Abstract
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.
Highlights
In the current biomathematical literature, prey-predator interaction has become an exciting subject matter due to its influence on the environment
Several researchers have focused on this concept to present a comprehensive rich dynamics of this phenomenon, including stability analysis of equilibria [22,23,24,25,26,27], Period-Doubling Bifurcation [28], Neimark–Sacker bifurcation [29], and chaos control [30]
A Period-Doubling Bifurcation (PDB) in a discrete dynamical system is a bifurcation where a small perturbation in a parameter value brings about the system moving to a new behaviour with the double period of the original system
Summary
In the current biomathematical literature, prey-predator interaction has become an exciting subject matter due to its influence on the environment. Many researchers [1,2,3,4] have devoted considerable time to explore several perspectives of the dynamical behaviour of this subject matter in ecology and the associated growth of population models [5,6,7]. Several researchers have focused on this concept to present a comprehensive rich dynamics of this phenomenon, including stability analysis of equilibria [22,23,24,25,26,27], Period-Doubling Bifurcation [28], Neimark–Sacker bifurcation [29], and chaos control [30]. A discrete-time prey-predator model with refuge is formulated in the second section. By the biological meaning of the model variables, we only consider the system in the region Ω (x, y): x ≥ 0, y ≥ 0 in the (x, y) − plane
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