Complex dynamics and chaos control in a fractional-order discrete prey-predator model incorporating an Ivlev functional response and Allee effect

This paper investigates predator–prey interactions to explore the complex dynamics arising from the Allee effect on prey populations, incorporating an Ivlev functional response. The analysis focuses on the occurrence and stability of equilibrium points. The study demonstrated that the system undergoes period-doubling (PD) and Neimark–Sacker (NS) bifurcations at the positive fixed point, utilizing center-manifold analysis and bifurcation theory. The chaotic behaviors of the system were identified using bifurcation diagrams and maximum Lyapunov exponent graphs. The system’s bifurcating and fluctuating behavior can be regulated using OGY control methods. Bifurcations in a discrete predator–prey model within a coupled network were examined. Numerical simulations demonstrated that chaotic behavior emerges in complex dynamical networks once the coupling strength parameter reaches a critical value. Furthermore, the Euler–Maruyama method was utilized for stochastic simulations to explore the system under environmental uncertainty, taking into account diverse environmental scenarios. All theoretical findings related to stability, bifurcations, and chaotic transitions in the coupled network were validated through numerical simulations. This study highlights the value of discrete models in capturing the full range of potential dynamics in ecological systems, as they can uncover complex phenomena — such as bifurcations and chaos — that may remain hidden in continuous-time frameworks. Such insights are crucial for advancing our understanding of ecological processes and for accurately modeling real-world species interactions.