Chaos and bifurcations of a discretized Holling-II prey-predator model including prey refuge and Allee effect

This study examines the dynamics of a discrete-time predator-prey system incorporating a Holling Type-II functional response, the Allee effect, and a refuge term. Through algebraic analysis, we establish the occurrence of Period-Doubling (PD) and Neimark-Sacker (NS) bifurcations in the positive quadrant, supported by the center manifold theorem and bifurcation theory. Numerical simulations reveal chaotic phenomena, including period-12 orbits, invariant closed curves, and strange attractors, further validated by Maximal Lyapunov exponents. To regulate chaos, we apply the OGY method to stabilize trajectories around an unstable equilibrium. Additionally, we investigate bifurcations in a coupled predator-prey complex network, demonstrating the emergence of chaos beyond a critical coupling threshold. Ecologically, the results indicate that the Allee effect is a key factor influencing predator-prey interactions. A moderate Allee effect contributes to stabilizing both populations, thereby supporting their long-term coexistence and survival within ecosystems. The findings provide an important understanding of how Allee effects, prey refuges, and network configurations shape ecological dynamics, shedding light on factors that contribute to population stability and the emergence of chaotic behavior in predator–prey interactions.