## Spatial Models: Stochastic and Deterministic

*Stephen M. Krone*
**Abstract**

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 integro-differential equations.
The first case falls under the heading of hydrodynamic limits and the second
case is an example of a mean-field limit theorem.