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In this paper we introduce a class of stochastic population models based on"patch dynamics". The size of the patch may be varied, and this allows one toquantify the departures of these stochastic models from various mean fieldtheories, which are generally valid as the patch size becomes very large. Thesemodels may be used to formulate a broad range of biological processes in bothspatial and non-spatial contexts. Here, we concentrate on two-speciescompetition. We present both a mathematical analysis of the patch model, inwhich we derive the precise form of the competition mean field equations (andtheir first order corrections in the non-spatial case), and simulation results.These mean field equations differ, in some important ways, from those which arenormally written down on phenomenological grounds. Our general conclusion isthat mean field theory is more robust for spatial models than for a singleisolated patch. This is due to the dilution of stochastic effects in a spatialsetting resulting from repeated rescue events mediated by inter-patchdiffusion. However, discrete effects due to modest patch sizes lead to strikingdeviations from mean field theory even in a spatial setting.

Stochastic models in population biology and their deterministic analogs