Complex Dynamics of a Fractional‐Order Population Dynamic Model Incorporating Ratio‐Dependent Ivlev Functional Response and Harvesting on the Predator Population

This research investigates the discrete‐time dynamics of a predator–prey model governed by a ratio‐dependent Ivlev functional response incorporating harvesting on the predator population. Through a detailed algebraic analysis, we show that the system experiences both period‐doubling (PD) and Neimark–Sacker (NS) bifurcations within the positive quadrant of the phase space. Using the center manifold theorem and bifurcation theory, we provide a theoretical understanding of these bifurcations. To support our theoretical results, numerical simulations are performed, revealing chaotic behavior such as phase portraits, period‐12 orbits, invariant closed curves, and attractor chaotic sets. Additionally, we compute the Lyapunov exponents to confirm the chaotic nature of the system. The results demonstrate that parameter values play a crucial role in shaping the model's dynamic behavior. Finally, we showcase the practical application of chaos control by employing state feedback and the OGY method to stabilize chaotic trajectories around an unstable equilibrium point. This study deepens our understanding of complex predator–prey dynamics and highlights the potential for managing chaos in ecological systems. Further, bifurcations are explored in a discrete predator–prey model within a coupled network, with numerical simulations showing that chaotic behavior emerges in such networks when the coupling strength reaches a critical threshold. Furthermore, we implemented the Euler–Maruyama method for stochastic simulations to analyze our system in the context of environmental uncertainties. We examined various cases to investigate different environmental scenarios. All theoretical results on stability, bifurcations, and chaotic transitions in the coupled network are validated through numerical simulations.