Mutual invadability implies coexistence in spatial models / Rick Durrett.

Format:

Book

Author/Creator:

Durrett, Richard, 1951- author.

Series:

Memoirs of the American Mathematical Society ; no. 740.

Memoirs of the American Mathematical Society, 0065-9266 ; number 740

Language:

English

Subjects (All):

Stochastic processes.

Reaction-diffusion equations.

Biology--Mathematical models.

Biology.

Physical Description:

1 online resource (133 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2002]

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Introduction""; ""Example 1. Predator-prey models""; ""Example 2. Epidemic models""; ""1. Perturbation of one-dimensional systems""; ""2. Two-species Examples""; ""Example 2.1. Linear competition with exclusion""; ""Example 2.2. Two-stage contact process""; ""Example 2.3. Diploid genetics""; ""Example 2.4. One-dimensional systems""; ""Example 2.5. Linear competition without exclusion""; ""3. Lower bounding lemmas for PDE""; ""4. Perturbation of higher-dimensional systems""; ""5. Lyapunov functions for Lotka Volterra systems""; ""6. Three species linear completion models""

""7. Three species predator-prey systems""""Example 7.1. Two-prey, one-predator model""; ""Example 7.2. Three species food chain""; ""Example 7.3. Two-predator, one-prey model""; ""Example 7.4. Two infection model""; ""8. Some asymptotic results for our ODE and PDE""; ""A List of the Invadability Conditions""; ""References""

Notes:

"March 2002, volume 156, number 740 (first of 5 numbers)."

Includes bibliographical references (pages 110-118).

Description based on print version record.

ISBN:

1-4704-0333-1