Bistability, Bifurcation, and Chaos in a Discrete Leslie–Gower Model with Predation-Driven Allee Effect

In this study, we investigate the dynamical behavior of a discrete-time Leslie–Gower predator–prey model incorporating a predation-driven Allee effect in the prey population. The presence of the Allee term introduces nonlinear feedback that significantly alters system stability and long-term dynamics. Through rigorous analytical and numerical approaches, we explore the existence and stability of equilibria, along with bifurcations that govern transitions between different dynamical regimes. Our results demonstrate that the system undergoes rich dynamical phenomena, including flip and Neimark–Sacker bifurcations, leading to periodic oscillations and quasi-periodic solutions. Numerical simulations further reveal that a gradual increase in prey growth rate can trigger the onset of chaos, whereas predation-driven Allee effects may suppress chaotic oscillations. However, these Allee effects also enlarge the basin of attraction of the prey extinction steady state, implying an increased likelihood of prey extinction. Moreover, the interplay between the Allee threshold and intrinsic growth rate induces complex behaviors such as bistability, where the system may converge to distinct attractors depending on initial conditions. These findings provide deeper insights into predator–prey interactions and emphasize the ecological significance of incorporating Allee effects in discrete models.