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The trade-off between mutual interference and time lags in predator-prey systems. (English) Zbl 0535.92024
This paper presents a non-linear 2 dimensional model of a predator-prey system. There is a delay $$\tau$$ in it, which corresponds to the time required to ”convert prey into predator”. Another feature of the model is the crowding counter-effect (or: mutual interference) in the predation. The paper deals with the local behavior near to the equilibrium corresponding to persistence of both species. Depending on the values of ecologically meaningful parameters, various results are shown:
In some cases (theorem 4.1) there is instability for all delays; in other cases, the delay is proved to act as a de-stabilizer (theorem 4.2). Similar results are obtained for asymptotic stability; formally, there is a Hopf bifurcation. The proofs combine the use of elementary calculus together with a lemma proved by G. J. Butler (which, strangely, does not seem to be ”classical”). Roughly speaking, this lemma says that: $$p(\lambda$$,$$\tau)$$ being analytic in $$\lambda$$, if $$R_{\tau}=\{Re \lambda:p(\lambda,\tau)=0\}$$ is $${\mathbb{R}}^*\!_-$$ for $$\tau =\tau_ 1$$ and has a nonempty intersection with $${\mathbb{R}}^*\!_+$$ for $$\tau =\tau_ 2$$, then: $$0\in R_{\tau}$$ for some $$\tau \in(\tau_ 1,\tau_ 2)$$.
Reviewer: O.Arino

MSC:
 92D40 Ecology 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
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References:
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