Limit cycles in a general Kolmogorov model
type: Item
instance of: scholarly article
title: Limit cycles in a general Kolmogorov model
zbMATH Open document ID: 1074.34035
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-05-12T00:00:00Z
review text: The authors study the planar system of differential equations \[ x' = \phi(x)[F(x) - \pi(y)], \quad y' = \rho(y)[-r + \psi(x) + \xi(y)], \] which contains many model predator-prey systems that appear in the literature. Their main result is a list of four conditions that guarantee that any limit cycle appearing in the first quadrant is unique. Rather than applying a transformation of coordinates that produces a Liénard system, they introduce auxiliary functions in order to estimate the integral of the divergence around any such cycle, thus determining its stability, and then produce a contradiction if more than one cycle is assumed to exist.
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 2167839
zbMATH Keywords: limit cycle
zbMATH Keywords: predator-prey systems
author: Xun-Cheng Huang
author: Le-Min Zhu
MaRDI profile type: Publication
full work available at URL: j.na.2004.11.003
OpenAlex ID: W1992529385
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DOI: 10.1016/J.NA.2004.11.003
description: scientific article; zbMATH DE number 2167839