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Dynamic consequences of reproductive delay in Leslie matrix models with nonlinear survival probabilities. (English) Zbl 0905.92025
Summary: 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.

##### 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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