Stability transitions and Neimark-Sacker bifurcation in a predation model with fear induced delay: A discrete-time analysis

Abstract

Discrete-time dynamical systems frequently exhibit complex and realistic behaviors, particularly in biological contexts. In this work, a continuous-time predator-prey ecosystem is discretized using the Euler discretization method, and the variation in system stability with different parameter values is investigated. Analysis of the fear-induced system addresses the persistence of populations and their mutual influences. Fear restricts prey foraging activities, thereby limiting predator-prey interactions and eventually leading to increased intraspecific competition among predators. Fear significantly influences the system behavior when there is no delay, and it can also be utilized as a chaos control parameter to stabilize the system. For the delayed case, stability switching and the existence of Neimark-Sacker bifurcation are examined. The findings indicate that fear can destabilize an otherwise stable system and conversely, stabilize an unstable one. The primary result showed that a low level of dread exerts a considerable psychological influence on the prey population, thus benefiting the predator population. However, as fear is increased further, its effect plateaus, and additional changes in population dynamics are minimal. To verify the theoretical conclusion and offer more understanding of the dynamics of the system, numerical simulations are carried out.