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Bifurcation analysis orchestrating isola and mushroom bifurcations of limit cycles in a predator–prey system

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  • Xu, Yancong
  • Yang, Yue
  • Rong, Libin
  • Aguirre, Pablo

Abstract

This paper investigates a predator–prey system with an additive Allee effect and a type IV functional response using a dynamical system approach. By means of a bifurcation analysis, we find the existence of codimension two and three Bogdanov–Takens bifurcation, codimension-three generalized Hopf bifurcation, and codimension-two cusp of limit cycles. In order to portray the dynamics of limit cycles, we first introduce the definitions of mushroom and isola bifurcations of limit cycles in a population dynamics model. Additionally, by means of geometrical illustrations, we explain the topological transition between the isola and mushroom bifurcations limit cycles. The model predicts that extinction of both populations may only occur if the Allee effect is strong. However, long term coexistence is possible in both weak and strong Allee regimes indicating that predation has a balancing role in the interaction dynamics. Nonetheless, a weak Allee effect can result in complex dynamics as well, including the presence of isolas, mushrooms and cusps of limit cycles.

Suggested Citation

  • Xu, Yancong & Yang, Yue & Rong, Libin & Aguirre, Pablo, 2025. "Bifurcation analysis orchestrating isola and mushroom bifurcations of limit cycles in a predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 200(P1).
    • Handle: RePEc:eee:chsofr:v:200:y:2025:i:p1:s0960077925008938
    DOI: 10.1016/j.chaos.2025.116880
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