Mutual Invadability Implies Coexistence in Spatial Models

by Richard Durrett

In (1994) Durrett and Levin proposed that the equilibrium behaviour 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 Durrett proves a general result in support of that picture. He gives 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, he shows 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.

118 pages, Mass Market Paperback

First published March 1, 2002

About the author

Richard Timothy Durrett is a mathematician known for his research and books on mathematical probability theory, stochastic processes and their application to mathematical ecology and population genetics.

He received his BS and MS at Emory University in 1972 and 1973 and his Ph.D. at Stanford University in 1976 under advisor Donald Iglehart. From 1985 until 2010 was on the faculty at Cornell University. Since 2010, Durrett has been a professor at Duke University.

He was elected to the United States National Academy of Sciences in 2007. In 2012 he became a fellow of the American Mathematical Society.[1]

Durrett is the founder of the Cornell Probability Summer Schools, and he is still its scientific organizer.