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

• Published in: Complexity (New York, N.Y.) Vol. 2022; no. 1
• Main Authors: Vinoth, S., Sivasamy, R., Sathiyanathan, K., Unyong, B., Vadivel, R., Gunasekaran, Nallappan
• Format: Journal Article
• Language: English
• Published: Hoboken Hindawi 2022; John Wiley & Sons, Inc; Wiley
• Subjects: Behavior; Bifurcation theory; Canonical forms; Comparative analysis; Continuous time systems; Control methods; Discrete systems; Discrete time systems; Equilibrium; Hybrid control; Mathematical models; Numerical analysis; Parameters; Pole placement; Population; Predator-prey simulation; Predators; Simulation methods; Stability; State feedback; Germany
• ISSN: 1076-2787; 1099-0526
• DOI: 10.1155/2022/6931354

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.