Exploring chaotic dynamics of a discrete-time prey-predator model with ratio-dependent functional response subject to weak Allee and slow-fast effect
TL;DR
The authors' bifurcation-theoretic analysis of the coexistence equilibrium reveals key local transitions, including period-doubling and Neimark–Sacker bifurcations, which highlight the significant influence of the Allee effect on the predator–prey dynamics.
Abstract
This research explores the dynamic behavior of a predator–prey system in which the functional response depends on the predator-to-prey ratio and the prey population is subject to a weak Allee effect. The main objective is to determine the equilibrium points and examine the bifurcations occurring near the positive equilibrium, with particular attention to their ecological implications. Our bifurcation-theoretic analysis of the coexistence equilibrium reveals key local transitions, including period-doubling and Neimark–Sacker bifurcations. To fully characterize these dynamical changes, we establish the corresponding nondegeneracy conditions and apply the Ott–Grebogi–Yorke (OGY) control strategy to suppress irregular and oscillatory behavior in the system. From an ecological standpoint, the results highlight the significant influence of the Allee effect on the predator–prey dynamics: a moderate Allee effect promotes stability, fostering coexistence and the long-term persistence of both species. The system’s dynamics are characterized using bifurcation diagrams, phase portraits, maximum Lyapunov exponent computations, and visualizations of the implemented control strategy.