Unveiling Complexity: A Discrete-Time Prey–Predator Model with Immigration Effects

This paper explores the qualitative dynamics of a discrete-time prey–predator model incorporating immigration effects. In this model, immigration is assumed to affect only a fraction of both prey and predator populations. We systematically analyze the equilibrium points of the system and assess their stability. Through the application of bifurcation theory, we demonstrate that the system undergoes a Neimark–Sacker bifurcation, leading to complex dynamical behavior. To address the emergence of chaos, a hybrid control strategy is implemented to stabilize the system. Numerical simulations reveal that immigration can intensify the system’s complex dynamics, suggesting that even small levels of immigration can have a profound impact on the behavior of prey–predator interactions.