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<p>In this paper, a Leslie–Gower predator–prey model with simplified Holling type IV functional response and Allee effect on predator is studied. We discuss the existence and stability of equilibria, and show that the system has at most three positive equilibria. Under sufficient conditions, the system can have a weak focus of order 3, a cusp of codimension 3, a nilpotent focus or an elliptic equilibrium of codimension 3, or a nilpotent elliptic equilibrium of codimension at least 4. Also, we prove that the degenerate Hopf bifurcation of codimension 3 occurs at the anti-saddle, cusp type degenerate Bogdanov–Takens bifurcation of codimension 3 occurs at the double equilibrium, and focus or elliptic type degenerate Bogdanov–Takens bifurcation of codimension 3 occurs at the triple equilibrium. Finally, some numerical simulations show that the system can exhibit two limit cycles, a semi-stable limit cycle or a limit cycle containing a homoclinic loop. Our results imply that the predator and prey can arrive a stable state of coexistence if Allee effect is small, or the predator will be extinct if Allee effect is large.</p>

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Bifurcation Analysis of a Leslie–Gower Predator–Prey Model with Allee Effect on Predator and Simplified Holling Type IV Functional Response

  • 期刊论文

  • 发表日期：

    2025年5月16日

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  • Mengxin Zhang,
  • Zhong Li,
  • Fengde Chen,
  • Lijuan Chen

摘要

In this paper, a Leslie–Gower predator–prey model with simplified Holling type IV functional response and Allee effect on predator is studied. We discuss the existence and stability of equilibria, and show that the system has at most three positive equilibria. Under sufficient conditions, the system can have a weak focus of order 3, a cusp of codimension 3, a nilpotent focus or an elliptic equilibrium of codimension 3, or a nilpotent elliptic equilibrium of codimension at least 4. Also, we prove that the degenerate Hopf bifurcation of codimension 3 occurs at the anti-saddle, cusp type degenerate Bogdanov–Takens bifurcation of codimension 3 occurs at the double equilibrium, and focus or elliptic type degenerate Bogdanov–Takens bifurcation of codimension 3 occurs at the triple equilibrium. Finally, some numerical simulations show that the system can exhibit two limit cycles, a semi-stable limit cycle or a limit cycle containing a homoclinic loop. Our results imply that the predator and prey can arrive a stable state of coexistence if Allee effect is small, or the predator will be extinct if Allee effect is large.