## 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)

### Keywords:

matrix equations; non-linearity; net reproductive number
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### References:

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