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Stability conditions for equilibria on nonlinear matrix population models. (English) Zbl 0920.92022

Let \(x(t)\) denote a vector of class densities and \(w(t)\) the weighted size of a population at times \(t=0,1,2,\dots\). For the matrix equation \(x(t+1)=A(w(t))x(t)\) the local stability of a positive equilibrium \(x\) is related to the net reproductive number \(n(w)\), i.e. the average number of offspring from an individual over its lifetime if the class distribution vector were held fixed at \(x\), as follows.
Under appropriate assumptions it has already been proved for Leslie age-structured models that \(n'(w)>0\) implies that \(x\) is unstable [cf. J.M. Cushing, Nat. Resour. Model. 2, No. 4, 539-580 (1988)]. Now this result is achieved under certain rather general conditions for general structured population models of the above form. Since \(n'(w)\leq 0\) turns out to be only a necessary condition for a positive equilibrium to be locally asymptotically stable, the authors also look for sufficient conditions and present a variety of models for which these conditions are fulfilled. Moreover, many examples for the application of the theoretical findings to models known from the literature are given.
Reviewer: D.Dorninger (Wien)

MSC:

92D25 Population dynamics (general)
39A11 Stability of difference equations (MSC2000)
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References:

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