Population dynamics under selection and mutation: Long-time behavior for differential equations in measure spaces

Azmy S. Ackleh, John Cleveland, Horst Thieme

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

We study the long-time behavior of solutions to a measure-valued selection-mutation model that we formulated in [14]. We establish permanence results for the full model, and we study the limiting behavior even when there is more than one strategy of a given fitness; a case that arises in applications. We show that for the pure selection case the solution of the dynamical system converges to a Dirac measure centered at the fittest strategy class provided that the support of the initial measure contains a fittest strategy; thus we term this Dirac measure an Asymptotically Stable Strategy. We also show that when the strategy space is discrete, the selection-mutation model with small mutation has a locally asymptotically stable equilibrium that attracts all initial conditions that are positive at the fittest strategy.

Original languageEnglish (US)
Pages (from-to)1472-1505
Number of pages34
JournalJournal of Differential Equations
Volume261
Issue number2
DOIs
StatePublished - Jul 15 2016

Keywords

  • Asymptotically stable strategy
  • Cone of nonnegative measures
  • Evolutionary game theory
  • Lyapunov functions
  • Persistence theory
  • Survival of the fittest

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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