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A Novel Discrete‐Time Leslie–Gower Model with the Impact of Allee Effect in Predator Population

Author

Listed:
  • S. Vinoth
  • R. Sivasamy
  • K. Sathiyanathan
  • B. Unyong
  • R. Vadivel
  • Nallappan Gunasekaran

Abstract

The discrete‐time system has more complex and chaotic dynamical behaviors as compared to the continuous‐time system. This paper extends a discrete Leslie–Gower predator‐prey system with the Allee effect in the predator’s population, whose dynamics are analyzed and explored. We have determined the equilibrium points and studied their local stability properties. We find that the system undergoes flip bifurcation and Neimark–Sacker bifurcation around the interior equilibrium point by choosing the Allee parameter as a bifurcation parameter. We discuss the stability and direction of both bifurcations with the help of the normal form theory and center manifold theorem. The flip bifurcation and Neimark–Sacker bifurcation are the most common routes to the chaotic orbit in the discrete system. Moreover, we utilize state feedback, pole placement, and hybrid control methods to control the chaos in the system. The work is complete with the numerical simulations to confirm the analytical findings.

Suggested Citation

  • S. Vinoth & R. Sivasamy & K. Sathiyanathan & B. Unyong & R. Vadivel & Nallappan Gunasekaran, 2022. "A Novel Discrete‐Time Leslie–Gower Model with the Impact of Allee Effect in Predator Population," Complexity, John Wiley & Sons, vol. 2022(1).
    • Handle: RePEc:wly:complx:v:2022:y:2022:i:1:n:6931354
    DOI: 10.1155/2022/6931354
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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. 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. Dhar, Joydip & Singh, Harkaran & Bhatti, Harbax Singh, 2015. "Discrete-time dynamics of a system with crowding effect and predator partially dependent on prey," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 324-335.
    4. Stephen Lynch, 2018. "Dynamical Systems with Applications using Python," Springer Books, Springer, number 978-3-319-78145-7, March.
    5. Liu, Xiaoli & Xiao, Dongmei, 2007. "Complex dynamic behaviors of a discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 80-94.
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    Cited by:

    1. Sarker Md. Sohel Rana & Md. Jasim Uddin, 2023. "Dynamics and Control of a Discrete Predator–Prey Model with Prey Refuge: Holling Type I Functional Response," Mathematical Problems in Engineering, John Wiley & Sons, vol. 2023(1).
    2. R. Lavanya & S. Vinoth & K. Sathiyanathan & Zeric Njitacke Tabekoueng & P. Hammachukiattikul & R. Vadivel, 2022. "Dynamical Behavior of a Delayed Holling Type‐II Predator‐Prey Model with Predator Cannibalism," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).

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