Dynamic consequences of reproductive delay in Leslie matrix models with nonlinear survival probabilities.

*(English)*Zbl 0905.92025Summary: The dynamic consequences of reproductive delay in Leslie matrix models with nonlinear survival probabilities \(p\) are analyzed. In consideration of two-age classes, a proof is presented for a wide range of \(p\) functions that, outside the strongly resonant cases, the transfer from stability to instability goes through a supercritical Hopf bifurcation and, moreover, that the nonlinear development has a strong resemblance of three or four cycles, either exact or approximate.

In three-age class models, the tendency toward four-periodical dynamics is shown to be even more pronounced, a qualitative finding that gradually disappears as we turn to the higher-dimensional cases. We also prove that for models of any dimension \(n> 1\) there are regions in parameter space where the equilibrium is unstable at its creation and we demonstrate that the dynamics in this age-class extinguishing case is \(2^k\cdot n\) cyclic.

In three-age class models, the tendency toward four-periodical dynamics is shown to be even more pronounced, a qualitative finding that gradually disappears as we turn to the higher-dimensional cases. We also prove that for models of any dimension \(n> 1\) there are regions in parameter space where the equilibrium is unstable at its creation and we demonstrate that the dynamics in this age-class extinguishing case is \(2^k\cdot n\) cyclic.

##### MSC:

92D25 | Population dynamics (general) |

91D20 | Mathematical geography and demography |

39A10 | Additive difference equations |

34K20 | Stability theory of functional-differential equations |

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