Bifurcation Analysis, Chaos Control, and Fast Approach for the Complex Dynamics of a Discrete-Time Predator-Prey System With a Weak Allee Effect

Abstract

In this paper, we analyze the complex dynamics of a discrete-time, Leslie-type predator-prey system that exhibits a weak Allee effect where the prey population has a mate-finding Allee effect. This discrete mathematical model has been obtained by applying the forward Euler scheme to its continuous-time counterpart. First, stability and bifurcation analyses are performed to explore the stability of positive equilibrium points and to identify critical points where dynamic behavioral transitions occur. Then, center manifold theory and normal form theory are used to classify bifurcations, including Flip bifurcation and Neimark-Sacker bifurcation, revealing the emergence of chaos and periodic orbits in the system by choosing integral step size as a bifurcation parameter. Specifically, numerical examples are presented to illustrate and support the theoretical results, thereby demonstrating the practical application of the theory. Next, we implement chaos control strategies, namely the feedback control method, to control chaotic dynamics. In addition, we apply the FAST method to improve the efficiency of our numerical simulations, allowing a more detailed exploration of the parameter space. Our results contribute to a better understanding of ecological interactions under the influence of the Allee effect and provide valuable insights for biodiversity management in predator-prey systems, especially for endangered populations.