Can adaptive prey refuge facilitate species coexistence in Bazykin's prey-predator model?

Bazykin's prey-predator system with constant and adaptive prey refuge is investigated in this paper. We examine Bazykin's resource consumer system with exponential growth rate and by employing constant prey refuge we demonstrate that refuge does promote species coexistence. The incorporation of constant prey refuge expands the stability zone for the interior equilibrium. Furthermore, the bifurcation diagram with reference to prey refuge (u(r)) shows how u(r) influences the system's behavior from unstable to periodic stability and then to equilibrium stability. Next, we provide a Bazykin's model with adaptive prey refuge and develop a fitness function for the prey population using refuge as a strategy and in order to obtain the prey's optimal response to the environment we determine evolutionary stable strategies (ESS). Our model consists of more than one ESS, thus we employ the best response dynamics for the prey strategy. Our analysis showcases that adaptive refuge used by the prey population promotes the coexistence of prey- predator dynamics. Our theoretical analysis is supported by extensive numerical simulations. Bifurcation diagrams with reference to the two most crucial parameters, namely, delta(2) (intraspecies competition rate among predators) and tau (the rate at which populations adapt to their environment), are included in the numerical analysis. Species cohabitation along a limit cycle or at an equilibrium is discovered to be dependent on the pace of strategy dynamics and the competition amongst predator species.