Complex dynamics of a discrete prey-predator model with complex network and stochastic modeling incorporating a ratio-dependent Ivlev functional response.

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.