This study examines the discrete prey-predator model in the sense of Caputo fractional derivative by incorporating harvesting on the predator population and immigration on the prey population. We establish the topological categories of the model’s fixed points. We show analytically that a fractional order prey-predator model supports both a Neimark–Sacker (NS) bifurcation and a period-doubling (PD) bifurcation under specific parametric circumstances. Using the central manifold and bifurcation theory, we provide evidence for NS and PD bifurcations. It has been discovered that the parameter values and the initial conditions have a significant influence on the dynamical behavior of the fractional order prey-predator model. Furthermore, two chaos management strategies are applied to eliminate the chaos that objectively exists in the model. Finally, numerical simulations are used to demonstrate complicated and chaotic behavior in order to support our theoretical and analytical discussions.

This research examines the predator–prey model’s discrete-time dynamics regulated by a ratio-dependent Ivlev functional response. Our comprehensive algebraic study demonstrates that the system undergoes both period-doubling bifurcation and Neimark–Sacker bifurcation in the positive quadrant of the phase space. We provide a theoretical framework to understand these bifurcations by employing the center manifold theorem and bifurcation theory. To substantiate our theoretical findings, we conduct numerical simulations that clearly illustrate chaotic phenomena, including phase portraits, period-11 orbits, invariant closed circles, and attractive chaotic sets. In addition, we compute Lyapunov exponents to validate the system’s chaotic characteristics. Moreover, we illustrate the practical implementation of chaos management through state feedback and Ott–Grebogi–Yorke approach to stabilize chaotic trajectories around an unstable equilibrium point. Bifurcations are analyzed in a discrete predator–prey model within a coupled network. Numerical simulations reveal that chaotic behavior arises in complex dynamical networks when the coupling strength parameter reaches a critical threshold. Furthermore, we employed the Euler–Maruyama approach for stochastic simulations to investigate our system under environmental uncertainty, analyzing realistic cases to encompass a variety of environmental conditions. All theoretical results concerning stability, bifurcation, and chaotic transitions in the coupled network are corroborated by numerical simulations.

This study explores a novel two-dimensional discrete-time ratio-dependent Holling-Tanner predator-prey model, incorporating the impact of the Fear effect on the prey population. The study focuses on identifying stationary points and analyzing bifurcations around the positive fixed point, with an emphasis on their biological significance. Our examination of bifurcations at the interior fixed point uncovers a variety of generic bifurcations, including one-parameter bifurcations, period-doubling, and Neimark-Sacker bifurcations. To further understand NS bifurcation, we establish non-degeneracy condition. The system’s bifurcating and fluctuating behavior is managed using Ott—Grebogi—Yorke (OGY) control technique. From an ecological perspective, these findings underscore the substantial role of the Fear effect in shaping predator-prey dynamics. The research is extended to a networked context, where interconnected prey-predator populations demonstrate the influence of coupling strength and network structure on the system’s dynamics. The theoretical results are validated through numerical simulations, which encompass local dynamical classifications, calculations of maximum Lyapunov exponents, phase portrait analyses, and bifurcation diagrams.