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A bifurcation theorem for evolutionary matrix models with multiple traits.

J M Cushing1,2, F Martins3, A A Pinto3

  • 1Department of Mathematics, University of Arizona, 617 N Santa Rita, Tucson, AZ, 85721, USA. cushing@math.arizona.edu.

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Summary
This summary is machine-generated.

This study extends bifurcation theory to evolutionary game models, revealing how population growth rates influence stability and persistence. It demonstrates backward bifurcation in an Ricker model, highlighting strong Allee effects.

Keywords:
BifurcationEquilibriaEvolutionary game theoryNonlinear matrix modelsStabilityStructured population dynamics

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Area of Science:

  • Population dynamics
  • Evolutionary game theory
  • Mathematical biology

Background:

  • Population stability and extinction are fundamental biological questions.
  • Nonlinear matrix models and bifurcation theorems address population dynamics.
  • Previous work focused on primitive matrix models for population persistence.

Purpose of the Study:

  • To extend the fundamental bifurcation theorem to evolutionary game theoretic nonlinear matrix models.
  • To analyze the dynamics of phenotypic traits under natural selection.
  • To investigate population persistence and extinction equilibria in evolutionary contexts.

Main Methods:

  • Developed an evolutionary game theoretic version of a general nonlinear matrix model.
  • Extended a fundamental bifurcation theorem to this new evolutionary model.
  • Applied the extended theorem to an evolutionary Ricker model with an Allee effect.

Main Results:

  • The study successfully extended bifurcation theory to evolutionary nonlinear matrix models.
  • Demonstrated the existence of bifurcating equilibria and characterized their stability.
  • Observed backward bifurcation and strong Allee effects in the Ricker model application.

Conclusions:

  • Bifurcation theory provides insights into population persistence and extinction in evolutionary models.
  • The extended theorem offers a framework for analyzing complex population dynamics.
  • Evolutionary dynamics, including Allee effects, can significantly impact population stability.