This dissertation starts with an introduction to the subject matter and an overview of some of the existing work in this field, followed by an explanation of the motives for developing a highly simplistic model of a river system, and a more detailed summary of the work done on it. The second chapter contains a description of the model developed, along with an explanation the choices made, and an overview of both the ways in which the result might be expected to resemble a real system, and how it might differ. This is followed by a description of the behaviour of this model in the region of linear behaviour, where it can be studied analytically, and then of the development of a simulation designed to extend the research into the non-linear regime. The fifth and sixth chapters go over the results obtained from this simulation, making comparisons where possible with physical systems. The fifth chapter covers naturally occurring examples: beaches and braided rivers, while the second is devoted entirely to a more detailed comparison with one specific experiment whose conditions are in some ways close to the simulation's. This reveals a number of resemblances. The chapter after this covers an alteration of the model that attempts to remedy the limitations imposed by the initial use of periodic boundary conditions, by allowing sediment to be created or destroyed in a controlled manner. It also describes the new aspects of linear behaviour resulting from this change, particularly the sharp transition that exists between erosional and depositional behaviour. Chapter eight extends this discussion to the non-linear behaviour, and the extent to which the two are similar. The last chapter summarises the ground covered, and outlines a number of possibilities for future use and improvement of the model.