Stability, Bifurcation, and Chaos Control in a Discrete-Time Predator–Prey Model With Gompertz Growth and Ivlev Functional Response Under Proportional Harvesting

This paper investigates the complex dynamics of a discrete-time predator–prey system incorporating proportionate prey harvesting. The model is derived from a continuous system using the forward Euler discretization method and extends a previously studied model by introducing a harvesting term. First, the positivity of solutions is established to ensure biological feasibility of the discretized system. We analyze the existence and local stability of biologically feasible fixed points, with particular focus on the interior fixed point. Through rigorous bifurcation analysis, we identify both Neimark–Sacker and period-doubling bifurcations, revealing transitions from stable equilibria to periodic and chaotic behavior. To manage these complex dynamics, we apply feedback control and hybrid control strategies, both of which are shown to effectively suppress bifurcation-induced chaos and stabilize the system. Numerical simulations are provided to validate the theoretical results and illustrate rich dynamical behavior, including quasi-periodic oscillations and strange attractors. Moreover, Codimension 2 bifurcations are identified, including 1:2, 1:3, and 1:4 resonance bifurcations that provide a clear explanation of the transition routes to complex dynamics in the model. The findings emphasize that a moderate level of harvesting can promote coexistence and stability of both prey and predator populations, whereas excessive harvesting may destabilize or collapse the system.