A global bifurcation theorem for Darwinian matrix models

Emily P. Meissen, Kehinde R. Salau, Jim M. Cushing

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

Motivated by models from evolutionary population dynamics, we study a general class of nonlinear difference equations called matrix models. Under the assumption that the projection matrix is non-negative and irreducible, we prove a theorem that establishes the global existence of a continuum with positive equilibria that bifurcates from an extinction equilibrium at a value of a model parameter at which the extinction equilibrium destabilizes. We give criteria for the global shape of the continuum, including local direction of bifurcation and its relationship to the local stability of the bifurcating positive equilibria. We discuss a relationship between backward bifurcations and Allee effects. Illustrative examples are given.

Original languageEnglish (US)
Pages (from-to)1114-1136
Number of pages23
JournalJournal of Difference Equations and Applications
Volume22
Issue number8
DOIs
StatePublished - Aug 2 2016

Keywords

  • Allee effects
  • Nonlinear matrix models
  • bifurcation
  • evolutionary population dynamics
  • stability

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

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