Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley–Martin Functional Response Incorporating Proportional Prey Refuge

Abstract

The paper investigates the dynamical behaviors of a two-species discrete predator-prey system with Crowley–Martin functional response incorporating prey refuge proportional to prey density. The existence of equilibrium points, stability of three fixed points, period-doubling bifurcation, Neimark–Sacker bifurcation, Marottos chaos, and Control Chaos are analyzed for the discrete-time domain. The time graphs, phase portraits, and bifurcation diagrams are obtained for different parameters of the model. Numerical simulations and graphics show that the discrete model exhibits rich dynamics, which also present that the system is a chaotic and complex one. This paper attempts to present a feedback control method which can stabilize chaotic orbits at an unstable equilibrium point.

The Dynamics of a Food Web System: Role of a Prey Refuge Depending on Both Species

“…Ghosh et al [14] studied the effects of extra food for predator with prey refuge. Santra et al [15] investigated the dynamical actions with Crowley-Martin functional response-associated prey refuge. Finally, Molla et al [16] suggested a model for Holling type-II prey-predator interactions in a prey refuge.…”

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

“…where the functions f 1 and g 1 denote the terms in model (9) in variables (x n , y n ) with the order at least two.…”

“…Nature offers some amount of protection to some prey populations in the form of refuge dimensions. In some instances, refugia give a prolonged prey-predator interaction [8,9] to reduce the probability of extinction due to predation activity. is phenomenon has been investigated extensively by many researchers in the discrete domain of refuse concepts [10][11][12].…”

Compensatory and overcompensatory dynamics in prey–predator systems exposed to harvest

“…Moreover, the interplay between the species may lead to dynamics of stunning complexity. There is a vast literature regarding such models and models which also include age or stage structure, see [15][16][17][18][19][20][21][22][23][24][25], and more recently, cf [26][27][28][29] and references therein, the question of how to control and stabilize chaotic behaviour in prey-predator systems has been addressed. The dynamics found in such systems strongly depends on compensatory or overcompensatory recruitment.…”