Existence of a polycycle in non-Lipschitz Gause-type predator-prey models
type: Item
instance of: scholarly article
title: Existence of a polycycle in non-Lipschitz Gause-type predator-prey models
zbMATH Open document ID: 1216.34038
author: D. Kharzeev
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-10-25T00:00:00Z
review text: This paper studies the Gause-type predator-prey model \[ x' = F(x) - y \phi(x),\quad y' = y (\psi(x) - \mu). \] The authors show that, when the interaction between predator and prey is not locally-Lipschitz continuous in the absence of one of them, there appears a polycycle, which affects the existence of limit cycles for the system. The authors assume: {\parindent8mm \begin{itemize}\item[(a] \(F(0) = 0\), and there exists \(k>0\) such that \(F(x) (x-k)<0\) for \(0<x\not=k\), and \(F'(k) < 0 < F'(0)\). \item[(a] \(\phi(0) = \psi(0) = 0\), and \(\phi'(x), \psi'(x) > 0\) for all \(x > 0\). \item[(a] \(\mu \in \psi((0,k))\). \item[(a] For some \(x > 0\), \(\int_0^x {d z \over \phi(z)} > +\infty\). \item[(a] \(\lim_{x\to 0^+} F(x)/\phi(x) = 0\), and there exist \[ \lim_{x\to 0^+} (F/\phi)'(x)\phi(x)\in {\mathbb R},\quad \lim_{x\to 0^+} \psi'(x)\phi(x)\in {\mathbb R}. \] \end{itemize}} Under the conditions above, the authors prove the existence of a unique value of \(\mu\) such that the equation possesses a positive heteroclinic connection \(H\) joining the equilibrium \((k,0)\) to the equilibrium \((0,0)\). In particular, for this value of \(\mu\), the above system has a polycycle consisting of \(H\) plus the linear segment that joins \((0,0)\) to \((k,0)\).
reviewed by: Jinzhi Lei
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5804526
zbMATH Keywords: Predator-prey
zbMATH Keywords: heteroclinic
zbMATH Keywords: polycycle
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.08.001
OpenAlex ID: W1964531216
Wikidata QID: Q115570424
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DOI: 10.1016/J.JMAA.2010.08.001
description: scientific article; zbMATH DE number 5804526