Stochastic Spatial Models: From Simulations to Mean Field and Local Structure Approximations

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A discrete stochastic spatial model for a single species is examined. First, detailed spatial simulations are performed using stochastic cellular automata. Then, several analytic approximations are made. First, two versions of mean field theory are presented: the infinite-dispersal mean field approximation, which is a metapopulation-like model, and the local-dispersal mean field approximation, which incorporates the locality of the cellular automaton model but assumes that no spatial correlations develop in the lattice. Next, the local-dispersal mean field theory is generalised into several varieties of local structure theory, in which one assumes that groups of nearby sites in the lattice are correlated, and tracks such correlations under the action of the cellular automaton rule. Assuming such local correlations allows one to predict patch occupancy as well as the degree of clustering in the cellular automaton model much more accurately than mean field theory, especially in parameter regimes where mean field theory does poorly. Simulation and mean field theory are seen to be two opposite extremes of an entire spectrum of methods that may be used to investigate discrete spatial models.