Global analysis in a predator-prey system with nonmonotonic functional response

Shigui Ruan, Dongmei Xiao

Research output: Contribution to journalArticle

332 Scopus citations

Abstract

A predator-prey system with nonmonotonic functional response is considered. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. The bifurcation analysis of the model depending on all parameters indicates that it exhibits numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the supercritical and subcritical Hopf bifurcations, and the homoclinic bifurcation. It is shown that there are different parameter values for which the model has a limit cycle or a homoclinic loop, or exhibits the so-called paradox of enrichment phenomenon. Moreover, a limit cycle cannot coexist with a homoclinic loop for all parameters. In the generic case, the model has the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation) but for some specific parameter values it has a multiple focus of multiplicity at least 2.

Original languageEnglish (US)
Pages (from-to)1445-1472
Number of pages28
JournalSIAM Journal on Applied Mathematics
Volume61
Issue number4
StatePublished - Dec 1 2000
Externally publishedYes

Keywords

  • Bogdanov-Takens bifurcation
  • Global analysis
  • Homoclinic orbit
  • Hopf bifurcation
  • Limit cycle
  • Paradox of enrichment
  • Predator-prey system

ASJC Scopus subject areas

  • Applied Mathematics

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