### Abstract

In this paper we consider the bifurcations that occur at the trivial equilibrium of a general class of nonlinear Leslie matrix models for the dynamics of a structured population in which only the oldest class is reproductive. Using the inherent net reproductive number n as a parameter, we show that a global branch of positive equilibria bifurcates from the trivial equilibrium at n = 1 despite the fact that the bifurcation is nongeneric. The bifurcation can be either supercritical or subcritical, but unlike the case of a generic transcritical bifurcation in iteroparous models, the stability of the bifurcating positive equilibria is not determined by the direction of bifurcation. In addition we show that a branch of single-class cycles also bifurcates from the trivial equilibrium at n = 1. In the case of two population classes, either the bifurcating equilibria or the bifurcating cycles are stable (but not both) depending on the relative strengths of the inter- and intra-class competition. Strong inter-class competition leads to stable cycles in which the two population classes are temporally separated. In the case of three or more classes the bifurcating cycles often lie on a bifurcating invariant loop whose structure is that of a cycle chain consisting of the different phases of a periodic cycle connected by heteroclinic orbits. Under certain circumstances, these bifurcating loops are attractors.

Original language | English (US) |
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Pages (from-to) | 17-36 |

Number of pages | 20 |

Journal | Mathematical Biosciences and Engineering |

Volume | 3 |

Issue number | 1 |

State | Published - Jan 1 2006 |

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### Keywords

- Bifurcation
- Cycles
- Leslie matrix
- Nonlinear matrix models
- Semelparity
- Stability

### ASJC Scopus subject areas

- Modeling and Simulation
- Agricultural and Biological Sciences(all)
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematical Biosciences and Engineering*,

*3*(1), 17-36.