect interactions that could
be examined in future work.
When negative and positive im are equally abundant, they tend to
cancel out one another’s effects on maximum observed entropy and
maximum observed system energy as expected, dampening the effects of
im connectance and strength except for the energy of prey-dependent
models with niche topology (Fig. 5.1). The exceptional behaviour of the
prey-dependent models may be partly because predators at higher trophic
105
Collective effect of interaction modifications
levels can have many prey species in a system with niche topology, and
can increase to particularly high abundances by collectively depressing
their numerous diet species to low-energy steady states when their
functional responses are only prey-dependent and enhanced by strong
positive im. In this case the negative im do not cancel out the positive im
because the potential population growth due to positive im has no upper
bound while population reduction due to negative im has a lower bound
of zero. In contrast to niche topology and prey dependence, random
topology averages out the number of prey species per predator, and
ratio dependence regulates predator population growth more tightly with
or without im by making the predator and prey populations fluctuate
proportionately (see Arditi & Ginzburg [180]). Nevertheless, caution must
be exercised in comparing prey and ratio dependence in terms of
quantitative absolute values, because the half-saturation parameter value
used in the functional response is different and because of the fundamental
structural differences in the models.
In addition, strong and abundant negative modifications are associated
with lower maximum observed evenness (Fig. 5.3a). This could be due to
negative im weakening the direct interaction strengths between species,
causing their abundances to be less dependent on one another, everything
else being equal. Interestingly, models with random topology have lower
maximum observed evenness (Fig. 5.3a). This may be related to the random
links between the metabolically scaled species rate equations giving rise to
a higher proportion of consumers with smaller niche values (body sizes)
than their resources, giving rise to a higher incidence of deviations from
the smooth energy pyramid. Such a scenario may apply to real systems
in which parasite diversity is on par with host diversity. In contrast,
niche-topology ratio-dependent models have higher maximum observed
evenness on average, and considerably higher minimum observed evenness
on average, than all the other model configurations (Fig. 5.3b). This
could be due to ratio dependence mandating a more even distribution
of energy among predator and prey because of their proportionate
population change, combined with the aforementioned energy-pyramid
106
Yangchen Lin
effect. Along with the results for system energy, the results for evenness
lean towards niche topology-ratio dependence as a high evenness-low
variability configuration that may be beneficial in terms of ecosystem
function and robustness.
For all im settings examined, the evenness-variability relationship is
generally higher evenness-higher variability for ratio-dependent models
while prey-dependent models are split into two clusters with opposite
relationships, the high evenness-low variability cluster representing those
systems that have reached steady state and resembling the relationship for
ratio-dependent models (Fig. 5.4). For ratio dependence especially, the
results appear to be in conflict with the current consensus on biodiversity
generally having stabilizing effects on various kinds of stability, including
stability in terms of ecosystem variability [241]. The solution to this
paradox may be that my measure of ‘diversity’ is restricted to the species
evenness component: when species abundances are more evenly distributed,
species are on average at greater liberty to ‘explore’ their dynamical state
spaces, as contrasted with a highly uneven system heavily dominated by
one species that monopolizes most of the resources leaving very little
leeway for the others. Indeed, the latter kind of system simulated in
this study may represent real systems under siege by invasive species
or threatened by increasing activity by the human species. This study
therefore provides fresh insights on the diversity-stability relationship,
given that most studies have considered the number of species but not
their relative abundances.
Notably, my results seem to depart qualitatively from those of Goudard
& Loreau [251] in terms of the maximum observed mean system energy
being positively associated or not associated with im connectance and
spread, depending on functional response (Fig. 5.1), opposing the rela-
tionship found by Goudard & Loreau for equilibrium biomass. Possible
factors include the use of more realistic network topology, functional
responses and stochasticity, and the lack of explicit nutrient limitation, in
my model; further work may shed light on how important these factors
are for building realistic ecological models.
107
Collective effect of interaction modifications
In reality, im are most likely to be antagonistic to each other [40]
such that multiple positive im are collectively less positive, and negative
im less negative, than their sum, but little is known empirically. My study
assumes, as in Goudard & Loreau [251], that the strengths of multiple im
are independent of one another, such that increasing the im connectance
causes the total im strength to increase additively. This may lead to
overestimation of the effect sizes, but I argue that the findings will be
qualitatively similar. This is because it is likely that the total im strength
would still accumulate in nature with increasing im connectance, only
that it would be less than purely additive. In addition to such ‘density
dependence’ of combined im strength, the variation of im strength over
time i.e. adaptive trait dynamics [364] also needs better understanding.
In conclusion, although individual im have been reported to have
limited impact on food webs [352], I found for a wide range of
parameter values that the multiple im that would be present in a real
ecosystem can have ecosystem-level effects depending on the strengths
and connectance of those modifications, but the effects pertain mostly to
the potential maximum boundary conditions rather than those in which
most of the model ecosystems reside. This suggests that multiple im
in a community tend to cancel out one another’s effects under average
conditions, supporting the view of Golubski & Abrams [40] that models
that lack im may still give ‘adequate predictions’ for more speciose
systems, in which im tend to have smaller net effects. In addition to
elucidating the effects of im, the model behaviours observed in my study
also underscore the importance of modulating the balance between prey-
and ratio-dependent functional responses appropriately. Prey-dependent
models are much more frequently used and, if used in food webs, most
commonly combined with niche topology. In such cases, model behaviour
may exhibit higher productivity and variability and lower evenness than
reality, if the community were actually more ratio-dependent. Finally,
the study of im can also provide insights into the role of ecosystem
engineers, or species that physically modify their environment. One of
the effects of ecosystem engineers is to ‘modulate’ the links between
108
Yangchen Lin
other species via the environment [365]; this process is essentially im.
Ecosystem engineering has been investigated only in simple food webs,
and much remains to be discovered about its influence on larger and
more complex ecological networks [365].
109
Collective effect of interaction modifications
5.5 appendix
5.5.1 supplementary figures
110
Y
angchen
L
in
0.2 0.4 0.6 0.8 1.0
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IM connectance IM
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IM connectance IM
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rea
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IM connectance IM
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IM connectance IM
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IM connectance IM
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IM connectance IM
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Niche topo, prey-dep. Random topo, prey-dep. Niche topo, ratio-dep. Random topo, ratio-dep.
Figure 5.5. Effect of im connectance and im spread on minimum observed means (upper row) and sd (lower row) of system
energy. The surface vertex at each connectance-spread setting denotes the mean response of the models with the lowest
5% of response values among about 100 independent replicate models with the given connectance and spread.
111
Collective
effect
of
interaction
m
odifications
0.2 0.4 0.6 0.8 1.0 −0.4
−0.3
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0
IM connectance IM
 m
ean
0.2 0.4 0.6 0.8 1.0 −0.4
−0.3
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−0.1
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0
IM connectance IM
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ean
0.2 0.4 0.6 0.8 1.0 −0.4
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0
IM connectance IM
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ean
0.2 0.4 0.6 0.8 1.0 −0.4
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IM connectance IM
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ean
0.2 0.4 0.6 0.8 1.0 −0.4
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IM connectance IM
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ean
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IM connectance IM
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ean
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IM connectance IM
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ean
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IM connectance IM
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ean
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Niche topo, prey-dep. Random topo, prey-dep. Niche topo, ratio-dep. Random topo, ratio-dep.
Figure 5.6. Effect of im connectance and mean im magnitude on minimum observed means (upper row) and sd (lower row)
of system energy. The surface vertex at each im connectance-mean setting denotes the mean response of the models with
the lowest 5% of response values among about 100 independent replicate models with the given im connectance and im
mean.
112
Yangchen Lin
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a
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log mean system energy
V
ar
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b
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 (
S
D
 s
y
st
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 e
n
er
gy
)
Figure 5.7. Relationships between mean and sd of system energy, by model con-
figuration (columns). Row a, both positive and negative im co-occurring; row b,
only negative im present. Data pooled across im connectances and strength levels.
PD, Prey-Dependent; RD, Ratio-Dependent.
5.5.2 plots of unpooled data
Relationships between species evenness and system variability, plotted
separately for each im connectance and strength level within each model
configuration within each simulation experiment (see §5.2 for methodolog-
ical details). Axes same as Fig. 5.4 in main text. Axis limits are different
(maximized) in each subplot.
113
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Species evenness (mean exp[Shannon entropy])
IM connectance
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
im spread experiment: niche-topology prey-dependent models
114
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angchen
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0.2
Species evenness (mean exp[Shannon entropy])
IM connectance
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
im spread experiment: random-topology prey-dependent models
115
Collective
effect
of
interaction
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Species evenness (mean exp[Shannon entropy])
IM connectance
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im spread experiment: niche-topology ratio-dependent models
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Species evenness (mean exp[Shannon entropy])
IM connectance
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im spread experiment: random-topology ratio-dependent models
117
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0.45
Species evenness (mean exp[Shannon entropy])
IM connectance
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
im magnitude experiment: niche-topology prey-dependent models
118
Y
angchen
L
in
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Species evenness (mean exp[Shannon entropy])
IM connectance
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im magnitude experiment: random-topology prey-dependent models
119
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Species evenness (mean exp[Shannon entropy])
IM connectance
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
im magnitude experiment: niche-topology ratio-dependent models
120
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angchen
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IM connectance
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
im magnitude experiment: random-topology ratio-dependent models
121

6 species decline
in compartmentalized networks
Abstract. The complex network approach is increasingly used to further
our understanding and management of complex systems in a wide range
of disciplines, including ecology. It has recently been recognized that
networks are often compartmentalized, and that this can have important
implications for their resilience to perturbations. I constructed stochastic
bioenergetic models comprising two compartments each containing trophic
and mutualistic interactions, with varying levels of intercompartment con-
nectivity. After steady state was reached, I simulated species declines in
one compartment and quantified their impacts on the biomass, diversity,
community composition and time series power spectra of both compart-
ments. Intercompartment trophic connectivity (density of feeding links
between the compartments) was positively correlated with the impacts
of species decline on the Shannon diversity of the compartments, but
negatively correlated with the impacts on power spectra of species time
series. Impacts on community composition were positively correlated with
connectivity when the network topology was based on the niche model
of food webs, but negatively correlated when the topology was random.
In contrast with trophic connectivity, the effects of intercompartment
mutualistic connectivity (density of mutualistic links) were much smaller.
The results show that differing intercompartment connectivity in networks
can cause the impacts of species decline to have different magnitudes and
directions depending on network topology and the ecosystem property
being measured. The variability found across replicate simulations also
highlights the context dependence of ecosystem behaviour and suggests
that caution should be exercised when applying conclusions from one
specific system to other systems.
6.1 introduction
In our increasingly interconnected world there is a growing interest in,
and relevance of, the network approach to understanding the complex
123
Species decline in compartmentalized networks
systems that pervade our environment, many of which have hitherto been
studied in a reductionistic way, in disciplines as disparate as neuroscience
and ecology. The scientific community is now appreciating the vulnerability
of human-designed or human-impacted natural networks to catastrophic
tipping points and need for multidisciplinary integrated global systems
science to address such problems [366]. To this end of capturing the
totality of interdependencies that govern the characteristics of real-world
complexity, the study of ‘networks of networks’ has been hailed as the
next frontier of complexity science [367, 368].
Real networks are usually highly compartmentalized [369, 370], ranging
from the World Wide Web [371] to the brain connectome [372], to
metabolic and signalling networks in cells [373, 374] and to ecological
networks of interacting species [375]. Compartmentalization can have
consequences for the behaviour or resilience of networks to perturbation
[376, 377], such as by limiting the extent of cascades [26, 378]. There
has been a corresponding development of network structural complexity
metrics that account for compartments and their nonplanarity [379].
In ecological networks of interacting species, a compartment can be
defined as some group of highly interconnected species that are highly
interconnected with one another but connected relatively sparsely to other
such groups. Pimm & Lawton [380] found little empirical evidence for
compartmentalization in food webs within relatively homogeneous habitats.
Since then, however, with more recent and more highly resolved data sets,
it has been found that many kinds of ecological networks do exhibit
compartmentalized topology, such as food webs [381, 382] and insect-
plant interaction networks [383, 384]. Furthermore, compartmentalization is
generally recognized to enhance stability and persistence in food webs
[48, 290].
In this study, I consider two-compartment networks, each compartment
being a spatially distinct ecological network that contains a different set
of species. Examples include canopy, understorey and soil compartments
in forest, and aquatic-terrestrial systems. Little research has been done on
compartmentalization at this scale, most studies having looked at either
124
Yangchen Lin
green or brown webs in isolation [385] or compartmentalization within
what is actually a functional group within a larger ecological network,
for example host-parasitoid communities in forest [262, 386]. Some recent
empirical studies have demonstrated that intercompartment interactions at
the habitat level play important roles in community dynamics. For example,
Giery et al. [387] demonstrated the existence of trophic interactions and
corresponding resource flux between canopy and understorey food webs
in woods. Trophic links between above-ground and below-ground webs
cause dynamic feedbacks [385], while those between elements of a spatial
mosaic of food webs in a floodplain affect the carrying capacity and
recovery of predator populations [388]. Facilitative interactions can also
occur between spatial compartments. For example, waterbodies created by
beavers support insects that, in turn, benefit terrestrial bats [389].
Very few studies have considered the explicit interaction topology of
compartmentalized networks at the ecosystem level. Pocock et al. [116]
found varying effects of species decline on the various subnetworks,
Valiente-Banuet & Verdú [295] reported that anthropogenic impacts can
act synergistically to cause network collapse, while Evans et al. [390]
found that the loss of certain habitats has disproportionate effects on
network integrity. These studies, however, did not investigate the effect
of compartmentalization per se. Furthermore, each compartment in these
studies collated a particular type of interaction of interest, rather than
the actual co-occurrence of interactions of multiple types between all
the closely interacting species in a given subhabitat, and considered only
network topology without dynamics.
The aforementioned studies provide important insights for specific
systems that are likely to be highly context-dependent; there thus re-
mains scope for more general theoretical understanding of the role of
compartmentalization in the dynamics of complex ecological networks, as
most models of ecological networks are not compartmentalized [391]. An
exception is Krause et al. [375], who examined the effects of removing
nodes from compartmentalized food webs constructed from empirical data,
and found that compartmentalization limited the extent of cascades, re-
125
Species decline in compartmentalized networks
sulting in greater stability. They, however, did not incorporate community
dynamics, which can be an important factor in mediating the stability of
the system [32]. Stouffer & Bascompte [290] came to similar conclusions
with compartmentalized food web models that incorporated community
dynamics, but did not incorporate nontrophic interactions or interaction
modifications.
Here, I use holistic models of ecological networks to investigate the
simplest case of compartmentalization, in which two such networks are
connected to each other by varying numbers of trophic and mutualistic
links, or connectivity. I ask how differing levels of intercompartment
trophic and mutualistic connectivity affect the impact and propagation of
species declines in one compartment both on itself and on the other
compartment. I assess the impact using an array of ecosystem metrics
that are indicative of ecosystem health, as well as of broad interest, such
as biomass, diversity and community composition.
6.2 methods
The network models comprised two independently generated (Chapter
2) 15-species compartments linked to each other, making a total of 30
species in the system. In a given model realization, the two compartments
had the same trophic topology, which was either random or based on
the generalized niche model (see Chapter 2), and the niche range was
kept constant at (0; 1) for both compartments. These are not ecologically
unreasonable simplifying assumptions, even though differences between
compartments have sometimes been observed, such as between forest soil
and above-ground food webs [392]. Ratio-dependent consumer-resource
functional responses were used. Interaction modification (Chapter 5) was
confined to within compartments, with connectance 0:1 and modification
strengths drawn from a uniform distribution in [−0:1; 0:1]. Environmental
stochasticity was implemented as pink noise with partial interspecific
synchrony (§2.3).
As there is little empirical information on the actual topology of inter-
compartment links, trophic and mutualistic links were randomly assigned
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between the compartments. The intercompartment trophic and mutualistic
connectivities were both varied from 0:05 to 0:3 in increments of 0:05
in a full factorial design, while the within-compartment trophic and mu-
tualistic connectances were kept constant at 0:3, near the upper limit of
the empirically observed range [158]. Assigning intercompartment links at
random means that species most highly interconnected with one another
(i.e. modules) may not correspond exactly to compartments in a given
model realization. The two main modules in each model were therefore
identified using the community detection algorithm in the igraph package
[393] in R, and models in which the modules did not correspond exactly
to the compartments were discarded.
The directedness of the intercompartment trophic links was decided
as follows: for niche-topology models, the species of higher niche value
in a species pair would be the consumer if both were non-basal species;
for random-topology models, one of the species was randomly assigned
as consumer. If one of the pair was a basal species, it was fed on
by the non-basal species regardless of niche value or network topology.
Basal species pairs were excluded from trophic interactions. Finally, the
Laplacian matrix of the entire network was computed, as in §2.1, to
ensure that there were no isolated components.
The models were simulated to 20 000 time steps to allow initial
transients to pass, and a set of four ecosystem metrics was calculated
using the simulation output from time steps 20 001 to 40 000 after steady
state had been reached. The ecosystem metrics (§4.2) were the average
total system energy over time, average exponent of Shannon entropy
over time, community composition8 and power spectrum. From time step
40 001 onwards, five randomly selected species (hereafter ‘declinees’) in one
compartment (hereafter ‘proximal compartment’, the other compartment
referred to as ‘distal’) were subjected to chronic decline of 0.01 of
the species states at each time step until the end of the simulation at
8This is the ‘community structure’ of previous chapters; the terminology is altered here to
avoid confusion with the term of the same name in network science.
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Species decline in compartmentalized networks
time step 80 000.9 The decline rate was chosen such that its effect was
considerably greater than that of the environmental noise.
The same ecosystem metrics were recalculated post-decline using time
steps 60 001 to 80 000. The impact of decline on each ecosystem metric
was then calculated as detailed in §4.2, separately for each compartment.
A sample of 128 independent model realizations was generated for each
intercompartment trophic-mutualistic connectivity treatment. Another set
of simulations and analyses was also run as already described, but with
the declinees being non-basal species with the five highest niche values
in the proximal compartment, representing the decline of large animals
or top predators.
Simulations were discarded where negative species states were pro-
duced due to the noise terms and intrinsic numerical integration errors.
In addition, extreme values occasionally occurred in impacts measured as
proportional change, as explained in §4.3, resulting in infinity values via
computer rounding error; these replicates were also taken out of the
analysis. As stated earlier, models in which modules did not correspond
to compartments were discarded. The final sample size for each intercom-
partment trophic-mutualistic connectivity combination ranged from 63 to
128.
Regression trees were constructed to identify the most important de-
terminants of decline impact among the following: compartment topology
(niche or random), niche range of the declinees in a given simulation
(either the full niche range of the community or the top five predators
only), intercompartment trophic and mutualistic connectivities, numbers of
intercompartment trophic and mutualistic links associated with declinees,
the number of declinees that are basal species, and what I term the
average trophic centrality and average mutualistic centrality of declinees.
The last two variables are defined as the average number of trophic and
mutualistic links, respectively, that each declinee has in a given network
realization; they are similar to degree centrality. The trophic centrality
9This implementation is more dynamically realistic than the instantaneous extinctions of pre-
vious chapters, although it is not absolute extinction.
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combines both predator and prey links to capture the essence of energy
flow through the declinees.
The main analysis then focuses on the relationships between inter-
compartment connectivity and decline impacts on the various ecosystem
metrics in the proximal and distal compartments.
6.3 results
The regression trees (Appendix 6.6.1) show that network topology explains
the greatest proportion of the deviance for all ecosystem metrics in both
compartments. There are small correlations (all Kendall | | < 0:2, most
p < 0:0125 after Bonferroni correction for four ecosystem metrics) between
intercompartment trophic connectivity and impacts on proximal or distal
compartments (Table 6.1); see below for details for each ecosystem metric.
Correlations for intercompartment mutualistic connectivity are generally
weaker (most | | < 0:1; Table 6.1).
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Table 6.1 (facing page). Kendall correlations between inter-
compartment connectivity and decline impacts on ecosystem
metrics.
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Compartment Ecosystem metric Declinee niche range Network topology .troph ptroph .mut pmut
proximal average energy full niche −0:031 0:00457 −0:0786 6:06× 10−13
random 0:0389 0:000426 −0:0424 0:000121
high niche −0:106 1:57× 10−22 −0:121 1:04× 10−28
random 0:0379 0:000561 −0:0366 0:000852
average entropy full niche 0:1 0 −0:0786 5:95× 10−13
random 0:118 0 −0:0394 0:000356
high niche 0:198 0 −0:0869 1:31× 10−15
random 0:122 0 −0:0495 6:64× 10−6
community composition full niche 0:0926 0 0:0367 0:000785
random −0:0386 0:000464 0:0045 0:683
high niche 0:135 0 0:0543 5:89× 10−7
random −0:0467 2:11× 10−5 0:0281 0:0105
power spectrum full niche −0:119 1:52× 10−27 −0:0587 7:52× 10−8
random −0:162 1:04× 10−48 −0:0182 0:099
high niche −0:105 4:18× 10−22 −0:114 9:99× 10−26
random −0:165 9:31× 10−51 −0:0332 0:00249
distal average energy full niche −0:0705 1:06× 10−10 −0:0368 0:000737
random −0:032 0:00374 −0:00228 0:836
high niche −0:114 1:6 × 10−25 −0:0456 2:74× 10−5
random −0:0157 0:153 0:0112 0:307
average entropy full niche 0:046 2:48× 10−5 −0:0312 0:00427
random −0:0274 0:0129 −0:00451 0:683
high niche 0:0882 4:44× 10−16 −0:00776 0:476
random −0:0124 0:258 −0:000492 0:964
community composition full niche 0:12 0 0:0284 0:00919
random −0:03 0:00644 −0:0212 0:0547
high niche 0:131 0 0:0393 0:000302
random −0:032 0:00353 −0:027 0:0139
power spectrum full niche −0:0674 6:76× 10−10 0:0767 2:12× 10−12
random −0:0996 1:66× 10−19 −0:00258 0:815
high niche 0:00908 0:404 0:176 0
random −0:107 2:95× 10−22 −0:00177 0:872
Compartments—proximal: where declines happen; distal: the other compartment. Declinee niche ranges—full: random
sample of all species in proximal compartment; high: species with five highest niche values. Subscripts: troph, intercompart-
ment trophic connectivity; mut, intercompartment mutualistic connectivity. Significance level  5 (:()25 after Bonferroni
correction for four ecosystem metrics.
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Intercompartment trophic connectivity levels
0.05 0.1 0.15 0.2 0.25 0.3
Figure 6.1. Decline impacts on average system energy for increasing intercom-
partment trophic connectivity with intercompartment mutualistic connectivity (:(5.
Results for other mutualistic connectivities are not illustrated as they show a similar
pattern. In each panel, the x-axis is the impact on system energy and the y-axis is
the rank of simulations sorted in increasing order of the impact on system energy
in the proximal compartment, demarcated by the red curve. Impacts on the distal
compartments of the respective simulations are marked by the free ends of the
black horizontal lines away from the red curve. A horizontal line thus measures
the difference between impacts on the proximal and distal compartments of a
given simulated network, and whether the distal impact is smaller (horizontal line
extends leftwards from curve) or greater (opposite direction) than the proximal
impact. Vertical dashed lines mark the value ( on the x-axis (no impact). Upper
row, niche-topology models; lower row, random-topology models; declinees in each
simulation were species with the five highest niche values. Axis ranges fixed across
all model configurations and intercompartment connectivities.
Niche-topology models undergo greater changes in average system
energy in the proximal compartment than random-topology models, es-
pecially in the niche-topology models where declinees are restricted to
species with the highest niche values (Fig. 6.1). The impact on average en-
tropy in the proximal compartment is relatively highly positively correlated
with intercompartment trophic connectivity for all model configurations
and declinee niche ranges (Table 6.1). Niche-topology models tend to
experience less negative and greater positive impacts in the proximal
compartment as intercompartment trophic connectivity increases, especially
when declinees are restricted to species with the highest niche values
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(Fig. 6.2). Unlike niche-topology models, random-topology models have
mostly negative changes in average entropy in the proximal compartment,
becoming decreasingly negative with increasing intercompartment trophic
connectivity (Fig. 6.2).
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0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0
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Impact on average Shannon entropy
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Intercompartment trophic connectivity levels
0.05 0.1 0.15 0.2 0.25 0.3
Figure 6.2. Pairwise comparisons between decline impacts on average entropy in
proximal versus distal compartments. Detailed explanation as for Fig. 6.1.
Niche-topology models undergo greater changes in community com-
position in the proximal compartment than random-topology models,
especially niche-topology models where the declinees are restricted to
species with the highest niche values (Fig. 6.3). Impacts on the community
composition of both proximal and distal compartments of niche-topology
models are positively correlated with intercompartment trophic connec-
tivity, while the corresponding impacts for random-topology models are
negatively correlated with intercompartment trophic connectivity (Table
6.1).
For all model configurations and declinee niche ranges, impacts on
the power spectrum of the proximal compartment are relatively highly
negatively correlated with intercompartment trophic connectivity (Table
6.1). For any given model, a large impact in the proximal compartment
is usually combined with a small impact in the distal compartment, while
a model experiencing a small impact in the proximal compartment may
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0 2 4 6 0 2 4 6 0 2 4 6 0 2 4 6 0 2 4 6
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Intercompartment trophic connectivity levels
0.05 0.1 0.15 0.2 0.25 0.3
Figure 6.3. Pairwise comparisons between decline impacts on community composition
in proximal versus distal compartments. Detailed explanation as for Fig. 6.1.
sometimes experience a large impact in the distal compartment (Fig.
6.4). Compared to niche-topology models, random-topology models have
a higher frequency of larger impacts on the power spectra of both
proximal and distal compartments overall (Fig. 6.4). These differences
between niche and random models are similar to those found in Chapter
4 (p. 76, Fig. 4.5a, extreme upper right panel).
0
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100 300 500 100 300 500 100 300 500 100 300 500 100 300 500
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Intercompartment trophic connectivity levels
0.05 0.1 0.15 0.2 0.25 0.3
Figure 6.4. Pairwise comparisons between decline impacts on power spectra in
proximal versus distal compartments. Detailed explanation as for Fig. 6.1.
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Finally, there do not generally appear to be meaningful correlations
between impacts in the proximal versus distal compartments, within
each model configuration and intercompartment connectivity setting (most
| | < 0:1)—the impact in the distal compartment does not seem to depend
on the impact in the proximal compartment (Figs. 6.1–6.4).
6.4 discussion
For all ecosystem metrics, compartmentalization has weak but systematic
effects on the response of the compartments to species decline, even
with the extensive parameter space of my simulations, although network
topology is a more important driver. Another interesting finding is that,
on average, the impact on the distal compartment usually changes little
with the impact on the proximal compartment for any given model
configuration and intercompartment connectivity, implying that cascading
effects do not propagate far. This may seem to downplay the danger of
cascading effects that have long been of concern for ecological prediction
and conservation, but complacency should not be allowed to set in—the
results of individual simulations (see also Appendix 6.6.2) clearly show
that, depending on the specific suite of parameter values (i.e. context),
the impact on the distal compartment may sometimes be relatively large
even when the impact on the proximal compartment is small, suggesting
a large cascade effect.
Negative impacts on the average system energy of proximal com-
partments of niche-topology models are greater when declinees are all
top predators than when they are randomly distributed in the trophic
spectrum (Appendix 6.6.2). This may be attributed to species of larger
body size having higher population biomass [177], and has implications
for the impact of biodiversity loss in real ecosystems, where top predators
often decline first. More explicitly, lower trophic levels do not appear
to increase appreciably in biomass compensating for top predator loss,
meaning that a ‘topless’ ecosystem may have a diminished quantity as
well as variety of ecological production, functions and socio-economically
valuable services.
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Species decline in compartmentalized networks
As entropy is a measure of the effective number of species, one would
expect the average entropy of the compartment in which the declines
took place to change more than the other compartment. The results
confirm this. Interestingly, when top predators decline in models with
niche topology, the impact on entropy in the proximal compartment
becomes less negative and more positive on average as intercompartment
trophic connectivity increases (Fig. 6.2). This may be attributable to the
randomly assigned intercompartment links ‘diluting’ the hierarchical niche
topology of the compartment, evening out the energy (biomass) flows
across the community.
In terms of community composition, greater changes occur in niche-
topology models where the declinees are top predators (Appendix 6.6.2),
suggesting that top-down effects and mechanisms such as mesopredator
release [394] are relatively important, at least in communities where top
predators have broader-than-random diets. This again has implications
for real-world scenarios where top predators are more vulnerable. The
opposing behaviours of models with niche and random topologies are
striking. This result underscores the importance of using appropriate
network topology when modeling a given system of interest for ecological
prediction and management.
The slightly lower occurrence of large impacts on power spectra
in either the proximal or distal compartment in models with higher
intercompartment connectivity could be partly because the relatively large
changes in periodicity regime caused by the perturbation in the proximal
compartment are dampened or diluted more by the species in the distal
compartment when the intercompartment connectivity is higher. There is
considerable variation in the magnitude of impacts on power spectra across
simulations, and the impacts in the proximal and distal compartments in
a given simulation can be very different from each other. The higher
frequency of larger impacts in random-topology models suggests that
random topology tends to have lower resilience to perturbations in terms
of the manner in which the community oscillates, as characterized by the
power spectrum. The variability of impacts, also observed in the other
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ecosystem metrics to a lesser extent, is a symptom of context dependency,
and again underscores the difficulty of predicting how perturbations will
propagate from one compartment to another. The details of species
traits and interaction structure mediate the exact pattern of amplitude
and frequency of oscillations. It also reminds us that caution should
be exercised when extrapolating ecological conclusions and management
recommendations for a specific system to other systems that, even if
superficially similar, may differ in important details.
There is considerable scope for future research into network com-
partmentalization in ecology and beyond. Firstly, it remains to be seen
whether the results would change with the number of compartments. A
particularly interesting aspect of this question is that the answer may
depend on how multiple compartments are connected to one another.
The two-compartment case is nevertheless useful for understanding how
perturbation propagates from one compartment to another, especially
when two-compartment systems can occur at some spatial scale in real-
ity, for example aquatic-terrestrial. Secondly, it is possible to combine
compartmentalizations at different scales for more realistic representa-
tions that mimic the nested or fractal structure of nature, somewhat
akin to the ‘super networks’ proposed by Olesen et al. [99]. For exam-
ple, Scotti et al. [100] modeled conspecific individual-individual networks
within species interacting in food webs that are in turn linked to one
another at the landscape level. Their model, however, lacked key inter-
actions such as individual-based interspecific interactions and nontrophic
interactions. Thirdly, the compartmentalized network approach could offer
new insights into the emerging concept of keystone communities [395]
whose removals have disproportionate dynamical impacts on metacommu-
nities. The network approach can accomplish this by enabling important
dynamic species interactions to be captured in ecosystem-level indicators
of metacommunity resilience.
The findings of this study contrast with some other findings about
other kinds of networks. For example, intermediate rather than minimum
or maximum connectivity between power grid networks has been found
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Species decline in compartmentalized networks
to minimize the risk and size of cascades [378]. This may be due either
to real differences in the processes governing different kinds of networks,
or possibly to my model assumptions such as equal average strengths of
intra- and intercompartment interactions. More mechanistic studies like
these will progressively give us a better idea of the degree of context
dependence of the relationship between compartmentalization and network
robustness to node failure. Furthermore, even deeper insights may be
gained by examining compartmentalization as an effect as well as a cause,
an ecological example being the effect of invasions on the way plant-
pollinator networks are compartmentalized [396]. Finally, pervading the
various proposed avenues for further inquiry is the challenge of acquiring
data on compartmentalization in ecological networks at the large scale
that has recently been efficiently accomplished [397] for compartments in
other kinds of large real-world networks, such as social networks and
the Internet.
6.5 conclusion
My results show that ecological network compartmentalization can mediate
the impacts of species decline in different ways depending on the
ecosystem property of interest, implying that there are no straightforward
rules governing how compartmentalization affects ecosystem stability or
resilience. In addition, the impact in the proximal compartment not
only exists as a direct effect of the decline, but also changes with the
intercompartment connectivity. These phenomena underscore an important
characteristic of the system, that of two-way feedback: perturbations in
one compartment are not only propagated to the other compartment
but also ‘bounced’ back to the originating compartment, sometimes as
a buffering effect. The results also suggest that both network topology
and declinee body size may be important in predicting the impacts of
declines on compartmentalized networks. My simulation configuration of
top predators declining from the niche-topology network is probably the
most frequent real-world scenario among those tested, based on empirical
observations of species loss [398] and food web structure (§2.1). It should
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also be appreciated that overarching all of the above observations is the
context dependence clearly manifested in my results as variation across
individual model simulations—an emergent observation not apparent in
isolated ecological studies of individual systems, of which only those
with ‘positive’ results tend to get published. I recommend that efforts
should continue to face up to the challenge of context dependence in
ecology, because it has implications not only for the understanding of
the complexity of ecological systems, but also for practical management.
There have been calls for combining food web theory and landscape
ecology to understand food webs at the landscape level [399, 400]; my
work takes a step in that direction, incorporating community dynamics
and different types of species interactions.
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Species decline in compartmentalized networks
6.6 appendix
6.6.1 regression trees
Regression tree for each of four ecosystem metrics, showing how vari-
ous explanatory variables mediate the impact of species decline in the
proximal (a) and distal (b) compartments of simulated ecological networks.
Explanatory variables: topo, network topology; trocon, intercompartment
trophic connectivity; mutcon, intercompartment mutualistic connectivity;
decnicherange, declinee niche range; decbasal, number of basal decli-
nee species; dectrolnk, number of declinee species with intercompartment
trophic links; decmutlnk, number of declinee species with intercompart-
ment mutualistic links; trocen, average trophic centrality of declinees;
mutcen, average mutualistic centrality of declinees. Only the nodes with
the greatest deviances are split, to show the most important explanatory
variables.
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in
topo: niche
decnicherange: full
trocen < 9:9
decbasal < 1:5
decbasal < 3:5
decmutlnk < 9:5
−0:140
−0:200
−0:140
−0:130 −0:160
−0:130 −0:170
−0:250
−0:240
−0:230 −0:250
−0:280
−0:079
Average energya topo: niche
decnicherange: full
decbasal < 1:5 trocon < 0:175
dectrolnk < 5:5 decbasal < 2:5
−0:0220
0:0087
−0:0120
−0:0260 −0:0053
0:0290
0:0140
−0:0005 0:0230
0:0440
0:0280 0:0530
−0:0520
Average entropy
topo: niche
decnicherange: full
decbasal < 1:5
trocen < 9:9
decbasal < 3:5
trocen < 11:5
0:44
0:70
0:53
0:40 0:58
0:50 0:67
0:88
0:81 1:10
0:86 1:20
0:18
Community composition
topo: niche
trocon < 0:075 dectrolnk < 9:5
trocen < 8:5
dectrolnk < 1:5
mutcen < 5:8
mutcen < 7:5
18
13
19 11
23
28
32 26
47
12 68
100 44
25
19
Power spectrum
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topo: niche
trocen < 11:3
dectrolnk < 7:5 trocen < 13:7
−0:0190
−0:0290
−0:0200
−0:0150 −0:0250
−0:0400
−0:0370 −0:0530
−0:0089
Average energyb topo: niche
trocen < 11:3
dectrolnk < 7:5
dectrolnk < 6:5
−0:0044
−0:0011
−0:0029 0:0011
−0:0032 0:0016
−0:0078
−0:0054 −0:0086
Average entropy
topo: niche
trocon < 0:175
trocon < 0:075 decbasal < 2:5
trocen < 14:3
0:180
0:260
0:240
0:220 0:250
0:290
0:270 0:310
0:300 0:360
0:099
Community composition
topo: niche
trocon < 0:075
decbasal < 2:5
trocon < 0:225
trocen < 5:9
mutcen < 8:1
15:0
11:0
17:0
21:0 13:0
9:5
19:0
21:0
51:0
79:0 24:0
21:0
16:0
Power spectrum
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6.6.2 plots from all treatments
In each subpanel, the x-axis is the impact on the ecosystem metric and
the y-axis is the rank of simulations sorted in increasing order of the
impact on the ecosystem metric in the proximal compartment, demarcated
by the thick curve. Impacts on the distal compartments of the respective
simulations are marked by the free ends of the grey horizontal lines away
from the curve. A horizontal line thus measures the difference between
impacts on the proximal and distal compartments of a given simulated
network, and whether the distal impact is smaller (horizontal line extends
leftwards from curve) or greater (opposite direction) than the proximal
impact. Vertical dashed lines mark the value 0 on the x-axis (no impact).
Axis ranges fixed across all model configurations and intercompartment
connectivities within each ecosystem metric.
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Average system energy: declinees from full niche range, niche topology.
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Impact on average system energy
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Average system energy: declinees from full niche range, random topology.
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Impact on average system energy
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im
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Intercompartment trophic connectivity levels
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Average system energy: declinees from high niche range, niche topology.
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Impact on average system energy
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Intercompartment trophic connectivity levels
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Average system energy: declinees from high niche range, random topology.
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Species decline in compartmentalized networks
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Impact on average Shannon entropy
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Average entropy: declinees from full niche range, niche topology.
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Average entropy: declinees from full niche range, random topology.
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Species decline in compartmentalized networks
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Impact on average Shannon entropy
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Average entropy: declinees from high niche range, niche topology.
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Average entropy: declinees from high niche range, random topology.
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Species decline in compartmentalized networks
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Impact on community composition (RMSD)
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Impact on community composition (RMSD)
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Intercompartment trophic connectivity levels
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Species decline in compartmentalized networks
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Impact on community composition (RMSD)
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Intercompartment trophic connectivity levels
In
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Community composition: declinees from high niche range, niche topology.
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Impact on community composition (RMSD)
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Intercompartment trophic connectivity levels
In
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Community composition: declinees from high niche range, random topology.
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Species decline in compartmentalized networks
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Impact on power spectrum (RMSD)
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Intercompartment trophic connectivity levels
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Power spectrum: declinees from full niche range, niche topology.
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Impact on power spectrum (RMSD)
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Intercompartment trophic connectivity levels
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Power spectrum: declinees from full niche range, random topology.
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Species decline in compartmentalized networks
0 200 400 600
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Impact on power spectrum (RMSD)
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Intercompartment trophic connectivity levels
In
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co
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Power spectrum: declinees from high niche range, niche topology.
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Impact on power spectrum (RMSD)
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ra
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Intercompartment trophic connectivity levels
In
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Power spectrum: declinees from high niche range, random topology.
159

The author's sanctum for inner contemplation. Mediæval brick
arch (1490) and timber columns (1912) beneath the oriel window of
the Long Gallery, Queens' College, Cambridge. Photograph by
the author exhibited at the Arts Festival (2014) in the Old Hall
(1449, visible at upper left), Queens' College.

7 coda
As ecologists, we are very good at finding linear relationships while
probing one component of a specific system under a specific set of con-
ditions, but when it comes to the totality of interdependent components
under diverse conditions, we hitherto have only a very blurry image.
This thesis, synthesizing a broad range of fundamental mechanisms of
species interaction in ecological network dynamics, has tried taking some
tentative steps towards beginning to clarify that image. There are two
overall revelations: that the structure, dynamics and perturbation of com-
plex ecological networks have effects that are highly context-dependent,
and that various interesting systematic patterns nevertheless emerge above
this context dependence. Where do we go next?
One obvious path is towards application to real-world conservation. A
systems approach can facilitate preventive proactive conservation, because
one is less likely to let problems develop unnoticed in one part of the
system while preoccupied with conserving another part. The public may
be reluctant to implement system-level conservation measures, however, if
such measures do not have direct and obvious benefits to their immediate
component (e.g. charismatic species) of interest [19, 116]. There is also the
challenge of identifying metrics can be easily measured in the field that
are indicative of ecosystem health [401]. Nevertheless, we are increasingly
conscious of the need for ‘system-level conservation ecology’ [38] and
‘conservation of networks’ [402], although some calls for exploiting the
power of systems approaches in conservation still focus on particular
species as the final deliverable [403]. The pioneering steps have been
taken, such as in a field study of the consequences of urbanization on the
structure of bird-plant networks [404], helping to understand demographic
impacts of environmental change. In time, scientific advancements and
changes in public perception will hopefully enable the full potential
of systems strategies to be realized. Meanwhile, given that theoretical
studies constitute a sizeable proportion of systems ecology research owing
to data scarcity, it might be worthwhile considering whether and how
163
Coda
theoretical modeling outputs could serve as evidence for the evidence-
based conservation paradigm championed by Sutherland et al. [405].
More specific to modeling methodology, some overarching issues remain
to be addressed in future work. It would be interesting to investigate
network topologies that are influenced more by phylogeny, or other drivers,
than by body size; research in this direction has so far considered different
interaction types in isolation [24, 406–409]. Yan & Zhang [410] also looked
at how interactions that do not increase monotonically with population
density affect community persistence, although they did it separately
for predator-prey and bipartite mutualistic dynamic networks under a
deterministic framework and did not subject their model communities
to exogenous perturbation; there is scope here for more realistic, yet
not drastically more complex, models of the type described in my work.
There is also a recent rise of an eco-evolutionary perspective on ecological
networks, ranging from the relationships between species’ taxonomies and
their topological positions and functions in food webs [285], to eco-
evolutionary dynamics and their implications for climate change [411],
and to using networks of interacting computer programmes to mimic and
understand species evolution in ecological networks [412]. Such research
has exciting prospects for not only ecology but also complexity science
at large, as it may extend to more general cases of broad scientific,
cultural and policy interest, such as the evolution of technology. Overall,
a strategy of using a hierarchy of models of varying levels of complexity
to more comprehensively understand the various aspects of the system
[413] may be employed to explore all these avenues.
Zooming out again to see both the details and the bigger picture at the
same time, one might envision a synthesis of general laws allying ecological
networks with complex networks in other disciplines (see §1.2). Knowledge
gleaned from the analysis of big data in other kinds of networks could
provide insights into properties of ecological networks for which data are
scarce. Conversely, ecological networks could potentially provide valuable
insights into combinatorial optimization problems in manmade networks,
analogous to the way that the adaptive spatial network evolution of
164
Yangchen Lin
slime molds has been found to help simultaneously optimize transport
networks for minimal cost, maximal efficiency and maximal fault tolerance
[414]. Such insights could even apply to the arts and entertainment.
For example, I would say that motion pictures with complex feedbacks
between multiple players resulting in unexpected nonlinear evolution of
the story line may be more exciting.
Going back to ecology, an interesting and useful research and man-
agement strategy is to look at ecological systems actually interacting
with other systems. In modeling terms, this could take the form of an
ecological network ‘plug-in’ to system dynamics models of the wider
environment. Indeed, linking ecological models with economic, social and
other kinds of models has been suggested as a way to make ecology
more useful to policy makers and practitioners [415]. Various researchers
have pioneered the so-called social-ecological or bio-economic models,
with conceptual models (see Milner-Gulland [416] for an overview) and
simulation models [417, 418] synthesizing ecological system dynamics, hu-
man behaviour and management policies. Complex systems science has
also been applied to the management of actual social-ecological systems
[419, 420]. These efforts, however, have not yet embraced the interaction
complexities of ecological networks.
In the policy arena, this multidisciplinary system dynamics approach
could prove useful for further boosting the effectiveness of horizon
scanning, with the aim of identifying issues of potentially far-reaching
future consequences for the global environment [421, 422]. The value of
horizon scanning lies not only in identifying a specific issue but also
in being able to infer its possible interactions with other components
of the system that could engender unexpected and disproportionate
consequences for seemingly unrelated components and for the system as
a whole. System dynamics and complexity science at large are poised
to tackle this challenge at a point in history that many have hailed
as the ‘Information Revolution’ (drawing inspiration from the Industrial
Revolution). Related fields such as risk analysis [330, 331], and info-gap
decision theory [423] where the lack of data and complexity of processes
165
Coda
make probability calculations impossible, also stand to gain from the
insights of complexity science.
Such advancements in networks and complex systems are setting the
stage for us to be more proactive with having an oversight and control
of everything to secure the well-being of mankind [366, 424]; recall the
discussion on network control (§4.4). But caution is needed here, for
in trying to suppress unfavourable but relatively small ‘earthquakes’ in
complex networks, we may inadvertently force a buildup of pressure
that triggers big catastrophes, although conversely we might judiciously
encourage small cascades in order to avoid a big one. Rigorous control
measures may also have the side effect of making the world more
tedious, more regimented and less exciting to live in, as well as possibly
reducing its adaptability to unexpected exogenous drivers. We should
use our understanding of complexity to exist harmoniously within nature,
acknowledging the fact that things cannot always be in our favour if
overall resilience is to be sustained. An ecosystem does not try to
maximize one overall goal, because its constituent agents engage in
conflicting selective processes [425]; this, I argue, is what gives rise to
diversity and resilience in the face of change. We are seeing something
unprecedented in the history of life on earth—an especially intelligent
species actually having global oversight rather than solely abiding by
local rules giving rise to self-organisation. Who knows where this will
lead us?
166
literature cited
[1] Lin, Y. & Sutherland, W. J. 2013. Color and degree of interspecific synchrony of
environmental noise affect the variability of complex ecological networks. Ecol. Model.
263:162–173.
[2] 2014. Extinction and invasion do not add up in noisy dynamic ecological
networks. Basic Appl. Ecol. 15:475–485.
[3] 2014. Interaction modification effects on ecological networks are affected by
ratio dependence and network topology. J. Theor. Biol. 363:151–157.
[4] 2015. Compartmentalization influences the response of bioenergetic ecological
networks to species declines. J. Complex Netw. doi:10.1093/comnet/cnv007.
[5] Assaneo, F. et al. 2013. Dynamics and coexistence in a system with intraguild mutualism.
Ecol. Complex. 14:64–74.
[6] Peh, K. S.-H. et al. 2014. Forest fragmentation and ecosystem function. In: Kettle,
C. & Koh, L. P., eds. Global Forest Fragmentation. cabi, United Kingdom pp. 96–114.
[7] Kauffman, S. 2000. Investigations. Oxford University Press, New York.
[8] Prigogine, I. 1997. The End of Certainty: Time, Chaos, and the New Laws of Nature. The
Free Press, New York.
[9] Kareiva, P. 1994. Higher order interactions as a foil to reductionist ecology. Ecology
75(6):1527–1528.
[10] Low-Décarie, E. et al. 2014. Rising complexity and falling explanatory power in
ecology. Front. Ecol. Environ. 12(7):412–418.
[11] Evans, M.R. et al. 2013. Predictive systems ecology. Proc. R. Soc. B 280:20131452.
[12] Strogatz, S.H. 2001. Exploring complex networks. Nature 410:268–276.
[13] Barabási, A.-L. 2012. The network takeover. Nat. Phys. 8:14–16.
[14] Schich, M. et al. 2014. A network framework of cultural history. Science 345:558–562.
[15] Solé, R. V. & Valverde, S. 2004. Information theory of complex networks: on evolu-
tion and architectural constraints. Lect. Notes Phys. 650:189–207.
[16] Proulx, S. R. et al. 2005. Network thinking in ecology and evolution. Trends Ecol.
Evol. 20(6):345–353.
[17] Bascompte, J. 2009. Disentangling the web of life. Science 325:416–419.
[18] Brose, U. 2010. Improving nature conservancy strategies by ecological network theory.
Basic Appl. Ecol. 11:1–5.
[19] Lewinsohn, T.M. & Cagnolo, L. 2012. Keystones in a tangled bank. Science 335:1449–
1451.
[20] Polis, G. A. 1991. Complex trophic interactions in deserts: an empirical critique of
food-web theory. Am. Nat. 138(1):123–155.
167
Literature cited
[21] Cardinale, B. et al. 2009. Towards a food web perspective on biodiversity and ecosys-
tem functioning. In: Naeem, S. et al., eds. Biodiversity, Ecosystem Functioning, & Human
Wellbeing. Oxford University Press pp. 105–120.
[22] Duffy, J. E. et al. 2007. The functional role of biodiversity in ecosystems: incorporat-
ing trophic complexity. Ecol. Lett. 10:522–538.
[23] Loreau, M. & Thébault, E. 2005. Food webs and the relationship between biodiver-
sity and ecosystem functioning. In: de Ruiter, P. C. et al., eds. Dynamic Food Webs:
Multispecies Assemblages, Ecosystem Development and Environmental Change. Academic Press,
Massachusetts pp. 270–282.
[24] Chiu, G. S. & Westveld, A.H. 2011. A unifying approach for food webs, phylogeny,
social networks, and statistics. P. Natl. Acad. Sci. usa 108(38):15881–15886.
[25] Brummitt, C.D. et al. 2013. Transdisciplinary electric power grid science. P. Natl.
Acad. Sci. usa 110(30):12159.
[26] Haldane, A.G. & May, R.M. 2011. Systemic risk in banking ecosystems.
Nature 469:351–355.
[27] Arinaminpathy, N. et al. 2012. Size and complexity in model financial systems. P. Natl.
Acad. Sci. usa 109(45):18338–18343.
[28] Shore, J. et al. 2013. Power laws and fragility in flow networks. Social Networks 35:116–
123.
[29] DasGupta, B. & Kaligounder, L. 2014. On global stability of financial networks. J.
Complex Netw. 2:313–354.
[30] Aufderheide, H. et al. 2013. How to predict community responses to perturbations in
the face of imperfect knowledge and network complexity. Proc. R. Soc. B 280:20132355.
[31] Rombouts, I. et al. 2013. Food web indicators under the Marine Strategy Framework
Directive: from complexity to simplicity? Ecol. Indic. 29:246–254.
[32] Wells, K. et al. 2014. Population fluctuations affect inference in ecological networks
of multi-species interactions. Oikos 123:589–598.
[33] Holme, P. & Saramäki, J. 2012. Temporal networks. Physics Reports 519:97–125.
[34] Vespignani, A. 2012. Modelling dynamical processes in complex socio-technical sys-
tems. Nat. Phys. 8:32–39.
[35] Barzel, B. & Barabási, A.-L. 2013. Universality in network dynamics. Nat. Phys. 9:673–
681.
[36] Bassett, D. S. et al. 2013. Robust detection of dynamic community structure in net-
works. Chaos 23:013142.
[37] Suweis, S. & D’Odorico, P. 2014. Early warning signs in social-ecological networks.
PLoS ONE 9(7):e101851.
[38] Bascompte, J. & Stouffer, D. B. 2009. The assembly and disassembly of ecological
networks. Phil. Trans. R. Soc. B 364:1781–1787.
168
Literature cited
[39] Eveleigh, E. S. et al. 2007. Fluctuations in density of an outbreak species drive diver-
sity cascades in food webs. P. Natl. Acad. Sci. usa 104(43):16976–16981.
[40] Golubski, A. J. & Abrams, P. A. 2011. Modifying modifiers: what happens when inter-
specific interactions interact? J. Anim. Ecol. 80:1097–1108.
[41] Kéfi, S. et al. 2012. More than a meal … integrating non-feeding interactions into
food webs. Ecol. Lett. 15:291–300.
[42] Melián, C. J. et al. 2009. Diversity in a complex ecological network with two interac-
tion types. Oikos 118:122–130.
[43] Blonder, B. et al. 2012. Temporal dynamics and network analysis. Methods Ecol. Evol.
3:958–972.
[44] Lawton, J.H. 1999. Are there general laws in ecology? Oikos 84:177–192.
[45] Scheffer, M. & Carpenter, S. R. 2003. Catastrophic regime shifts in ecosystems:
linking theory to observation. Trends Ecol. Evol. 18(12):648–656.
[46] Pimm, S. L. & Lawton, J.H. 1977. Number of trophic levels in ecological communities.
Nature 268:329–331.
[47] 1978. On feeding on more than one trophic level. Nature 275:542–544.
[48] Pimm, S. L. 1979. The structure of food webs. Theor. Pop. Biol. 16:144–158.
[49] Yodzis, P. 1981. The stability of real ecosystems. Nature 289:674–676.
[50] Dambacher, J.M. et al. 2003. Qualitative predictions in model ecosystems. Ecol. Model.
161:79–93.
[51] Allesina, S. & Tang, S. 2012. Stability criteria for complex ecosystems. Nature 483:205–
208.
[52] Melbourne-Thomas, J. et al. 2012. Comprehensive evaluation of model uncertainty
in qualitative network analyses. Ecol. Monogr. 82(4):505–519.
[53] Gross, T. et al. 2009. Generalized models reveal stabilizing factors in food webs.
Science 325:747–750.
[54] Yeakel, J. D. et al. 2011. Generalized modeling of ecological population dynamics.
Theor. Ecol. 4:179–194.
[55] Berlow, E. L. et al. 2004. Interaction strengths in food webs: issues and opportunities.
J. Anim. Ecol. 73:585–598.
[56] Janson, S. 2011. Networking—smoothly does it. Science 333:298–299.
[57] Rohr, R. P. et al. 2014. On the structural stability of mutualistic systems.
Science 345:1253497.
[58] McCann, K. S. 2000. The diversity-stability debate. Nature 405:228–233.
[59] Rohde, K. 2005. Nonequilibrium Ecology. Cambridge University Press.
169
Literature cited
[60] Evans, M.R. et al. 2013. Do simple models lead to generality in ecology? Trends Ecol.
Evol. 28(10):578–583.
[61] Amano, T. 2012. Unravelling the dynamics of organisms in a changing world using
ecological modelling. Ecol. Res. 27:495–507.
[62] Evans, M.R. 2012. Modelling ecological systems in a changing world. Phil. Trans. R.
Soc. B 367:181–190.
[63] Evans, M.R. et al. 2014. Data availability and model complexity, generality, and
utility: a reply to Lonergan. Trends Ecol. Evol. 29(6):302–303.
[64] Wollrab, S. et al. 2012. Simple rules describe bottom-up and top-down control in
food webs with alternative energy pathways. Ecol. Lett. 15:935–946.
[65] Condie, S. A. et al. 2014. Relating food web structure to resilience, keystone status
and uncertainty in ecological responses. Ecosphere 5(7):81.
[66] Lang, B. et al. 2014. Effects of environmental warming and drought on size-structured
soil food webs. Oikos 123:1224–1233.
[67] Brose, U. et al. 2005. Scaling up keystone effects from simple to complex ecological
networks. Ecol. Lett. 8:1317–1325.
[68] May, R.M. 1972. Will a large complex system be stable? Nature 238:413–414.
[69] 1974. Stability and Complexity in Model Ecosystems 2nd ed. Monographs in Popu-
lation Biology 6. Princeton University Press, New Jersey.
[70] McNaughton, S. J. 1978. Stability and diversity of ecological communities.
Nature 274:251–253.
[71] Pimm, S. L. 1984. The complexity and stability of ecosystems. Nature 307:321–326.
[72] Polis, G. A. 1998. Stability is woven by complex webs. Nature 395:744–745.
[73] Rudolf, V.H.W. & Lafferty, K.D. 2011. Stage structure alters how complexity affects
stability of ecological networks. Ecol. Lett. 14:75–79.
[74] Rooney, N. & McCann, K. S. 2012. Integrating food web diversity, structure and
stability. Trends Ecol. Evol. 27(1):40–46.
[75] Neutel, A.-M. & Thorne, M.A. S. 2014. Interaction strengths in balanced carbon
cycles and the absence of a relation between ecosystem complexity and stability. Ecol.
Lett. 17:651–661.
[76] Lewis, O.T. 2009. Biodiversity change and ecosystem function in tropical forests.
Basic Appl. Ecol. 10:97–102.
[77] Mason, P.H. 2014. Degeneracy: demystifying and destigmatizing a core concept in
systems biology. Complexity 20(3):12–21.
[78] Coreau, A. et al. 2010. Exploring the difficulties of studying futures in ecology: what
do ecological scientists think? Oikos 119:1364–1376.
[79] Patten, B. C. 1992. Energy, emergy and environs. Ecol. Model. 62:29–69.
170
Literature cited
[80] Ulanowicz, R. E. 2004. Quantitative methods for ecological network analysis. Comput.
Biol. Chem. 28:321–339.
[81] Fath, B.D. et al. 2007. Ecological network analysis: network construction. Ecol. Model.
208:49–55.
[82] Olff, H. et al. 2009. Parallel ecological networks in ecosystems. Phil. Trans. R. Soc. B
364:1755–1779.
[83] Raffaelli, D. et al. 2005. Do marine and terrestrial ecologists do it differently? Mar.
Ecol. Prog. Ser. 304:283–289.
[84] Vasas, V. & Jordán, F. 2006. Topological keystone species in ecological interaction
networks: considering link quality and non-trophic effects. Ecol. Model. 196:365–378.
[85] Dame, R. F. & Patten, B. C. 1981. Analysis of energy flows in an intertidal oyster reef.
Mar. Ecol. Prog. Ser. 5:115–124.
[86] Baird, D. & Ulanowicz, R. E. 1989. The seasonal dynamics of the Chesapeake Bay
ecosystem. Ecol. Monogr. 59(4):329–364.
[87] Cross, W. F. et al. 2013. Food-web dynamics in a large river discontinuum. Ecol. Monogr.
83(3):311–337.
[88] Hoover, C. et al. 2013. Effects of hunting, fishing and climate change on the Hudson
Bay marine ecosystem: II. Ecosystem model future projections. Ecol. Model. 264:143–
156.
[89] Schaubroeck, T. et al. 2012. Improved ecological network analysis for environmental
sustainability assessment: a case study on a forest ecosystem. Ecol. Model. 247:144–156.
[90] Libralato, S. et al. 2010. Food-web traits of protected and exploited areas of the
Adriatic Sea. Biol. Conserv. 143:2182–2194.
[91] Fath, B.D. & Patten, B. C. 1998. Network synergism: emergence of positive relations
in ecological systems. Ecol. Model. 107:127–143.
[92] Fath, B.D. 2007. Network mutualism: positive community-level relations in ecosys-
tems. Ecol. Model. 208:56–67.
[93] Yodzis, P. & Innes, S. 1992. Body size and consumer-resource dynamics. Am. Nat.
139(6):1151–1175.
[94] Williams, R. J. & Martinez, N.D. 2004. Stabilization of chaotic and non-permanent
food-web dynamics. Eur. Phys. J. B 38:297–303.
[95] Brose, U. et al. 2006. Allometric scaling enhances stability in complex food webs. Ecol.
Lett. 9:1228–1236.
[96] Williams, R. J. & Martinez, N.D. 2008. Success and its limits among structural
models of complex food webs. J. Anim. Ecol. 77:512–519.
[97] Gilbert, B. et al. 2014. A bioenergetic framework for the temperature dependence
of trophic interactions. Ecol. Lett. 17:902–914.
171
Literature cited
[98] Garay-Narváez, L. et al. 2014. Food web modularity and biodiversity promote species
persistence in polluted environments. Oikos 123:583–588.
[99] Olesen, J.M. et al. 2010. From Broadstone to Zackenberg: space, time and hierarchies
in ecological networks. Adv. Ecol. Res. 42:1–69.
[100] Scotti, M. et al. 2013. Social and landscape effects on food webs: a multi-level network
simulation model. J. Complex Netw. 1:160–182.
[101] Carmichael, T. & Hadzikadic, M. 2013. Emergent features in a general food web
simulation: Lotka-Volterra, Gause’s law, and the paradox of enrichment. Advances in
Complex Systems 16(8):1350014.
[102] Dyer, L. A. et al. 2010. Diversity of interactions: a metric for studies of biodiversity.
Biotropica 42(3):281–289.
[103] Novotny, V. et al. 2010. Guild-specific patterns of species richness and host special-
ization in plant-herbivore food webs from a tropical forest. J. Anim. Ecol. 79:1193–
1203.
[104] Yeakel, J. D. et al. 2014. Collapse of an ecological network in Ancient Egypt. P. Natl.
Acad. Sci. usa 111(40):14472–14477.
[105] Lafferty, K.D. et al. 2008. Parasites in food webs: the ultimate missing links. Ecol.
Lett. 11:533–546.
[106] Sukhdeo, M. V.K. 2010. Food webs for parasitologists: a review. J. Parasitol. 96(2):273–
284.
[107] Preston, D.L. et al. 2012. Food web including infectious agents for a California
freshwater pond. Ecology 93(7):1760.
[108] Reagan, D. P. & Waide, R. B., eds. 1996. The Food Web of a Tropical Rain Forest. Uni-
versity of Chicago Press, Chicago.
[109] Bascompte, J. et al. 2005. Interaction strength combinations and the overfishing of a
marine food web. P. Natl. Acad. Sci. usa 102(15):5443–5447.
[110] Jacob, U. et al. 2011. The role of body size in complex food webs: a cold case. Adv.
Ecol. Res. 45:181–223.
[111] Planque, B. et al. 2014. Who eats whom in the Barents Sea: a food web topology
from plankton to whales. Ecology 95(5):1430.
[112] Wirta, H.K. et al. 2014. Complementary molecular information changes our percep-
tion of food web structure. P. Natl. Acad. Sci. usa 111(5):1885–1890.
[113] Vinagre, C. & Costa, M. J. 2014. Estuarine-coastal gradient in food web network
structure and properties. Mar. Ecol. Prog. Ser. 503:11–21.
[114] Schmitz, O. 2010. Resolving Ecosystem Complexity. Monographs in Population Biology
47. Princeton University Press, New Jersey.
[115] Hampton, S. E. et al. 2013. Big data and the future of ecology. Front. Ecol. Environ.
11(3):156–162.
172
Literature cited
[116] Pocock, M. J.O. et al. 2012. The robustness and restoration of a network of ecological
networks. Science 335:973–977.
[117] Kéfi, S. et al. 2015. Network structure beyond food webs: mapping non-trophic and
trophic interactions on Chilean rocky shores. Ecology 96(1):291–303.
[118] Novak, M. et al. 2011. Predicting community responses to perturbations in the face
of imperfect knowledge and network complexity. Ecology 92(4):836–846.
[119] Yodzis, P. 1988. The indeterminacy of ecological interactions as perceived through
perturbation experiments. Ecology 69(2):508–515.
[120] Kuwae, T. et al. 2012. Variable and complex food web structures revealed by exploring
missing trophic links between birds and biofilm. Ecol. Lett. 15:347–356.
[121] Boit, A. et al. 2012. Mechanistic theory and modelling of complex food-web dynamics
in Lake Constance. Ecol. Lett. 15:594–602.
[122] Hudson, L.N. & Reuman, D.C. 2013. A cure for the plague of parameters: con-
straining models of complex population dynamics with allometries. Proc. R. Soc. B
280:20131901.
[123] Lonergan, M. 2014. Data availability constrains model complexity, generality, and
utility: a response to Evans et al. Trends Ecol. Evol. 29(6):301–302.
[124] Gardner, T. A. et al. 2007. Predicting the uncertain future of tropical forest species
in a data vacuum. Biotropica 39(1):25–30.
[125] Berlow, E. L. et al. 2009. Simple prediction of interaction strengths in complex food
webs. P. Natl. Acad. Sci. usa 106(1):187–191.
[126] Romanuk, T.N. et al. 2009. Predicting invasion success in complex ecological networks.
Phil. Trans. R. Soc. B 364:1743–1754.
[127] Macilwain, C. 2014. A touch of the random. Science 344:1221–1223.
[128] Boyd, I. L. 2012. The art of ecological modeling. Science 337:306–307.
[129] Carpenter, S. R. 2002. Ecological futures: building an ecology of the long now.
Ecology 83(8):2069–2083.
[130] Scheffer, M. 2009. Critical Transitions in Nature and Society. Princeton Studies in Com-
plexity. Princeton University Press, New Jersey.
[131] Purves, D. et al. 2013. Time to model all life on earth. Nature 493:295–297.
[132] Ings, T. C. et al. 2009. Ecological networks—beyond food webs. J. Anim. Ecol. 78:253–
269.
[133] Sterman, J.D. 2002. All models are wrong: reflections on becoming a systems scientist.
Syst. Dyn. Rev. 18(4):501–531.
[134] Patten, B. C. 2013. Systems ecology and environmentalism: getting the science right.
Ecol. Eng. 61:446–455.
173
Literature cited
[135] Patten, B. C. 2014. Systems ecology and environmentalism: getting the science right.
Part I: facets for a more holistic Nature Book of ecology. Ecol. Model. 293:4–21.
[136] Grimm, V. & Railsback, S. F. 2012. Pattern-oriented modelling: a ‘multi-scope’ for
predictive systems ecology. Phil. Trans. R. Soc. B 367:298–310.
[137] RCore Team 2013. R: a Language and Environment for Statistical Computing. R Foundation
for Statistical Computing.
[138] Williams, R. J. & Martinez, N.D. 2000. Simple rules yield complex food webs. Nature
404:180–183.
[139] Allesina, S. et al. 2008. A general model for food web structure. Science 320:658–661.
[140] Williams, R. J. 2008. Effects of network and dynamical model structure on species
persistence in large model food webs. Theor. Ecol. 1:141–151.
[141] Stouffer, D. B. 2010. Scaling from individuals to networks in food webs. Funct. Ecol.
24:44–51.
[142] Williams, R. J. et al. 2010. The probabilistic niche model reveals the niche structure
and role of body size in a complex food web. PLoS ONE 5(8):e12092.
[143] Milo, R. et al. 2002. Network motifs: simple building blocks of complex networks.
Science 298:824–827.
[144] Bascompte, J. & Melián, C. J. 2005. Simple trophic modules for complex food webs.
Ecology 86(11):2868–2873.
[145] Stouffer, D. B. et al. 2007. Evidence for the existence of a robust pattern of prey
selection in food webs. Proc. R. Soc. B 274:1931–1940.
[146] Stouffer, D. B. et al. 2006. A robust measure of food web intervality. P. Natl. Acad.
Sci. usa 103(50):19015–19020.
[147] Halnes, G. et al. 2007. The modified niche model: including detritus in simple struc-
tural food web models. Ecol. Model. 208:9–16.
[148] Warren, C. P. et al. 2010. The inverse niche model for food webs with parasites. Theor.
Ecol. 3:285–294.
[149] Williams, R. J. & Purves, D. 2011. The probabilistic niche model reveals substantial
variation in the niche structure of empirical food webs. Ecology 92(9):1849–1857.
[150] Petchey, O.L. et al. 2011. Fit, efficiency, and biology: some thoughts on judging food
web models. J. Theor. Biol. 279:169–171.
[151] Pierce, G. J. & Ollason, J.G. 1987. Eight reasons why optimal foraging theory is a
complete waste of time. Oikos 49(1):111–118.
[152] Carmel, Y. & Ben-Haim, Y. 2005. Info-gap robust-satisficing model of foraging
behavior: do foragers optimize or satisfice? Am. Nat. 166(5):633–641.
[153] Rossberg, A.G. et al. 2010. Food-web structure in low- and high-dimensional trophic
niche spaces. J. R. Soc. Interface 7:1735–1743.
174
Literature cited
[154] Lafferty, K.D. et al. 2006. Parasites dominate food web links. P. Natl. Acad. Sci. usa
103(30):11211–11216.
[155] Thieltges, D.W. et al. 2013. Parasites as prey in aquatic food webs: implications for
predator infection and parasite transmission. Oikos 122:1473–1482.
[156] Selakovic, S. et al. 2014. Infectious disease agents mediate interaction in food webs
and ecosystems. Proc. R. Soc. B 281:20132709.
[157] Martinez, N.D. 1992. Constant connectance in community food webs.
Am. Nat. 139(6):1208–1218.
[158] Dunne, J. A. et al. 2002. Network structure and biodiversity loss in food webs: robust-
ness increases with connectance. Ecol. Lett. 5:558–567.
[159] Rojas-Echenique, J. & Allesina, S. 2011. Interaction rules affect species coexistence
in intransitive networks. Ecology 92(5):1174–1180.
[160] Charnov, E. L. et al. 1976. Ecological implications of resource depression. Am. Nat.
110(972):247–259.
[161] Gross, K. 2008. Positive interactions among competitors can produce species-rich
communities. Ecol. Lett. 11:929–936.
[162] Crowley, P.H. & Cox, J. J. 2011. Intraguild mutualism. Trends Ecol. Evol. 26(12):627–
633.
[163] Roopin, M. et al. 2008. Nutrient transfer in a marine mutualism: patterns of ammonia
excretion by anemonefish and uptake by giant sea anemones. Mar. Biol. 154:547–556.
[164] Rossignol, P. A. et al. 1985. Enhanced mosquito blood-finding success on parasitemic
hosts: evidence for vector-parasite mutualism. P. Natl. Acad. Sci. usa 82:7725–7727.
[165] Donatti, C. I. et al. 2011. Analysis of a hyper-diverse seed dispersal network: modu-
larity and underlying mechanisms. Ecol. Lett. 14:773–781.
[166] Pires, M.M. et al. 2011. Do food web models reproduce the structure of mutualistic
networks? PLoS ONE 6(11):e27280.
[167] Suweis, S. et al. 2013. Emergence of structural and dynamical properties of ecological
mutualistic networks. Nature 500:449–452.
[168] Holbrook, S. J. et al. 2008. Effects of sheltering fish on growth of their host corals.
Mar. Biol. 155:521–530.
[169] James, A. et al. 2012. Disentangling nestedness from models of Ecol. Complex. Nature
487:227–230.
[170] Jordano, P. 1987. Patterns of mutualistic interactions in pollination and seed dispersal:
connectance, dependence asymmetries, and coevolution. Am. Nat. 129(5):657–677.
[171] Burkholder, P. R. 1952. Cooperation and conflict among primitive organisms. Am. Sci.
40(4):601–631.
[172] Newman, M.E. J. 2010. Networks: an Introduction. Oxford University Press.
175
Literature cited
[173] Sukhdeo, M. V.K. 2012. Where are the parasites in food webs? Parasites and Vectors
5:239.
[174] Fontaine, C. et al. 2011. The ecological and evolutionary implications of merging
different types of networks. Ecol. Lett. 14:1170–1181.
[175] Ulanowicz, R. E. et al. 2014. Limits on ecosystem trophic complexity: insights from
ecological network analysis. Ecol. Lett. 17:127–136.
[176] Kleiber, M. 1932. Body size and metabolism. Hilgardia 6(11):315–353.
[177] Brown, J.H. et al. 2004. Toward a metabolic theory of ecology. Ecology 85(7):1771–1789.
[178] Calder III, W.A. 1983. Ecological scaling: mammals and birds. Annu. Rev. Ecol. Syst.
14:213–230.
[179] Pomeroy, D. 1990. Why fly? The possible benefits for lower mortality. Biol. J. Linn.
Soc. 40:53–65.
[180] Arditi, R. & Ginzburg, L.R. 1989. Coupling in predator-prey dynamics: ratio-depen-
dence. J. Theor. Biol. 139:311–326.
[181] Akçakaya, H.R. et al. 1995. Ratio-dependent predation: an abstraction that works.
Ecology 76(3):995–1004.
[182] Abrams, P. A. & Ginzburg, L.R. 2000. The nature of predation: prey dependent,
ratio dependent or neither? Trends Ecol. Evol. 15(8):337–341.
[183] Arditi, R. & Ginzburg, L.R. 2012. How Species Interact: Altering the Standard View on
Trophic Ecology. Oxford University Press, New York.
[184] Piana, P. A. et al. 2006. Comparison of predator-prey interaction models for fish
assemblages from the neotropical region. Ecol. Model. 192:259–270.
[185] Kratina, P. et al. 2009. Functional responses modified by predator density. Oecologia
159:425–433.
[186] Berg, H. A. van den 1998. Propagation of permanent perturbations in food chains and
food webs. Ecol. Model. 107:225–235.
[187] Novak, M. 2013. Trophic omnivory across a productivity gradient: intraguild preda-
tion theory and the structure and strength of species interactions. Proc. R. Soc. B
280:20131415.
[188] Preston, D.L. et al. 2013. Biomass and productivity of trematode parasites in pond
ecosystems. J. Anim. Ecol. 82:509–517.
[189] Staniczenko, P. P. A. et al. 2010. Structural dynamics and robustness of food webs.
Ecol. Lett. 13:891–899.
[190] Valdovinos, F. S. et al. 2010. Consequences of adaptive behaviour for the structure
and dynamics of food webs. Ecol. Lett. 13:1546–1559.
[191] Thierry, A. et al. 2011. Adaptive foraging and the rewiring of size-structured food
webs following extinctions. Basic Appl. Ecol. 12:562–570.
176
Literature cited
[192] Ramos-Jiliberto, R. et al. 2012. Topological plasticity increases robustness of mutu-
alistic networks. J. Anim. Ecol. 81:896–904.
[193] Cohen, A. E. et al. 1998. A novel experimental apparatus to study the impact of white
noise and 1/f noise on animal populations. Proc. R. Soc. B 265:11–15.
[194] Cohen, J. E. et al. 1999. Spectral mimicry: a method of synthesizing matching time
series with different fourier spectra. Circuits Syst. Signal Process. 18(3):431–442.
[195] Fowler, M. S. & Ruokolainen, L. 2013. Confounding environmental colour and dis-
tribution shape leads to underestimation of population extinction risk. PLoS ONE
8(2):e55855.
[196] Roughgarden, J. 1975. A simple model for population dynamics in stochastic environ-
ments. Am. Nat. 109(970):713–736.
[197] Sæther, B.-E. et al. 1998. Environmental stochasticity and extinction risk in a popu-
lation of a small songbird, the Great Tit. Am. Nat. 151(5):441–450.
[198] Neubert, M.G. et al. 2004. Reactivity and transient dynamics of predator-prey and
food web models. Ecol. Model. 179:29–38.
[199] Ruokolainen, L. et al. 2009. Ecological and evolutionary dynamics under coloured
environmental variation. Trends Ecol. Evol. 24(10):555–563.
[200] Sutherland, W. J. et al. 2013. Identification of 100 fundamental ecological questions.
J. Ecol. 101:58–67.
[201] Solé, R. V. et al. 2002. Self-organized instability in complex ecosystems. Phil. Trans.
R. Soc. B 357:667–681.
[202] Ruokolainen, L. & Fowler, M. S. 2008. Community extinction patterns in coloured
environments. Proc. R. Soc. B 275:1775–1783.
[203] Levins, R. 1969. The effect of random variations of different types on population
growth. P. Natl. Acad. Sci. usa 62:1061–1065.
[204] Lewontin, R. C. & Cohen, D. 1969. On population growth in a randomly varying
environment. P. Natl. Acad. Sci. usa 62:1056–1060.
[205] Dennis, B. 2002. Allee effects in stochastic populations. Oikos 96(3):389–401.
[206] Greenman, J. V. & Benton, T.G. 2005. The impact of environmental fluctuations
on structured discrete time population models: resonance, synchrony and threshold
behaviour. Theor. Pop. Biol. 68:217–235.
[207] Ebenman, B. & Jonsson, T. 2005. Using community viability analysis to identify
fragile systems and keystone species. Trends Ecol. Evol. 20(10):568–575.
[208] Ripa, J. et al. 1998. A general theory of environmental noise in ecological food webs.
Am. Nat. 151(3):256–263.
[209] Spagnolo, B. & La Barbera, A. 2002. Role of the noise on the transient dynamics of
an ecosystem of interacting species. Physica A 315:114–124.
177
Literature cited
[210] Ripa, J. & Ives, A.R. 2003. Food web dynamics in correlated and autocorrelated
environments. Theor. Pop. Biol. 64:369–384.
[211] Gravel, D. et al. 2011. Species coexistence in a variable world. Ecol. Lett. 14:828–839.
[212] Gjata, N. et al. 2012. The strength of simulated indirect interaction modules in a real
food web. Ecol. Complex. 11:160–164.
[213] Vasseur, D. A. & Fox, J.W. 2007. Environmental fluctuations can stabilize food web
dynamics by increasing synchrony. Ecol. Lett. 10:1066–1074.
[214] Steele, J.H. 1985. A comparison of terrestrial and marine ecological systems. Nature
313:355–358.
[215] Pimm, S. L. & Redfearn, A. 1988. The variability of population densities. Nature
334:613–614.
[216] Halley, J.M. 1996. Ecology, evolution and )=f-noise. Trends Ecol. Evol. 11(1):33–37.
[217] Morales, J.M. 1999. Viability in a pink environment: why “white noise” models can
be dangerous. Ecol. Lett. 2:228–232.
[218] Halley, J.M. & Inchausti, P. 2004. The increasing importance of 1/f-noises as models
of ecological variability. Fluct. Noise Lett. 4(2):r1–r26.
[219] Vasseur, D. A. & Yodzis, P. 2004. The color of environmental noise. Ecology 85(4):1146–
1152.
[220] Hennig, H. 2014. Synchronization in human musical rhythms and mutually interacting
complex systems. P. Natl. Acad. Sci. usa 111(36):12974–12979.
[221] Petchey, O.L. et al. 1997. Effects on population persistence: the interaction be-
tween environmental noise colour, intraspecific competition and space. Proc. R. Soc. B
264:1841–1847.
[222] Cuddington, K.M. & Yodzis, P. 1999. Black noise and population persistence. Proc.
R. Soc. B 266:969–973.
[223] Ripa, J. & Heino, M. 1999. Linear analysis solves two puzzles in population dynamics:
the route to extinction and extinction in coloured environments. Ecol. Lett. 2:219–222.
[224] Greenman, J. V. & Benton, T.G. 2003. The amplification of environmental noise in
population models: causes and consequences. Am. Nat. 161(2):225–239.
[225] Reuman, D.C. et al. 2006. Power spectra reveal the influence of stochasticity on
nonlinear population dynamics. P. Natl. Acad. Sci. usa 103(49):18860–18865.
[226] Reuman, D.C. et al. 2008. Colour of environmental noise affects the nonlinear dy-
namics of cycling, stage-structured populations. Ecol. Lett. 11:820–830.
[227] Ezard, T.H.G. & Coulson, T. 2010. How sensitive are elasticities of long-run stochas-
tic growth to how environmental variability is modelled? Ecol. Model. 221:191–200.
[228] Lögdberg, F. & Wennergren, U. 2012. Spectral color, synchrony, and extinction risk.
Theor. Ecol. 5(4):545–554.
178
Literature cited
[229] Vasseur, D. A. 2007. Populations embedded in trophic communities respond differ-
ently to coloured environmental noise. Theor. Pop. Biol. 72:186–196.
[230] Ruokolainen, L. et al. 2007. Extinctions in competitive communities forced by coloured
environmental variation. Oikos 116:439–448.
[231] Loreau, M. & Mazancourt, C. de 2008. Species synchrony and its drivers: neutral and
nonneutral community dynamics in fluctuating environments. Am. Nat. 172(2):e48–e66.
[232] Moran, P. A. P. 1953. The statistical analysis of the Canadian lynx cycle. Australian J.
Zool. 1(3):291–298.
[233] Raimondo, S. et al. 2004. Population synchrony within and among Lepidoptera species
in relation to weather, phylogeny, and larval phenology. Ecol. Entomol. 29:96–105.
[234] Bowman, J. et al. 2008. Spatial and temporal dynamics of small mammals at a regional
scale in Canadian boreal forest. J. Mammal. 89(2):381–387.
[235] Kvasnes, M. A. J. et al. 2010. Spatial dynamics of Norwegian tetraonid populations.
Ecol. Res. 25:367–374.
[236] Tachiki, Y. et al. 2010. Pollinator coupling can induce synchronized flowering in
different plant species. J. Theor. Biol. 267:153–163.
[237] Chevalier, M. et al. 2014. Spatial synchrony in stream fish populations: influence of
species traits. Ecography 37(10):960–968.
[238] Yachi, S. & Loreau, M. 1999. Biodiversity and ecosystem productivity in a fluctuating
environment: the insurance hypothesis. P. Natl. Acad. Sci. usa 96:1463–1468.
[239] Morin, X. et al. 2014. Temporal stability in forest productivity increases with tree
diversity due to asynchrony in species dynamics. Ecol. Lett. 17:1526–1535.
[240] Borrvall, C. & Ebenman, B. 2008. Biodiversity and persistence of ecological commu-
nities in variable environments. Ecol. Complex. 5:99–105.
[241] Loreau, M. & Mazancourt, C. de 2013. Biodiversity and ecosystem stability: a syn-
thesis of underlying mechanisms. Ecol. Lett. 16:106–115.
[242] Kokkoris, G.D. et al. 2002. Variability in interaction strength and implications for
biodiversity. J. Anim. Ecol. 71:362–371.
[243] Allesina, S. & Levine, J.M. 2011. A competitive network theory of species diversity.
P. Natl. Acad. Sci. usa 108(14):5638–5642.
[244] Bruno, J. F. et al. 2003. Inclusion of facilitation into ecological theory. Trends Ecol.
Evol. 18(3):119–125.
[245] Xie, J. et al. 2011. Ecological mechanisms underlying the sustainability of the agricul-
tural heritage rice-fish coculture system. P. Natl. Acad. Sci. usa 108(50):e1381–e1387.
[246] He, Q. et al. 2013. Global shifts towards positive species interactions with increasing
environmental stress. Ecol. Lett. 16:695–706.
[247] Okuyama, T. & Holland, J.N. 2008. Network structural properties mediate the sta-
bility of mutualistic communities. Ecol. Lett. 11:208–216.
179
Literature cited
[248] Kondoh, M. et al. 2010. Food webs are built up with nested subwebs. Ecology 91(11):3123–
3130.
[249] Thébault, E. & Fontaine, C. 2010. Stability of ecological communities and the
architecture of mutualistic and trophic networks. Science 329:853–856.
[250] Bascompte, J. 2010. Structure and dynamics of ecological networks. Science 329:765–
766.
[251] Goudard, A. & Loreau, M. 2008. Nontrophic interactions, biodiversity, and ecosystem
functioning: an interaction web model. Am. Nat. 171(1):91–106.
[252] Visser, S.N. de et al. 2011. The Serengeti food web: empirical quantification and
analysis of topological changes under increasing human impact. J. Anim. Ecol. 80:484–
494.
[253] Orlofske, S. A. et al. 2012. Parasite transmission in complex communities: predators
and alternative hosts alter pathogenic infections in amphibians. Ecology 93(6):1247–1253.
[254] Filotas, E. et al. 2010. Positive interactions and the emergence of community structure
in metacommunities. J. Theor. Biol. 266:419–429.
[255] Lee, K.-M. et al. 2014. Multiplex networks. In: D’Agostino, G. & Scala, A., eds.
Networks of Networks: The Last Frontier of Complexity. Springer pp. 53–72.
[256] Sauve, A.M.C. et al. 2014. Structure-stability relationships in networks combining
mutualistic and antagonistic interactions. Oikos 123:378–384.
[257] Yoshikawa, T. & Isagi, Y. 2014. Determination of temperate bird-flower interactions
as entangled mutualistic and antagonistic sub-networks: characterization at the net-
work and species levels. J. Anim. Ecol. 83:651–660.
[258] Scheffer, M. et al. 2001. Catastrophic shifts in ecosystems. Nature 413:591–596.
[259] Folke, C. et al. 2004. Regime shifts, resilience, and biodiversity in ecosystem man-
agement. Annu. Rev. Ecol. Evol. Syst. 35:557–581.
[260] Marcogliese, D. J. & Cone, D.K. 1997. Food webs: a plea for parasites. Trends Ecol.
Evol. 12(8):320–325.
[261] Wood, M. J. 2006. Parasites entangled in food webs. Trends Parasitol. 23(1):8–10.
[262] Lewis, O.T. et al. 2002. Structure of a diverse tropical forest insect-parasitoid com-
munity. J. Anim. Ecol. 71:855–873.
[263] Maunsell, S. C. et al. 2014. Changes in host-parasitoid food web structure with ele-
vation. J. Anim. Ecol. 84(2):353–363.
[264] Byers, J. E. 2008. Including parasites in food webs. Trends Parasitol. 25(2):55–57.
[265] Chen, H.-W. et al. 2011. The reduction of food web robustness by parasitism: fact
and artefact. Int. J. Parasitol. 41:627–634.
[266] Sato, T. et al. 2012. Nematomorph parasites indirectly alter the food web and ecosys-
tem function of streams through behavioural manipulation of their cricket hosts. Ecol.
Lett. 15:786–793.
180
Literature cited
[267] Dunne, J. A. et al. 2013. Parasites affect food web structure primarily through increased
diversity and complexity. PLoS Biol. 11(6):e1001579.
[268] Holland, J.N. et al. 2013. Consumer-resource dynamics of indirect interactions in a
mutualism-parasitism food web module. Theor. Ecol. 6:475–493.
[269] Stouffer, D. B. et al. 2011. The role of body mass in diet contiguity and food-web
structure. J. Anim. Ecol. 80:632–639.
[270] Zook, A. E. et al. 2011. Food webs: ordering species according to body size yields
high degree of intervality. J. Theor. Biol. 271:106–113.
[271] Digel, C. et al. 2011. Body sizes, cumulative and allometric degree distributions across
natural food webs. Oikos 120:503–509.
[272] Lehman, C. L. & Tilman, D. 2000. Biodiversity, stability, and productivity in com-
petitive communities. Am. Nat. 156(5):534–552.
[273] Jost, L. 2006. Entropy and diversity. Oikos 113(2):363–375.
[274] Ellis, M.M. & Crone, E. E. 2013. The role of transient dynamics in stochastic pop-
ulation growth for nine perennial plants. Ecology 94(8):1681–1686.
[275] Ripley, B. 2013. tree: Classification and regression trees. http: / /CRAN.R- project.org/
package=tree.
[276] Andersen, K.H. et al. 2009. Trophic and individual efficiencies of size-structured
communities. Proc. R. Soc. B 276:109–114.
[277] Scheffer, M. et al. 2012. Anticipating critical transitions. Science 338:344–348.
[278] Thébault, E. & Loreau, M. 2003. Food-web constraints on biodiversity–ecosystem
functioning relationships. P. Natl. Acad. Sci. usa 100(25):14949–14954.
[279] 2005. Trophic interactions and the relationship between species diversity and
ecosystem stability. Am. Nat. 166(4):e95–e114.
[280] 2006. The relationship between biodiversity and ecosystem functioning in
food webs. Ecol. Res. 21:17–25.
[281] Mougi, A. & Kondoh, M. 2012. Diversity of interaction types and ecological commu-
nity stability. Science 337:349–351.
[282] Otto, S. B. et al. 2007. Allometric degree distributions facilitate food-web stability.
Nature 450:1226–1229.
[283] Walker, B. 1995. Conserving biological diversity through ecosystem resilience. Conserv.
Biol. 9(4):747–752.
[284] Elmqvist, T. et al. 2003. Response diversity, ecosystem change, and resilience. Front.
Ecol. Environ. 1(9):488–494.
[285] Stouffer, D. B. et al. 2012. Evolutionary conservation of species’ roles in food webs.
Science 335:1489–1492.
181
Literature cited
[286] Hooper, D.U. et al. 2012. A global synthesis reveals biodiversity loss as a major driver
of ecosystem change. Nature 486:105–108.
[287] Tylianakis, J.M. et al. 2008. Global change and species interactions in terrestrial
ecosystems. Ecol. Lett. 11:1351–1363.
[288] Curtsdotter, A. et al. 2011. Robustness to secondary extinctions: comparing trait-
based sequential deletions in static and dynamic food webs. Basic Appl. Ecol. 12:571–
580.
[289] Sahasrabudhe, S. & Motter, A. E. 2011. Rescuing ecosystems from extinction cascades
through compensatory perturbations. Nat. Commun. 2:170.
[290] Stouffer, D. B. & Bascompte, J. 2011. Compartmentalization increases food-web
persistence. P. Natl. Acad. Sci. usa 108(9):3648–3652.
[291] Schneider, F.D. et al. 2012. Body mass constraints on feeding rates determine the
consequences of predator loss. Ecol. Lett. 15:436–443.
[292] Säterberg, T. et al. 2013. High frequency of functional extinctions in ecological
networks. Nature 499:468–470.
[293] Riede, J.O. et al. 2011. Size-based food web characteristics govern the response to
species extinctions. Basic Appl. Ecol. 12:581–589.
[294] Torres-Alruiz, M.D. & Rodríguez, D. J. 2013. A topo-dynamical perspective to
evaluate indirect interactions in trophic webs: new indexes. Ecol. Model. 250:363–369.
[295] Valiente-Banuet, A. & Verdú, M. 2013. Human impacts on multiple ecological net-
works act synergistically to drive ecosystem collapse. Front. Ecol. Environ. 11(8):408–
413.
[296] Woodward, G. & Hildrew, A.G. 2001. Invasion of a stream food web by a new top
predator. J. Anim. Ecol. 70:273–288.
[297] Layer, K. et al. 2011. Long-term dynamics of a well-characterised food web: four
decades of acidification and recovery in the Broadstone Stream model system. Adv.
Ecol. Res. 44:69–117.
[298] Galiana, N. et al. 2014. Invasions cause biodiversity loss and community simplification
in vertebrate food webs. Oikos 123:721–728.
[299] Traveset, A. & Richardson, D.M. 2006. Biol. Invasions as disruptors of plant repro-
ductive mutualisms. Trends Ecol. Evol. 21(4):208–216.
[300] Memmott, J. 2009. Food webs: a ladder for picking strawberries or a practical tool
for practical problems? Phil. Trans. R. Soc. B 364:1693–1699.
[301] Burkle, L. A. & Alarcón, R. 2011. The future of plant-pollinator diversity: under-
standing interaction networks across time, space, and global change. Am. J. Bot.
98(3):528–538.
[302] Rodriguez-Cabal, M. A. et al. 2013. Node-by-node disassembly of a mutualistic in-
teraction web driven by species introductions. P. Natl. Acad. Sci. usa 110(41):16503–
16507.
182
Literature cited
[303] Traveset, A. et al. 2013. Invaders of pollination networks in the Galápagos Islands:
emergence of novel communities. Proc. R. Soc. B 280:20123040.
[304] Stouffer, D. B. et al. 2014. How exotic plants integrate into pollination networks. J.
Ecol. 102:1442–1450.
[305] Sugiura, S. 2010. Species interactions-area relationships: biological invasions and
network structure in relation to island area. Proc. R. Soc. B 277:1807–1815.
[306] Vacher, C. et al. 2010. Ecological integration of alien species into a tree-parasitic
fungus network. Biol. Invasions 12:3249–3259.
[307] Strayer, D. L. 2012. Eight questions about invasions and ecosystem functioning. Ecol.
Lett. 15:1199–1210.
[308] Gurevitch, J. & Padilla, D.K. 2004. Are invasive species a major cause of extinctions?
Trends Ecol. Evol. 19(9):470–474.
[309] Wardle, D. A. et al. 2011. Terrestrial ecosystem responses to species gains and losses.
Science 332:1273–1277.
[310] Forys, E. A. & Allen, C.R. 2002. Functional group change within and across scales
following invasions and extinctions in the Everglades ecosystem. Ecosystems 5:339–347.
[311] Jackson, S. T. & Sax, D. F. 2010. Balancing biodiversity in a changing environment:
extinction debt, immigration credit and species turnover. Trends Ecol. Evol. 25(3):153–
160.
[312] Cucherousset, J. et al. 2012. Non-native species promote trophic dispersion of food
webs. Front. Ecol. Environ. 10:406–408.
[313] Estes, J. A. et al. 2011. Trophic downgrading of Planet Earth. Science 333:301–306.
[314] Kardol, P. & Wardle, D. A. 2010. How understanding aboveground-belowground
linkages can assist restoration ecology. Trends Ecol. Evol. 25(11):670–679.
[315] Lundberg, P. et al. 2000. Species loss leads to community closure. Ecol. Lett. 3:465–468.
[316] Zavaleta, E. S. et al. 2001. Viewing invasive species removal in a whole-ecosystem
context. Trends Ecol. Evol. 16(8):454–459.
[317] Carvalheiro, L.G. et al. 2008. Pollinator networks, alien species and the conservation
of rare plants: Trinia glauca as a case study. J. Appl. Ecol. 45:1419–1427.
[318] Raymond, B. et al. 2011. Qualitative modelling of invasive species eradication on
subantarctic Macquarie Island. J. Appl. Ecol. 48:181–191.
[319] Cardinale, B. 2012. Impacts of biodiversity loss. Science 336:552–553.
[320] Hulme, P. E. et al. 2013. Bias and error in understanding plant invasion impacts. Trends
Ecol. Evol. 28(4):212–218.
[321] Donohue, I. et al. 2013. On the dimensionality of ecological stability. Ecol. Lett. 16:421–
429.
183
Literature cited
[322] Joo, J. et al. 2007. Effects of noise on ecological invasion processes: bacteriophage-
mediated competition in bacteria. J. Stat. Phys. 128:229–256.
[323] Simberloff, D. 2013. Biol. Invasions: what’s worth fighting and what can be won?
Ecol. Eng. 65:112–121.
[324] Sobol’, I.M. 1967. On the distribution of points in a cube and the approximate eval-
uation of integrals. USSR Computational Mathematics and Mathematical Physics 7(4):86–
112.
[325] Greenman, J. V. & Benton, T.G. 2005. The frequency spectrum of structured discrete
time population models: its properties and their ecological implications. Oikos 110:369–
389.
[326] O’Neill, R. V. 2001. Is it time to bury the ecosystem concept? Ecology 82(12):3275–3284.
[327] Zaccarelli, N. et al. 2013. Order and disorder in ecological time-series: introducing
normalized spectral entropy. Ecol. Indic. 28:22–30.
[328] Gross, N. et al. 2013. Functional differences between alien and native species: do
biotic interactions determine the functional structure of highly invaded grasslands?
Funct. Ecol. 27(5):1262–1272.
[329] Gallardo, B. & Aldridge, D.C. 2013. Priority setting for invasive species manage-
ment: risk assessment of Ponto-Caspian invasive species into Great Britain. Ecol. Appl.
23(2):352–364.
[330] Franklin, J. et al. 2008. Evaluating extreme risks in invasion ecology: learning from
banking compliance. Divers. Distrib. 14:581–591.
[331] Burgman, M.A. et al. 2010. Reconciling uncertain costs and benefits in Bayes nets
for invasive species management. Risk Anal. 30(2):277–284.
[332] Vilcinskas, A. et al. 2013. Invasive harlequin ladybird carries biological weapons against
native competitors. Science 340:862–863.
[333] Hobbs, R. J. et al. 2009. Novel ecosystems: implications for conservation and restora-
tion. Trends Ecol. Evol. 24(11):599–605.
[334] Larson, B.M.H. et al. 2013. Managing invasive species amidst high uncertainty and
novelty. Trends Ecol. Evol. 28(5):255–256.
[335] Simberloff, D. et al. 2013. Impacts of Biol. Invasions: what’s what and the way forward.
Trends Ecol. Evol. 28(1):58–66.
[336] Yelenik, S.G. & D’Antonio, C.M. 2013. Self-reinforcing impacts of plant invasions
change over time. Nature 503:517–520.
[337] Ripple, W. J. & Beschta, R. L. 2004. Wolves and the ecology of fear: can predation
risk structure ecosystems? BioScience 54(8):755–766.
[338] Nicholls, H. 2013. The 18-km2 rat trap. Nature 497:306–308.
[339] Lampert, A. et al. 2014. Optimal approaches for balancing invasive species eradication
and endangered species management. Science 344:1028–1031.
184
Literature cited
[340] Russell, J. C. et al. 2014. Over-invasion by functionally equivalent invasive species.
Ecology 95(8):2268–2276.
[341] Poisot, T. et al. 2013. Trophic complementarity drives the biodiversity-ecosystem
functioning relationship in food webs. Ecol. Lett. 16:853–861.
[342] O’Connor, N.E. & Donohue, I. 2013. Environmental context determines multi-
trophic effects of consumer species loss. Glob. Change Biol. 19:431–440.
[343] Cornelius, S. P. et al. 2013. Realistic control of network dynamics. Nat. Commun. 4:1942.
[344] Isbell, F. & Loreau, M. 2013. Human impacts on minimum subsets of species critical
for maintaining ecosystem structure. Basic Appl. Ecol. 14:623–629.
[345] Ghedini, C.G. & Ribeiro, C.H.C. 2011. Rethinking failure and attack tolerance
assessment in complex networks. Physica A 390:4684–4691.
[346] Mones, E. et al. 2014. Shock waves on complex networks. Sci. Rep. 4:4949.
[347] Cowan, N. J. et al. 2012. Nodal dynamics, not degree distributions, determine the
structural controllability of complex networks. PLoS ONE 7(6):e38398.
[348] Quax, R. et al. 2013. The diminishing role of hubs in dynamical processes on complex
networks. J. R. Soc. Interface 10:20130568.
[349] Galbiati, M. et al. 2013. The power to control. Nat. Phys. 9:126–128.
[350] Wuchty, S. 2014. Controllability in protein interaction networks. P. Natl. Acad. Sci.
usa 111(19):7156–7160.
[351] Hsieh, H.-Y. & Perfecto, I. 2012. Trait-mediated indirect effects of phorid flies on
ants. Psyche 2012:380474.
[352] Kamran-Disfani, A.R. & Golubski, A. J. 2013. Lateral cascade of indirect effects in
food webs with different types of adaptive behavior. J. Theor. Biol. 339:58–69.
[353] Wootton, J. T. 1993. Indirect effects and habitat use in an intertidal community:
interaction chains and interaction modifications. Am. Nat. 141(1):71–89.
[354] 1994. The nature and consequences of indirect effects in ecological commu-
nities. Annu. Rev. Ecol. Syst. 25:443–466.
[355] Dambacher, J.M. & Ramos-Jiliberto, R. 2007. Understanding and predicting effects
of modified interactions through a qualitative analysis of community structure. Q.
Rev. Biol. 82(3):227–250.
[356] Werner, E. E. & Peacor, S.D. 2003. A review of trait-mediated indirect interactions
in ecological communities. Ecology 84(5):1083–1100.
[357] Krenek, L. & Rudolf, V.H.W. 2014. Allometric scaling of indirect effects: body
size ratios predict non-consumptive effects in multi-predator systems. J. Anim. Ecol.
83:1461–1468.
[358] Arditi, R. et al. 2005. Rheagogies: modelling non-trophic effects in food webs. Ecol.
Complex. 2:249–258.
185
Literature cited
[359] Ulanowicz, R. E. & Puccia, C. J. 1990. Mixed trophic impacts in ecosystems. Coenoses
5(1):7–16.
[360] Schmitz, O. et al. 2004. Trophic cascades: the primacy of trait-mediated indirect
interactions. Ecol. Lett. 7:153–163.
[361] Bolker, B. et al. 2003. Connecting theoretical and empirical studies of trait-mediated
interactions. Ecology 84(5):1101–1114.
[362] Hsieh, H.-Y. et al. 2012. Cascading trait-mediated interactions induced by ant phero-
mones. Ecol. Evol. 2(9):2181–2191.
[363] Roode, J. C. de et al. 2011. Aphids indirectly increase virulence and transmission
potential of a monarch butterfly parasite by reducing defensive chemistry of a shared
food plant. Ecol. Lett. 14:453–461.
[364] Abrams, P. A. 1995. Implications of dynamically variable traits for identifying, classi-
fying, and measuring direct and indirect effects in ecological communities. Am. Nat.
146(1):112–134.
[365] Sanders, D. et al. 2014. Integrating ecosystem engineering and food webs. Oikos
123(5):513–524.
[366] Helbing, D. 2013. Globally networked risks and how to respond. Nature 497:51–59.
[367] D’Agostino, G. & Scala, A., eds. 2014. Networks of Networks: The Last Frontier of
Complexity. Springer.
[368] Kivelä, M. et al. 2014. Multilayer networks. J. Complex Netw. 2:203–271.
[369] Gao, J. et al. 2012. Networks formed from interdependent networks. Nat. Phys. 8:40–
48.
[370] Newman, M.E. J. 2012. Communities, modules and large-scale structure in networks.
Nat. Phys. 8:25–31.
[371] Flake, G.W. et al. 2002. Self-organization and identification of web communities.
ieee Comput. 35:66–71.
[372] Betzel, R. F. et al. 2013. Multi-scale community organization of the human structural
connectome and its relationship with resting-state functional connectivity. Network
Science 1(3):353–373.
[373] Hartwell, L.H. et al. 1999. From molecular to modular cell biology. Nature 402:c47–
c52.
[374] Wagner, A. & Fell, D. A. 2001. The small world inside large metabolic networks. Proc.
R. Soc. B 268:1803–1810.
[375] Krause, A. E. et al. 2003. Compartments revealed in food-web structure. Nature 426:282–
285.
[376] Variano, E. A. et al. 2004. Networks, dynamics, and modularity.
Phys. Rev. Lett. 92(18):188701.
186
Literature cited
[377] Domenico, M. de et al. 2014. Navigability of interconnected networks under random
failures. P. Natl. Acad. Sci. usa 111(23):8351–8356.
[378] Brummitt, C.D. et al. 2012. Suppressing cascades of load in interdependent networks.
P. Natl. Acad. Sci. usa 109(12):e680–e689.
[379] Rigterink, D. & Singer, D. J. 2014. A network complexity metric based on planarity
and community structure. J. Complex Netw. 2:299–312.
[380] Pimm, S. L. & Lawton, J.H. 1980. Are food webs divided into compartments? J. Anim.
Ecol. 49:879–898.
[381] Montoya, J.M. et al. 2006. Ecological networks and their fragility. Nature 442:259–
264.
[382] Raffaelli, D. & Hall, S. J. 1992. Compartments and predation in an estuarine food
web. J. Anim. Ecol. 61:551–560.
[383] Fonseca, C.R. & Ganade, G. 1996. Asymmetries, compartments and null interactions
in an Amazonian ant-plant community. J. Anim. Ecol. 65:339–347.
[384] Prado, P. I. & Lewinsohn, T.M. 2004. Compartments in insect-plant associations and
their consequences for community structure. J. Anim. Ecol. 73:1168–1178.
[385] Hines, J. & Gessner, M.O. 2012. Consumer trophic diversity as a fundamental mech-
anism linking predation and ecosystem functioning. J. Anim. Ecol. 81:1146–1153.
[386] Memmott, J. et al. 1994. The structure of a tropical host-parasitoid community. J.
Anim. Ecol. 63:521–540.
[387] Giery, S. T. et al. 2013. Bidirectional trophic linkages couple canopy and understorey
food webs. Funct. Ecol. 27:1436–1441.
[388] Bellmore, J. R. et al. 2013. The floodplain food web mosaic: a study of its importance
to salmon and steelhead with implications for their recovery. Ecol. Appl. 23(1):189–207.
[389] Nummi, P. et al. 2011. Bats benefit from beavers: a facilitative link between aquatic
and terrestrial food webs. Biodivers. Conserv. 20:851–859.
[390] Evans, D.M. et al. 2013. The robustness of a network of ecological networks to habitat
loss. Ecol. Lett. 16:844–852.
[391] Allesina, S. & Pascual, M. 2009. Food web models: a plea for groups. Ecol. Lett.
12:652–662.
[392] Digel, C. et al. 2014. Unravelling the complex structure of forest soil food webs:
higher omnivory and more trophic levels. Oikos 123:1157–1172.
[393] Csardi, G. & Nepusz, T. 2006. The igraph software package for complex network
research. InterJournal Complex Systems:1695.
[394] Crooks, K.R. & Soulé, M.E. 1999. Mesopredator release and avifaunal extinctions
in a fragmented system. Nature 400:563–566.
[395] Mouquet, N. et al. 2012. Extending the concept of keystone species to communities
and ecosystems. Ecol. Lett. 16:1–8.
187
Literature cited
[396] Albrecht, M. et al. 2014. Consequences of plant invasions on compartmentalization
and species’ roles in plant-pollinator networks. Proc. R. Soc. B 281:20140773.
[397] Gopalan, P. K. & Blei, D.M. 2013. Efficient discovery of overlapping communities in
massive networks. P. Natl. Acad. Sci. usa 110(36):14534–14539.
[398] Woodroffe, R. & Ginsberg, J. R. 1998. Edge effects and the extinction of populations
inside protected areas. Science 280:2126–2128.
[399] Huxel, G.R. et al. 2004. At the frontier of the integration of food web ecology
and landscape ecology. In: Polis, G. A. et al., eds. Food Webs at the Landscape Level.
University of Chicago Press, Chicago pp. 434–451.
[400] Polis, G. A. et al., eds. 2004. Food Webs at the Landscape Level. University of Chicago
Press, Chicago.
[401] Palmer, M.A. & Febria, C.M. 2012. The heartbeat of ecosystems. Science 336:1393–
1394.
[402] Tylianakis, J.M. et al. 2010. Conservation of species interaction networks. Biol. Conserv.
143:2270–2279.
[403] Norris, K. 2012. Biodiversity in the context of ecosystem services: the applied need
for systems approaches. Phil. Trans. R. Soc. B 367:191–199.
[404] Rodewald, A.D. et al. 2014. Community-level demographic consequences of urban-
ization: an ecological network approach. J. Anim. Ecol. 83:1409–1417.
[405] Sutherland, W. J. et al. 2004. The need for evidence-based conservation. Trends Ecol.
Evol. 19(6):305–308.
[406] Eklöf, A. et al. 2012. Relevance of evolutionary history for food web structure. Proc.
R. Soc. B 279:1588–1596.
[407] Naisbit, R. E. et al. 2012. Phylogeny versus body size as determinants of food web
structure. Proc. R. Soc. B 282:20120327.
[408] Chamberlain, S. et al. 2014. Phylogenetic tree shape and the structure of mutualistic
networks. J. Ecol. 102:1234–1243.
[409] Minoarivelo, H.O. et al. 2014. Detecting phylogenetic signal in mutualistic interac-
tion networks using a Markov process model. Oikos 123:1250–1260.
[410] Yan, C. & Zhang, Z. 2014. Specific non-monotonous interactions increase persistence
of ecological networks. Proc. R. Soc. B 281:20132797.
[411] Moya-Laraño, J. et al. 2012. Climate change and eco-evolutionary dynamics in food
webs. Adv. Ecol. Res. 47:1–80.
[412] Fortuna, M. A. et al. 2013. Evolving digital ecological networks. PLoS Comput. Biol.
9(3):e1002928.
[413] Held, I. 2014. Simplicity amid complexity. Science 343:1206–1207.
[414] Tero, A. et al. 2010. Rules for biologically inspired adaptive network design. Science
327:439–442.
188
Literature cited
[415] Sutherland, W. J. & Freckleton, R. P. 2012. Making predictive ecology more relevant
to policy makers and practitioners. Phil. Trans. R. Soc. B 367:322–330.
[416] Milner-Gulland, E. J. 2012. Interactions between human behaviour and ecological
systems. Phil. Trans. R. Soc. B 367:270–278.
[417] Horan, R.D. et al. 2011. Managing ecological thresholds in coupled environmental-
human systems. P. Natl. Acad. Sci. usa 108(18):7333–7338.
[418] Soltani, A. et al. 2014. A dynamic bio-economic model for community management
of goat and oak forests in Zagros, Iran. Ecol. Econ. 106:174–185.
[419] Parrott, L. & Meyer, W. S. 2012. Future landscapes: managing within complexity.
Front. Ecol. Environ. 10(7):382–389.
[420] Filotas, E. et al. 2014. Viewing forests through the lens of complex systems science.
Ecosphere 5(1):1.
[421] Sutherland, W. J. & Woodroof, H. J. 2009. The need for environmental horizon
scanning. Trends Ecol. Evol. 24(10):523–527.
[422] Sutherland, W. J. et al. 2014. A horizon scan of global conservation issues for 2014.
Trends Ecol. Evol. 29(1):15–22.
[423] Ben-Haim, Y. 2006. Info-Gap Decision Theory: Decisions under Severe Uncertainty. Aca-
demic Press, Oxford.
[424] Helbing, D. et al. 2014. Saving human lives: what complexity science and information
systems can contribute. J. Stat. Phys. 158(3):735–781.
[425] Loreau, M. 2010. From Populations to Ecosystems: Theoretical Foundations for a New Ecolog-
ical Synthesis. Monographs in Population Biology 46. Princeton University Press, New
Jersey.
189