Bifurcation, chaos, multistability, and pattern formation of a discrete predator–prey model with Allee effect

This study constructs a discrete-time model featuring the Holling type II functional response to investigate how the Allee effect drives the appearance of both periodic and chaotic behaviors. Through rigorous algebraic analysis, it is demonstrated that Neimark–Sacker (NS) bifurcations occur when varying the Allee parameter n and the prey’s intrinsic growth rate r within the positive quadrant. By applying the center manifold theorem and core results from bifurcation theory, a solid theoretical framework is established to track these qualitative changes. Extensive numerical simulations validate that the system undergoes a subcritical NS bifurcation as n and r change, marking a shift from stable equilibria into chaotic oscillations with increasing Allee intensity. Moreover, the model reveals the coexistence of periodic attractors of periods 11 and 34 over certain (n, r) pairs, underscoring how initial population levels critically shape long-term outcomes. A parallel analysis contrasts this Holling II formulation with its Holling type I analog. By varying the Allee threshold n, it becomes apparent that the type I system collapses at a lower Allee value than the type II system. This disparity suggests that predation saturation—a hallmark of Holling II—confers greater resilience under strong Allee effects. Finally, a comparative evaluation of both functional responses’ advantages and limitations highlights the need for adaptive management strategies to preserve biodiversity and maintain ecological stability.