Memoirs of the American Mathematical Society 2002; 118 pp; softcover Volume: 156 ISBN10: 0821827685 ISBN13: 9780821827680 List Price: US$62 Individual Members: US$37.20 Institutional Members: US$49.60 Order Code: MEMO/156/740
 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 reactiondiffusion 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, predatorprey systems, and various competition models. Readership Graduate students and research mathematicians interested in probability theory, stochastic processes, and differential equations. Table of Contents  Introduction
 Perturbations of onedimensional systems
 Twospecies examples
 Lower bounding lemmas for PDE
 Perturbations of higherdimensional systems
 Lyapunov functions for twospecies Lotka Volterra systems
 Three species linear competition models
 Three species predatorprey systems
 Some asymptotic results for our ODE and PDE
 A list of the invadability conditions
 References
