Qualitative Analysis of a Leslie–Gower Discrete Prey–Predator System With Allee Effect Under Deterministic and Stochastic Environment

This work examines a two-dimensional discrete Leslie-type predator–prey system in which the prey dynamics are subject to the Allee effect. The primary objective is to determine equilibrium states and examine bifurcation phenomena in the neighborhood of the positive fixed point, highlighting their ecological significance. The analysis of the interior equilibrium confirms that the model undergoes several essential dynamical changes, including one-parameter bifurcations together with period-doubling and Neimark–Sacker bifurcation mechanisms. To clarify the structure of these transitions, the associated nondegeneracy requirements are established and the critical normal form coefficients are explicitly evaluated. The chaotic responses and bifurcation-generated oscillations are subsequently suppressed through the combined use of state–feedback control and the Ott–Grebogi–Yorke (OGY) strategy. Biologically, the results emphasize the central role of the Allee effect in regulating predator–prey behavior. More precisely, a moderate Allee influence tends to stabilize the interacting populations, promote their coexistence, and strengthen the long-term viability of the ecosystem. Moreover, the bifurcation properties of the discrete predator–prey system are also explored in a coupled network framework. The numerical results show that when the coupling intensity surpasses a critical threshold, the network dynamics become chaotic and display intricate, irregular oscillatory patterns. In addition, stochastic simulations based on the Euler–Maruyama scheme were performed to examine system behavior under environmental variability, allowing the model to reflect diverse ecological conditions.