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This research analyzes the dynamics of a discrete-time prey–predator model to understand stability and bifurcation behaviors influenced by the Allee effect.

May 28, 2026

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

Key Points

This research analyzes the dynamics of a discrete-time prey–predator model to understand stability and bifurcation behaviors influenced by the Allee effect.
Constructed a discrete-time model from a continuous-time predator-prey system using forward Euler discretization.
Derived conditions for equilibrium stability and explored bifurcation mechanisms such as saddle-node and Neimark-Sacker.
Conducted numerical simulations including bifurcation diagrams and Lyapunov exponent plots to illustrate results.
Identified conditions under which coexistence equilibrium loses stability, leading to oscillatory and chaotic behavior.
Demonstrated that the Allee effect plays a significant role in extinction scenarios and species coexistence.
Presenting a variety of dynamical behaviors shows the complex interaction of predation and population thresholds.

Abstract

This paper deals with a discrete-time prey–predator model incorporating a ratiodependent 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.

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