### Abstract

Using the methodology of evolutionary game theory (EGT), I study a class of Darwinian matrix models which are derived from a class of nonlinear matrix models for structured populations that are known to possess stable (normalized) distributions. Utilizing the limiting equations that result from this ergodic property, I prove extinction and stability results for the limiting equations of the EGT versions of these kinds of structured population models. This is done in a bifurcation theory context. The results provide conditions sufficient for a branch of non-extinction equilibria to bifurcate from the branch of extinction equilibria. When this bifurcation is supercritical (explicit criteria are given), these equilibria are stable and represent stable non-extinction equilibria (which are also candidate ESS equilibria). These kinds of matrix models are motivated by applications to size structured populations, and I give an application of this type. Besides illustrating the formal theory, this application shows the importance of trade-offs among life history parameters that are necessary for the existence of an evolutionarily stable equilibrium.

Original language | English (US) |
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Pages (from-to) | 103-116 |

Number of pages | 14 |

Journal | Nonlinear Dynamics and Systems Theory |

Volume | 10 |

Issue number | 2 |

State | Published - Jul 9 2010 |

### Keywords

- Bifurcation
- Equilibrium
- Evolutionary game theory
- Limiting equation
- Nonlinear matrix model
- Stability
- Stable distribution
- Structured population dynamics

### ASJC Scopus subject areas

- Mathematical Physics
- Applied Mathematics

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## Cite this

*Nonlinear Dynamics and Systems Theory*,

*10*(2), 103-116.