| eBook ISBN: | 978-1-4704-0333-1 |
| Product Code: | MEMO/156/740.E |
| List Price: | $62.00 |
| MAA Member Price: | $55.80 |
| AMS Member Price: | $37.20 |
| eBook ISBN: | 978-1-4704-0333-1 |
| Product Code: | MEMO/156/740.E |
| List Price: | $62.00 |
| MAA Member Price: | $55.80 |
| AMS Member Price: | $37.20 |
-
Book DetailsMemoirs of the American Mathematical SocietyVolume: 156; 2002; 118 ppMSC: Primary 60; Secondary 34; 35
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.
ReadershipGraduate students and research mathematicians interested in probability theory, stochastic processes, and differential equations.
-
Table of Contents
-
Chapters
-
Introduction
-
1. Perturbation of one-dimensional systems
-
2. Two-species examples
-
3. Lower bounding lemmas for PDE
-
4. Perturbation of higher-dimensional systems
-
5. Lyapunov functions for Lotka Volterra systems
-
6. Three species linear competion models
-
7. Three species predator-prey systems
-
8. Some asymptotic results for our ODE and PDE
-
-
RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Requests
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.
Graduate students and research mathematicians interested in probability theory, stochastic processes, and differential equations.
-
Chapters
-
Introduction
-
1. Perturbation of one-dimensional systems
-
2. Two-species examples
-
3. Lower bounding lemmas for PDE
-
4. Perturbation of higher-dimensional systems
-
5. Lyapunov functions for Lotka Volterra systems
-
6. Three species linear competion models
-
7. Three species predator-prey systems
-
8. Some asymptotic results for our ODE and PDE
