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 coefficient of prey refuge on the predator population has also been performed.