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Bifurcations in a Leslie–Gower predator–prey model with strong Allee effects and constant prey refuges

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  • Chen, Fengde
  • Li, Zhong
  • Pan, Qin
  • Zhu, Qun

Abstract

In this paper, we study a Leslie–Gower predator–prey model with strong Allee effects and constant prey refuges. It is shown that the model can undergo a cusp type degenerate Bogdanov–Takens bifurcation of codimension 4, focus and elliptic types degenerate Bogdanov–Takens bifurcations of codimension 3, and degenerate Hopf bifurcation of codimension 3 as the parameters vary. The model can exhibit the coexistence of multiple positive steady states, multiple limit cycles, and homoclinic loops. Our results indicate that a larger prey refuge contributes to the coexistence of both species. Numerical simulations, including three limit cycles, quadristability, a large-amplitude limit cycle enclosing three positive steady states and a homoclinic loop, two large-amplitude limit cycles enclosing three positive steady states, are presented to illustrate the theoretical results.

Suggested Citation

  • Chen, Fengde & Li, Zhong & Pan, Qin & Zhu, Qun, 2025. "Bifurcations in a Leslie–Gower predator–prey model with strong Allee effects and constant prey refuges," Chaos, Solitons & Fractals, Elsevier, vol. 192(C).
    • Handle: RePEc:eee:chsofr:v:192:y:2025:i:c:s0960077925000074
    DOI: 10.1016/j.chaos.2025.115994
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    References listed on IDEAS

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    1. Arancibia–Ibarra, Claudio & Flores, José, 2021. "Dynamics of a Leslie–Gower predator–prey model with Holling type II functional response, Allee effect and a generalist predator," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 1-22.
    2. Shang, Zuchong & Qiao, Yuanhua, 2023. "Multiple bifurcations in a predator–prey system of modified Holling and Leslie type with double Allee effect and nonlinear harvesting," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 745-764.
    3. Zhang, Huisen & Cai, Yongli & Fu, Shengmao & Wang, Weiming, 2019. "Impact of the fear effect in a prey-predator model incorporating a prey refuge," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 328-337.
    4. Liyun Lai & Zhenliang Zhu & Fengde Chen, 2020. "Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect," Mathematics, MDPI, vol. 8(8), pages 1-21, August.
    5. Li, Yajing & He, Mengxin & Li, Zhong, 2022. "Dynamics of a ratio-dependent Leslie–Gower predator–prey model with Allee effect and fear effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 201(C), pages 417-439.
    6. Sarangi, B.P. & Raw, S.N., 2023. "Dynamics of a spatially explicit eco-epidemic model with double Allee effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 241-263.
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