Dynamics on Effect of Prey Refuge Proportional to Predator in Discrete-Time Prey-Predator Model

Prey-predator models with refuge effect have great importance in the context of ecology. Constant refuge and refuge proportional to prey are the most popular concepts of refuge in the existing literature. Now, there are new different types of refuge concepts attracting researchers. This study considers a refuge concept proportional to the predator due to the fear induced by predators. When predators increase, fears also increase and that is why prey refuges also increase. Here, we examine the influence of prey refuge proportional to predator effect in a discrete prey-predator interaction with the Holling type II functional response model. Is this refuge stabilizing or destabilizing the system? That is the central question of this study. The existence and stability of fixed points, Period-Doubling Bifurcation, Neimark–Sacker Bifurcation, the influence of prey refuge, and chaos are analyzed. This work provides the bifurcation diagrams and Lyapunov exponents to analyze the refuge parameter of the model. The proposed discrete model indicates rich dynamics as the effect of prey refuge through numerical simulations.