An Analysis of Stability and Bifurcation in a Delayed Leslie–Gower Predator–Prey Model with Linear Harvesting and the Fear Effect

Abstract

This study examines a delayed predator-prey paradigm wherein a discrete time delay influences the evolution of the predator. The model incorporates nonlinear interaction variables, harvesting, and the fear effect. This study's originality is in the simultaneous examination of prey harvesting, fear-induced behavioral changes, and predator reproductive delays within a cohesive Leslie–Gower framework an area not previously investigated in the literature. The model effectively represents complex ecological dynamics by integrating delay-induced feedback in predator reproduction with a rational functional response. Our model offers novel insights into the interplay of prey harvesting, fear-induced behavioral modifications, and temporal delay factors typically examined in isolation on predator-prey dynamics in a hitherto underexplored area of ecological modelling. By demonstrating the boundedness and invariance of solutions, we ensure the biological relevance of the system. Equilibrium points are computed for both the non-delayed and delayed versions of the model, and their local stability is examined. The impact of delay on the stability of coexistence equilibrium is investigated by a comprehensive Hopf bifurcation examination. We ascertain the period, stability, and orientation of bifurcating periodic orbits by the use of centre manifold and normal form theory. The theoretical findings are corroborated by numerical simulations, encompassing bifurcation diagrams for harvesting rate and terror impact without delay. These findings underscore the significance of behavioral responses and harvesting strategies in affecting population dynamics in the presence of temporal delays.