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Mutual Invadability Implies Coexistence in Spatial Models
 
Rick Durrett Cornell University, Ithaca, NY
Mutual Invadability Implies Coexistence in Spatial Models
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
Mutual Invadability Implies Coexistence in Spatial Models
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Mutual Invadability Implies Coexistence in Spatial Models
Rick Durrett Cornell University, Ithaca, NY
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 Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1562002; 118 pp
    MSC: 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.

    Readership

    Graduate 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1562002; 118 pp
MSC: 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.

Readership

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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.