Incorporating Memory Effects in Population Ecology Using Fractional Derivatives: Stability Perspectives, Bifurcations, and Chaos Control

This study investigates a discrete‐time predator–prey model that includes both prey refuge and memory effects. The research identifies the conditions under which fixed points exist and remain stable. A key focus is placed on analyzing different types of bifurcation—such as period doubling (PD), Neimark–Sacker (NS), and strong resonances (1 : 2, 1 : 3, and 1 : 4)—occurring at the positive fixed point to uncover their ecological significance. Bifurcation theory is applied to study these dynamics, and the theoretical findings are validated through numerical simulations performed with the MATLAB tool MatContM. In addition, a control mechanism is introduced to mitigate severe instabilities within the system. The results show that predation rate is key to ecological balance, while prey refuge has limited impact on stability. The study offers important insights for conserving biodiversity and managing ecosystems.