Spatial models: stochastic and deterministic

Theoretical studies of biological populations via analysis and/or simulation of deterministic and stochastic systems sometimes end up drawing conflicting conclusions. Papers purporting to investigate the same dynamics, albeit through different methods, often cannot agree on essential properties of the system being modeled. This problem often arises when trying to compare results that were obtained from different kinds of mathematical models, say those based on differential equations and individual-based stochastic models. While such models can successfully represent or characterize different views of the same phenomena, it is important to know when two different approaches are comparable, as well as any limitations that may be inherent in such a comparison. This survey paper is directed primarily to mathematical biologists whose primary mode of operation is partial differential equations. More generally, we seek to illuminate connections between the two main realms of spatial modeling. We begin by presenting a quick introduction to a class of stochastic spatial models, known as interacting particle systems, which are readily applicable to biological (and many other) systems. We then give examples of how various scaled limits of these models give rise to reaction-diffusion equations and integrodifferential equations. The first case falls under the heading of hydrodynamic limits and the second case is an example of a mean-field limit theorem.