Bifurcation of Codimension 3 in a Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response

Jicai Huang, Xiaojing Xia, Xinan Zhang, Shigui Ruan

Research output: Contribution to journalArticle

14 Scopus citations

Abstract

It was shown in [Li & Xiao, 2007] that in a predator-prey model of Leslie type with simplified Holling type IV functional response some complex bifurcations can occur simultaneously for some values of parameters, such as codimension 1 subcritical Hopf bifurcation and codimension 2 Bogdanov-Takens bifurcation. In this paper, we show that for the same model there exists a unique degenerate positive equilibrium which is a degenerate Bogdanov-Takens singularity (focus case) of codimension 3 for other values of parameters. We prove that the model exhibits degenerate focus type Bogdanov-Takens bifurcation of codimension 3 around the unique degenerate positive equilibrium. Numerical simulations, including the coexistence of three hyperbolic positive equilibria, two limit cycles, bistability states (one stable equilibrium and one stable limit cycle, or two stable equilibria), tristability states (two stable equilibria and one stable limit cycle), a stable limit cycle enclosing a homoclinic loop, a homoclinic loop enclosing an unstable limit cycle, or a stable limit cycle enclosing three unstable hyperbolic positive equilibria for various parameter values, confirm the theoretical results.

Original languageEnglish (US)
Article number1650034
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Volume26
Issue number2
DOIs
StatePublished - Feb 1 2016

Keywords

  • degenerate focus type Bogdanov-Takens bifurcation of codimension 3
  • Predator-prey model of Leslie type
  • simplified Holling type IV functional response

ASJC Scopus subject areas

  • Applied Mathematics
  • General
  • Engineering(all)
  • Modeling and Simulation

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