Stochastic spatial models

In the models we will consider, space is represented by a grid of sites tha t can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites. Our main aim here is to e xplain an idea of Durrett and Levin (1994): the behavior of these models ca n be predicted from the properties of the mean field ODE, i.e., the equatio ns for the densities of the various types that result from pretending that all sites are always independent. We will illustrate this picture through a discussion of eight families of examples from statistical mechanics, genet ics, population biology, epidemiology, and ecology. Some of our findings ar e only conjectures based on simulation, but in a number of cases we are abl e to prove results for systems with "fast stirring" by exploiting connectio ns between the spatial model and an associated reaction diffusion equation.