Local and global stability for population models. (English) Zbl 0607.92018
The author considers a population model of the form \(X_{t+1}=f(X_ t)\), where f is a continuous function such that \(f(0)=0\) and there is a unique positive equilibrium point \(\bar X\) such that \(f(\bar X)=\bar X\), \(f(X)>X\) for \(0<X<\bar X\) and \(f(X)<X\) for \(\bar X<X\). Moreover, if f has a maximum \(X_ M\) in \((0,\bar X)\), then f is decreasing for all \(X>X_ M\) such that \(f(X)>0.\)
The main result of the paper is the following theorem: A population model is globally stable iff it has no cycles of period 2. Also some sufficient conditions for global stability are presented.
At the end of the paper some models are studied which have appeared in the literature. Among others, it is shown that these models are globally stable.
The main result of the paper is the following theorem: A population model is globally stable iff it has no cycles of period 2. Also some sufficient conditions for global stability are presented.
At the end of the paper some models are studied which have appeared in the literature. Among others, it is shown that these models are globally stable.
Reviewer: U.Wilczyńska
MSC:
| 92D25 | Population dynamics (general) |
Keywords:
deterministic population models; local stability; positive equilibrium point; cycles of period 2; sufficient conditions for global stabilityReferences:
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