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Some Bifurcations of Codimensions 1 and 2 in a Discrete Predator–Prey Model with Non-Linear Harvesting

Author

Listed:
  • Ming Liu

    (School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China)

  • Linyi Ma

    (School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China)

  • Dongpo Hu

    (School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China)

Abstract

This paper delves into the dynamics of a discrete-time predator–prey system. Initially, it presents the existence and stability conditions of the fixed points. Subsequently, by employing the center manifold theorem and bifurcation theory, the conditions for the occurrence of four types of codimension 1 bifurcations (transcritical bifurcation, fold bifurcation, flip bifurcation, and Neimark–Sacker bifurcation) are examined. Then, through several variable substitutions and the introduction of new parameters, the conditions for the existence of codimension 2 bifurcations (fold–flip bifurcation, 1:2 and 1:4 strong resonances) are derived. Finally, some numerical analyses of two-parameter planes are provided. The two-parameter plane plots showcase interesting dynamical behaviors of the discrete system as the integral step size and other parameters vary. These results unveil much richer dynamics of the discrete-time model in comparison to the continuous model.

Suggested Citation

  • Ming Liu & Linyi Ma & Dongpo Hu, 2024. "Some Bifurcations of Codimensions 1 and 2 in a Discrete Predator–Prey Model with Non-Linear Harvesting," Mathematics, MDPI, vol. 12(18), pages 1-44, September.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:18:p:2872-:d:1478579
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    References listed on IDEAS

    as
    1. Wenjie Qin & Sanyi Tang & Robert A. Cheke, 2014. "The Effects of Resource Limitation on a Predator-Prey Model with Control Measures as Nonlinear Pulses," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-13, March.
    2. Hunki Baek, 2014. "Spatiotemporal Dynamics of a Predator-Prey System with Linear Harvesting Rate," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-9, April.
    3. Reynolds-Hogland, Melissa J. & Hogland, John S. & Mitchell, Michael S., 2008. "Evaluating intercepts from demographic models to understand resource limitation and resource thresholds," Ecological Modelling, Elsevier, vol. 211(3), pages 424-432.
    4. Dongpo Hu & Xuexue Liu & Kun Li & Ming Liu & Xiao Yu, 2023. "Codimension-Two Bifurcations of a Simplified Discrete-Time SIR Model with Nonlinear Incidence and Recovery Rates," Mathematics, MDPI, vol. 11(19), pages 1-24, September.
    5. Wang, Yong & Zhou, Xu & Jiang, Weihua, 2023. "Bifurcations in a diffusive predator–prey system with linear harvesting," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
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