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Bifurcations of a predator–prey system of Holling and Leslie types

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  • Li, Yilong
  • Xiao, Dongmei

Abstract

A predator–prey model with simplified Holling type-IV functional response and Leslie type predator’s numerical response is considered. It is shown that the model has two non-hyperbolic positive equilibria for some values of parameters, one is a cusp of co-dimension 2 and the other is a multiple focus of multiplicity one. When parameters vary in a small neighborhood of the values of parameters, the model undergoes the Bogdanov–Takens bifurcation and the subcritical Hopf bifurcation in two small neighborhoods of these two equilibria, respectively. And it is further shown that by choosing different values of parameters the model can have a stable limit cycle enclosing two equilibria, or a unstable limit cycle enclosing a hyperbolic equilibrium, or two limit cycles enclosing a hyperbolic equilibrium. However, the model never has two limit cycles enclosing a hyperbolic equilibrium each for all values of parameters. Some computer simulation are presented to illustrate the conclusions.

Suggested Citation

  • Li, Yilong & Xiao, Dongmei, 2007. "Bifurcations of a predator–prey system of Holling and Leslie types," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 606-620.
    • Handle: RePEc:eee:chsofr:v:34:y:2007:i:2:p:606-620
    DOI: 10.1016/j.chaos.2006.03.068
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    Cited by:

    1. Li, Xinxin & Yu, Hengguo & Dai, Chuanjun & Ma, Zengling & Wang, Qi & Zhao, Min, 2021. "Bifurcation analysis of a new aquatic ecological model with aggregation effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 75-96.
    2. Hu, Guang-Ping & Li, Wan-Tong & Yan, Xiang-Ping, 2009. "Hopf bifurcations in a predator–prey system with multiple delays," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1273-1285.
    3. Yin, Hongwei & Zhou, Jiaxing & Xiao, Xiaoyong & Wen, Xiaoqing, 2014. "Analysis of a diffusive Leslie–Gower predator–prey model with nonmonotonic functional response," Chaos, Solitons & Fractals, Elsevier, vol. 65(C), pages 51-61.
    4. 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.
    5. Chen, Mengxin & Wu, Ranchao, 2023. "Steady states and spatiotemporal evolution of a diffusive predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    6. Wang, Shufan & Tang, Haopeng & Ma, Zhihui, 2021. "Hopf bifurcation of a multiple-delayed predator–prey system with habitat complexity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 180(C), pages 1-23.
    7. Xu, Chaoqun, 2020. "Probabilistic mechanisms of the noise-induced oscillatory transitions in a Leslie type predator-prey model," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    8. Jiao, Xubin & Li, Xiaodi & Yang, Youping, 2022. "Dynamics and bifurcations of a Filippov Leslie-Gower predator-prey model with group defense and time delay," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).

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