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A Study of Stability and Bifurcation in a Discretized Predator–Prey Model With Holling Type III Response and Prey Refuge Via Piecewise Constant Argument Method

Author

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  • Faisal Alsharif
  • Rizwan Ahmed
  • Ibrahim Alraddadi
  • Mohammed Alsubhi
  • Muhammad Amer
  • Md. Jasim Uddin

Abstract

This study explores the dynamics of a discrete‐time predator–prey system, incorporating a Holling type III functional response and prey refuge, through the piecewise constant argument method. This method keeps things consistent and prevents negative population values, which is often a drawback of older techniques. We take a closer look at fixed points, exploring their existence and stability. We identify the conditions that lead to period‐doubling and Neimark–Sacker bifurcations, and we back up our findings with numerical simulations. Our findings emphasize how important the predation coefficient is for keeping ecological balance and show that, in this model setup, the refuge for prey has a minimal effect on the stability of the system. These insights help us better understand the relationships between predators and their prey, providing valuable guidance for conserving biodiversity and managing ecosystems.

Suggested Citation

  • Faisal Alsharif & Rizwan Ahmed & Ibrahim Alraddadi & Mohammed Alsubhi & Muhammad Amer & Md. Jasim Uddin, 2025. "A Study of Stability and Bifurcation in a Discretized Predator–Prey Model With Holling Type III Response and Prey Refuge Via Piecewise Constant Argument Method," Complexity, John Wiley & Sons, vol. 2025(1).
    • Handle: RePEc:wly:complx:v:2025:y:2025:i:1:n:4542190
    DOI: 10.1155/cplx/4542190
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    References listed on IDEAS

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    1. Jing, Zhujun & Yang, Jianping, 2006. "Bifurcation and chaos in discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 259-277.
    2. Arancibia-Ibarra, Claudio & Aguirre, Pablo & Flores, José & van Heijster, Peter, 2021. "Bifurcation analysis of a predator-prey model with predator intraspecific interactions and ratio-dependent functional response," Applied Mathematics and Computation, Elsevier, vol. 402(C).
    3. Khajanchi, Subhas, 2017. "Modeling the dynamics of stage-structure predator-prey system with Monod–Haldane type response function," Applied Mathematics and Computation, Elsevier, vol. 302(C), pages 122-143.
    4. Akhtar, S. & Ahmed, R. & Batool, M. & Shah, Nehad Ali & Chung, Jae Dong, 2021. "Stability, bifurcation and chaos control of a discretized Leslie prey-predator model," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    5. Binhao Hong & Chunrui Zhang, 2023. "Neimark–Sacker Bifurcation of a Discrete-Time Predator–Prey Model with Prey Refuge Effect," Mathematics, MDPI, vol. 11(6), pages 1-13, March.
    6. Weijie Lu & Yonghui Xia, 2022. "Multiple Periodicity in a Predator–Prey Model with Prey Refuge," Mathematics, MDPI, vol. 10(3), pages 1-13, January.
    7. Ghosh, Joydev & Sahoo, Banshidhar & Poria, Swarup, 2017. "Prey-predator dynamics with prey refuge providing additional food to predator," Chaos, Solitons & Fractals, Elsevier, vol. 96(C), pages 110-119.
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    Cited by:

    1. Mussa A. Stephano, 2025. "Dynamics of Allee and Refuge Effects in Prey–Predator System With Holling′s Type III Functional Response," Journal of Applied Mathematics, John Wiley & Sons, vol. 2025(1).

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