### Abstract

We prove a general theorem for nonlinear matrix models of the type used in structured population dynamics that describes the bifurcation that occurs when the extinction equilibrium destabilizes as a model parameter is varied. The existence of a bifurcating continuum of positive equilibria is established, and their local stability is related to the direction of bifurcation. Our theorem generalizes existing theorems found in the literature in two ways. First, it allows for a general appearance of the bifurcation parameter (existing theorems require the parameter to appear linearly). This significantly widens the applicability of the theorem to population models. Second, our theorem describes circumstances in which a backward bifurcation can produce stable positive equilibria (existing theorems allow for stability only when the bifurcation is forward). The signs of two diagnostic quantities determine the stability of the bifurcating equilibrium and the direction of bifurcation. We give examples that illustrate these features.

Original language | English (US) |
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Pages (from-to) | 25-44 |

Number of pages | 20 |

Journal | Journal of Difference Equations and Applications |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2 2020 |

### Keywords

- 37G35
- 39A28
- 39A30
- Nonlinear difference equations
- bifurcation
- equilibrium
- matrix equations
- population dynamics
- stability

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Applied Mathematics

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## Cite this

*Journal of Difference Equations and Applications*,

*26*(1), 25-44. https://doi.org/10.1080/10236198.2019.1699916