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Multiple bifurcations and managing chaos: A discretized ratio-dependent Holling–Tanner predator–prey model with Allee effect in prey

Author

Listed:
  • Uddin, Md. Jasim
  • Boora, Savita
  • Rana, Sarker Md. Sohel
  • Malik, Pradeep

Abstract

This work introduces a ratio-dependent Holling–Tanner predator–prey model with the Allee effect in prey and then discretizes the introduced model through the Euler forward scheme. A brief discussion is held on the stability analysis for several fixed points in the discretized model. Several types of bifurcations, including codimension one and two bifurcations, are demonstrated in this study. Codimension-1 bifurcation, which covers Neimark–Sacker and flip bifurcations, and codimension-2 bifurcations, which include strong resonance 1:2, 1:3, and 1:4 at a positive fixed point. Various critical states under non-degeneracy conditions are computed using the critical normal form coefficient approach for each bifurcation. The model displays complex dynamical behaviours, like quasi-periodic orbits and chaotic sets. Additionally, the system’s chaos was managed by the development of control mechanisms, such as the OGY methodology. It has been established that bifurcation and chaos can be stabilized under certain circumstances. A thorough numerical simulation further supports our analytical findings, which include stability regions, bifurcation curves in 2D & 3D, phase plots, and the maximal Lyapunov exponent, etc.

Suggested Citation

  • Uddin, Md. Jasim & Boora, Savita & Rana, Sarker Md. Sohel & Malik, Pradeep, 2026. "Multiple bifurcations and managing chaos: A discretized ratio-dependent Holling–Tanner predator–prey model with Allee effect in prey," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 243(C), pages 95-120.
    • Handle: RePEc:eee:matcom:v:243:y:2026:i:c:p:95-120
    DOI: 10.1016/j.matcom.2025.11.024
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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. 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.
    3. Md. Jasim Uddin & Sarker Md. Sohel Rana & Hiroki Sayama, 2024. "Qualitative Analysis of the Discretization of a Continuous Fractional Order Prey-Predator Model with the Effects of Harvesting and Immigration in the Population," Complexity, Hindawi, vol. 2024, pages 1-27, February.
    4. Panday, Pijush & Samanta, Sudip & Pal, Nikhil & Chattopadhyay, Joydev, 2020. "Delay induced multiple stability switch and chaos in a predator–prey model with fear effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 134-158.
    5. Uddin, Md. Jasim & Rana, Sarker Md. Sohel & Işık, Seval & Kangalgil, Figen, 2023. "On the qualitative study of a discrete fractional order prey–predator model with the effects of harvesting on predator population," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    6. Lajmiri, Z. & Khoshsiar Ghaziani, R. & Orak, Iman, 2018. "Bifurcation and stability analysis of a ratio-dependent predator-prey model with predator harvesting rate," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 193-200.
    7. 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.
    8. Abdul Qadeer Khan & Syeda Noor-ul-Huda Naqvi & Shaimaa A. A. Ahmed & Waleed A. I. El-Morsi & Carlos Aguilar-Ibanez, 2024. "Chaos Control, Codimension-One and Codimension-Two 1 : 2 Strong Resonance Bifurcation Analysis of a Predator-Prey Model with Holling Types I and III Functional Responses," Complexity, Hindawi, vol. 2024, pages 1-30, September.
    9. R. Khoshsiar Ghaziani & W. Govaerts & C. Sonck, 2011. "Codimension-Two Bifurcations of Fixed Points in a Class of Discrete Prey-Predator Systems," Discrete Dynamics in Nature and Society, Hindawi, vol. 2011, pages 1-27, June.
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