The trade-off between mutual interference and time lags in predator-prey systems
type: Item
instance of: scholarly article
title: The trade-off between mutual interference and time lags in predator-prey systems
zbMATH Open document ID: 0535.92024
DOI: 10.1007/BF02458826
author: V. Sree Hari Rao
author: H. I. Freedman
published in: Bulletin of Mathematical Biology
publication date: 1983-01-01T00:00:00Z
review text: 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 \textit{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 \(\tau \in(\tau_ 1,\tau_ 2)\).
Mathematics Subject Classification ID: 92D40
Mathematics Subject Classification ID: 34K99
zbMATH DE Number: 3850893
zbMATH Keywords: time lags
zbMATH Keywords: Gause predator-prey model
zbMATH Keywords: periodic solutions
zbMATH Keywords: non-linear 2 dimensional model
zbMATH Keywords: crowding counter-effect
zbMATH Keywords: mutual interference
zbMATH Keywords: local behavior near to the equilibrium corresponding to persistence of both species
zbMATH Keywords: asymptotic stability
zbMATH Keywords: Hopf bifurcation
reviewed by: Ovide Arino
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02458826
OpenAlex ID: W4230198350
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description: scientific article; zbMATH DE number 3850893