Depensation of Supplementary Food in a System of Interacting Species with Refuge

The dynamics of a predator–prey interacting model involving the occurrence of refuge and Beverton–Holt-like food supplement has its own interest in the realm of ecological scaffold. While the prey species is persuaded by the refuge phenomena, the generalist predator species looks for alternative food source followed by Beverton–Holt-like functional form. The current study investigates a predator–prey refuge model involving a generalized predator in which the nonlinear phenomenon in presence of the hyperbolic functional response is thoroughly explored. The model system reveals its dynamical complexity in terms of bi-stability along with local and global bifurcations of diverse nature like saddle-node, Hopf, generalized Hopf (Bautin), cusp, Bogdanov–Takens and homoclinic bifurcations as well. For the sake of comparison, the present model without any food supplement is also studied so as to ensure the strong impact of prey refuge on temporal behaviour of the solution. The entire parametric space is divided into several regions based on the selection of two system parameters of significance at a time. Attempts are also made to obtain the number of ecologically feasible equilibria in each region together with their dynamical response. All the analytical outcomes are duly validated graphically through numerical simulations in order to substantiate the aptness of the proposed model.