A bifurcation theorem for evolutionary matrix models with multiple traits

J. M. Cushing, F. Martins, A. A. Pinto, Amy Veprauskas

Research output: Research - peer-reviewArticle

Abstract

One fundamental question in biology is population extinction and persistence, i.e., stability/instability of the extinction equilibrium and of non-extinction equilibria. In the case of nonlinear matrix models for structured populations, a bifurcation theorem answers this question when the projection matrix is primitive by showing the existence of a continuum of positive equilibria that bifurcates from the extinction equilibrium as the inherent population growth rate passes through 1. This theorem also characterizes the stability properties of the bifurcating equilibria by relating them to the direction of bifurcation, which is forward (backward) if, near the bifurcation point, the positive equilibria exist for inherent growth rates greater (less) than 1. In this paper we consider an evolutionary game theoretic version of a general nonlinear matrix model that includes the dynamics of a vector of mean phenotypic traits subject to natural selection. We extend the fundamental bifurcation theorem to this evolutionary model. We apply the results to an evolutionary version of a Ricker model with an added Allee component. This application illustrates the theoretical results and, in addition, several other interesting dynamic phenomena, such as backward bifurcation induced strong Allee effects.

LanguageEnglish (US)
Pages1-30
Number of pages30
JournalJournal of Mathematical Biology
DOIs
StateAccepted/In press - Jan 6 2017

Fingerprint

Matrix Models
Bifurcation
Theorem
Nonlinear Dynamics
Population
extinction
Extinction
Genetic Selection
Population Growth
Growth
Direction compound
Nonlinear Model
Model
dynamic models
natural selection
population growth
Biological Sciences
Backward Bifurcation
Allee Effect
Projection Matrix

Keywords

  • Bifurcation
  • Equilibria
  • Evolutionary game theory
  • Nonlinear matrix models
  • Stability
  • Structured population dynamics

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics

Cite this

A bifurcation theorem for evolutionary matrix models with multiple traits. / Cushing, J. M.; Martins, F.; Pinto, A. A.; Veprauskas, Amy.

In: Journal of Mathematical Biology, 06.01.2017, p. 1-30.

Research output: Research - peer-reviewArticle

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