ABSTRACT: Stochastic Spatial Models: From Simulations to
Mean Field and Local Structure Approximations
by David Hiebeler
Center for Applied Math, Cornell University
Journal of Theoretical Biology 187 (1997), 307--319.
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
generalized 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.
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Last modified: Wed May 19 20:37:00 1999