Modeling the Dynamics of a Ratio-Dependent Leslie–Gower Predator–Prey System with Strong Allee Effect

Hainan Xu, Daihai He

Research output: Journal article publicationJournal articleAcademic researchpeer-review

12 Citations (Scopus)


In this paper, the dynamics of a ratio-dependent predator–prey model with strong Allee effect and Holling IV functional response is investigated by using dynamical analysis. The model is shown to have complex dynamical behaviors including subcritical or supercritical Hopf bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension-2, a nilpotent focus or cusp of codimension-2. The codimension-2 Bogdanov–Takens bifurcation point acts as an organizing center for the whole bifurcation set. The coexistence of stable and unstable positive equilibria, homoclinic cycle is also found. Our analysis shows that the ratio-dependent model may collapse suddenly due to certain parameter variation, i.e. the numbers of predator and prey will decrease sharply to zeroes after undergoing a short time of sustained oscillations with small amplitudes. Of particular interest is that the coalescence of saddle-node bifurcation point and Hopf bifurcation point may indicate the occurrence of relaxation oscillations and the critical state of extinction of predator and prey. Numerical simulations and phase portraits including one-parameter bifurcation curve and two-parameter bifurcation curves are given to illustrate the theoretical results.

Original languageEnglish
Article number2250239
Pages (from-to)2250239-1 - 2250239-18
Number of pages18
JournalInternational Journal of Bifurcation and Chaos
Issue number16
Publication statusPublished - 30 Dec 2022


  • Bogdanov–Takens bifurcation
  • Holling IV functional response
  • Hopf bifurcation
  • Ratio-dependent predator–prey model
  • strong Allee effect

ASJC Scopus subject areas

  • Modelling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics


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