In (1994) Durrett and Levin proposed that the equilibrium behavior of stochastic spatial models could be determined from properties of the solution of the mean field ordinary differential equation (ODE) that is obtained by pretending that all sites are always independent. Here we prove a general result in support of that picture. We give a condition on an ordinary differential equation which implies that densities stay bounded away from 0 in the associated reaction-diffusion equation, and that coexistence occurs in the stochastic spatial model with fast stirring. Then using biologists' notion of invadability as a guide, we show how this condition can be checked in a wide variety of examples that involve two or three species: epidemics, diploid genetics models, predator-prey systems, and various competition models.
Introduction Perturbations of one-dimensional systems Two-species examples Lower bounding lemmas for PDE Perturbations of higher-dimensional systems Lyapunov functions for two-species Lotka Volterra systems Three species linear competition models Three species predator-prey systems Some asymptotic results for our ODE and PDE A list of the invadability conditions References.