Multiparametric bifurcation analysis of a basic two-stage population model.

*(English)*Zbl 1106.34030Summary: We investigate long-term dynamics of the most basic model for stage-structured populations, in which the per capita transition from the juvenile into the adult class is density dependent. The model is represented by an autonomous system of two nonlinear differential equations with four parameters for a single population. We find that the interaction of intra-adult competition and intra-juvenile competition gives rise to multiple attractors, one of which can be oscillatory. A detailed numerical study reveals a rich bifurcation structure for this two-dimensional system, originating from a degenerate Bogdanov-Takens (BT) bifurcation point when one parameter is kept constant. Depending on the value of this fixed parameter, the corresponding triple critical equilibrium has either an elliptic sector or it is a topological focus, which is demonstrated by the numerical normal form analysis. It is shown that the canonical unfolding of the codimension-three BT point reveals the underlying dynamics of the model. Certain new features of this unfolding in the elliptic case, which are important in applications but have been overlooked in available theoretical studies, are established. Various three-, two-, and one-parameter bifurcation diagrams of the model are presented and interpreted in biological terms.

Reviewer: Josef Hainzl (Freiburg)

##### MSC:

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

34C23 | Bifurcation theory for ordinary differential equations |

92D25 | Population dynamics (general) |

37G05 | Normal forms for dynamical systems |

37G10 | Bifurcations of singular points in dynamical systems |

37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |

34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |