Dynamics of a Predator–Prey Model with Holling Type II Functional Response Incorporating a Prey Refuge Depending on Both the Species

Abstract

We propose a mathematical model for prey–predator interactions allowing prey refuge. A prey–predator model is considered in the present investigation with the inclusion of Holling type-II response function incorporating a prey refuge depending on both prey and predator species. We have analyzed the system for different interesting dynamical behaviors, such as, persistent, permanent, uniform boundedness, existence, feasibility of equilibria and their stability. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The system exhibits Hopf-bifurcation around the unique interior equilibrium point of the system. The explicit formula for determining the stability, direction and periodicity of bifurcating periodic solutions are also derived with the use of both the normal form and the center manifold theory. The theoretical findings of this study are substantially validated by enough numerical simulations. The ecological implications of the obtained results are discussed as well.

Dynamical Analysis of a Predator-Prey Model Incorporating Predator Cannibalism and Refuge

“…The parameter k 1 represents half-saturation constant of predation. Holling type II functional response is more realistic than Holling type I because the rate of the prey predation is saturated [23]. This is in accordance with the real conditions, that it is impossible to predators to eat the prey continuously.…”

Predator-prey dynamics with Allee effect on predator species subject to intra-specific competition and nonlinear prey refuge

“…In this study, we have considered x, y as the prey and predator species respectively and δxy as the amount of nonlinear prey refuge (cf. [17,[19][20][21]) admissible for t 0 with 0 x(1 − δy) x. The factor y y+h comes into predator growth function to decrease the predator growth rate because, if a predator eats more, it will be less likely to die from starvation or from the consequences of weakness due to hunger; and it is positive for any value of functional response, because a predator is highly unlikely to live forever.…”

“…Biological description and dimension of the parameters used in the model (2.3) have been chosen from[19] and[21]: V stands for volume of the species and T for time.…”

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

“…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. In this paper, a Lotka-Volttera food web model with a prey refuge that consists of prey and predator is proposed and studied.…”