Coexistence in stochastic spatial models

TL;DR

This paper reviews twenty years of research on coexistence in stochastic spatial models, emphasizing the role of mean-field ODEs and highlighting open problems in the field.

Contribution

It synthesizes existing rigorous results and identifies nine key open problems related to coexistence in stochastic spatial models.

Findings

Coexistence can be analyzed through mean-field ODEs.

Rigorous results support the mean-field approach.

Nine important open problems are outlined.

Abstract

In this paper I will review twenty years of work on the question: When is there coexistence in stochastic spatial models? The answer, announced in Durrett and Levin [Theor. Pop. Biol. 46 (1994) 363--394], and that we explain in this paper is that this can be determined by examining the mean-field ODE. There are a number of rigorous results in support of this picture, but we will state nine challenging and important open problems, most of which date from the 1990's.