**Memoirs of the American Mathematical Society**

2002;
118 pp;
Softcover

MSC: Primary 60;
Secondary 34; 35

Print ISBN: 978-0-8218-2768-0

Product Code: MEMO/156/740

List Price: $62.00

Individual Member Price: $37.20

**Electronic ISBN: 978-1-4704-0333-1
Product Code: MEMO/156/740.E**

List Price: $62.00

Individual Member Price: $37.20

# Mutual Invadability Implies Coexistence in Spatial Models

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*Rick Durrett*

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.

#### Table of Contents

# Table of Contents

## Mutual Invadability Implies Coexistence in Spatial Models

- Contents vii8 free
- Introduction 110 free
- 1. Perturbation of one-dimensional systems 1221
- 2. Two-species Examples 1726
- 3. Lower bounding lemmas for PDE 3443
- 4. Perturbation of higher-dimensional systems 4049
- 5. Lyapunov functions for Lotka Volterra systems 4857
- 6. Three species linear competion models 6069
- 7. Three species predator-prey systems 7584
- 8. Some asymptotic results for our ODE and PDE 102111
- A List of the Invadability Conditions 109118
- References 110119

#### Readership

Graduate students and research mathematicians interested in probability theory, stochastic processes, and differential equations.