Dynamics of a delayed predator–prey interaction incorporating nonlinear prey refuge under the influence of fear effect and additional food

Abstract

The main object of this work is to analyse the impacts of fear effect on the dynamics of predator-prey interaction incorporating non linear prey refuge to control the extinction of predator. We have taken a Holling type II functional response in presence of additional food. The positivity and uniform boundedness of solutions have been discussed. The feasibility conditions of all equilibrium points and their stability behaviours are derived. Next, we have studied the existence of local bifurcation (transcritical bifurcation) of codimension 1. Moreover, impacts of fear effect have been studied analytically and noticed that the predator biomass can not only be decreased at the interior equilibrium by the effect of fear but the system also be stabilized by excluding the existence of periodic solutions through Hopf-bifurcation. In addition, uniform persistence of our proposed system has been discussed analytically. Further, we have analysed the stability behaviours of the interior equilibrium for all combinations of the delay factors (τ and τ ′ ) and observed that the delay parameters may produce oscillating behaviours through a Hopf-bifurcation. Numerical simulations have been illustrated using MATLAB to validate all the analytical results. Numerically, the impact of coef cient of prey refuge on the predator population has also been performed.

Dynamics of a Predator‐Prey Population in the Presence of Resource Subsidy under the Influence of Nonlinear Prey Refuge and Fear Effect

“…Existence Here d [TC] criteria and stable behaviour of all the biologically meaningful equilibria have been discussed. It has to be noted that Hopf bifurcations are exhibited around E 3 (subsidy free) and E * (interior) of system (2) considering k as a bifurcating parameter (see 9,11,and 12). Also, observing Figures 6 and 12, it can be concluded that high levels of fear can stabilize system (2) by excluding the existence of periodic solutions.…”

“…In 2020, Mondal and Samanta [11] analyzed the dynamics of a delayed predator-prey interaction incorporating nonlinear prey refuge function under the influence of fear effect and additional food. Motivated by the work of Mondal and Samanta [11], we have first considered a predator-preysubsidy model with nonlinear prey refuge function where the prey and subsidy occur in the same habitat and they are both consumed by a single generalist predator according to the following differential equations:…”

“…It involves a parameter k denoting the level of fear to represent the antipredator behaviour of the prey. Some research works have already been done on the ecological system under the influence of predation fear felt by prey species [6][7][8][9][10][11][12][13][14][15][16][17]. Moreover, the impact of fear in a two-species predator-prey model with prey refuge was analyzed by many researchers [11,18,19].…”

“…Recently, Mondal and Samanta have studied the dynamics of the predator-prey system with prey refuge dependent on both species (that is, mxy amounts of prey are free from the predator risk, where 0 < 1 − my < 1, m is prey refuge coefficient, x is the biomass of prey population, and y is the biomass of predator population) in the presence of additional food (for details, see [23]). In 2020, Mondal and Samanta [11] have also analyzed the dynamics of predatorprey interaction having nonlinear prey refuge function Φ(x, y) � (mxy/(a + y)) which is the amount of prey that are free from predation, where a is half saturation constant and y is the biomass of predator population.…”

Global Bifurcation Structure of a Predator‐Prey System with a Spatial Degeneracy and B‐D Functional Response

“…It is used to describe the dynamic distribution of population density produced by the interaction of the species in the ecosystem or the surrounding environment. In particular, the interaction of the predator-prey system is one of the basic structures in complex ecosystems, and such models have been widely studied (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]) under the uniform condition of the space, in other words, all coefficients of the model are positive. Because the natural environment of most species is spatially heterogeneous, biologists and mathematicians believe that the inhomogeneity of the spatial environment has a significant impact on the dynamic behavior of the biological population system, which is confirmed by the biologist C. B. Huffaker's biological experiment.…”