Dynamics and bifurcation analysis of a discrete-time prey–predator model with ratio-dependent Holling type III response and strong Allee effect

Abstract

This paper deals with a discrete-time prey–predator model incorporating a ratio-dependent Holling type III functional response and a strong Allee effect in the prey population. The model is obtained from an underlying continuous-time predator–prey system by means of the forward Euler discretization, yielding a two-dimensional nonlinear map. We investigate the existence of equilibrium points and derive conditions for their local stability. We then analyze the main bifurcation mechanisms of the model and show that the coexistence equilibrium may lose stability through saddle-node, period-doubling and Neimark–Sacker bifurcations, giving rise to oscillatory and chaotic dynamics. Particular attention is paid to the role of the Allee threshold in shaping extinction and coexistence scenarios. Numerical simulations, including bifurcation diagrams, maximum Lyapunov exponent plots and phase portraits, are presented to support the analytical results and to illustrate the wide spectrum of dynamical behaviors generated by the model. The results show that the combined effects of ratio-dependent predation, predator interference and a strong Allee threshold substantially enrich the dynamics of the system.