Coexistence of Limit Cycles and Homoclinic Loops in a Leslie-Gower Model with Allee Effect

• Zhifei Guo ^[1] ; Dongping He ^[2]
  1. [1] Hefei University

     Hefei University

     China

     
  2. [2] Guangxi Normal University

     Guangxi Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 2, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • It was shown that a multiplicative Allee effect induces the rise of either a limit cycle or a homoclinic loop in a Leslie-Gower predator–prey model. In this work, we continue to explore its complicated dynamics caused by the Allee effect, i.e., the coexistence of multiple limit cycles and the homoclinic loops. Applying the resultant elimination method, we determine that the maximum multiplicity of a weak focus is four, and verify that four limit cycles bifurcating from degenerate Hopf bifurcation can coexist.

    Furthermore, we prove that a stable limit cycle can coexist with a homoclinic loop, which arise from a degenerate Bogdanov-Takens bifurcation of codimension three.

    Numerical simulations are also given to illustrate the two coexistence phenomena.

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