Theoretical and Numerical Bifurcation Analysis of a Discrete Predator–Prey System of Ricker Type with Weak Allee Effect

Abstract

This study aims to explore the complexity of a discrete-time predator–prey system with a weak Allee effect. The existence and stability of fixed points, as well as period-doubling and Neimark–Sacker bifurcations, are all investigated. The system’s bifurcating and fluctuating behavior is controlled using feedback and hybrid control techniques. Additionally, numerical simulations are performed as evidence to support theoretical results. From an ecological perspective, these findings suggest that the Allee effect plays a pivotal role in shaping predator–prey dynamics. The moderate Allee effect fosters stability in both predator and prey populations, promoting coexistence and persistence within ecosystems. However, the disproportionate impact on predator populations underscores predators’ vulnerability to changes in prey behavior and availability, highlighting the importance of considering indirect effects in ecological modeling and conservation efforts.