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Dynamical transitions and chaos control in a discrete predator–prey model with Gompertz growth and Holling type III response

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  • Almatrafi, M.B.
  • Berkal, Messaoud
  • Ahmed, Rizwan

Abstract

This study investigates the dynamics of a discrete predator–prey model derived from a continuous system using the forward Euler method. The model features Gompertz growth for the prey and a Holling type III functional response for the predator. We analyze the existence and local stability of equilibria and identify conditions under which the system exhibits period-doubling and Neimark–Sacker bifurcations, marking the onset of oscillatory and complex dynamics. To detect chaos, we apply the 0−1 chaos test, confirming irregular behavior in specific parameter regimes. Numerical simulations, including bifurcation diagrams, Lyapunov exponents, phase portraits, and time series plots, demonstrate the model’s rich dynamical behavior. Two control strategies, hybrid control and state feedback control, are implemented to control bifurcation and chaos. Our findings highlight how nonlinear responses and discrete-time dynamics can generate unpredictable population patterns and provide tools for stabilizing ecological systems.

Suggested Citation

  • Almatrafi, M.B. & Berkal, Messaoud & Ahmed, Rizwan, 2025. "Dynamical transitions and chaos control in a discrete predator–prey model with Gompertz growth and Holling type III response," Chaos, Solitons & Fractals, Elsevier, vol. 200(P2).
    • Handle: RePEc:eee:chsofr:v:200:y:2025:i:p2:s0960077925010847
    DOI: 10.1016/j.chaos.2025.117071
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    References listed on IDEAS

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    1. Rizwan Ahmed & Muhammad Rafaqat & Imran Siddique & Mohammad Asif Arefin & A. E. Matouk, 2023. "Complex Dynamics and Chaos Control of a Discrete-Time Predator-Prey Model," Discrete Dynamics in Nature and Society, Hindawi, vol. 2023, pages 1-20, July.
    2. Çelik, Canan & Duman, Oktay, 2009. "Allee effect in a discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1956-1962.
    3. Pati, N.C. & Layek, G.C. & Pal, Nikhil, 2020. "Bifurcations and organized structures in a predator-prey model with hunting cooperation," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    4. Mokni, Karima & Ch-Chaoui, Mohamed & Mondal, Bapin & Ghosh, Uttam, 2024. "Rich dynamics of a discrete two dimensional predator–prey model using the NSFD scheme," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 992-1018.
    5. 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).
    6. 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).
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