Dynamical Study of a Predator-Prey Interaction Incorporating Fear Effect with Saturated Fear Cost and Prey Refuge

Ecological balance is significantly impacted by predator-prey interactions, and particularly, predation risk is well-known to play a crucial role in influencing ecological and evolutionary processes as it brings a reduction in the predation rate which in turn, may help in conservation and management of the ecosystem. The impact of the fear effect on prey with a saturated fear cost and prey refuge in the dynamics of the predator-prey model, incorporating Holling type II foraging strategy, is investigated here. The conditions that allow all biologically feasible steady states to exist, with their local and global stability, uniform persistence, and bifurcation analysis, are thoroughly discussed. We observe that an increasing level of saturated fear cost in prey may destabilize the system and periodic oscillations appear via Hopf bifurcation. Moreover, periodic oscillations can be prevented by increasing the level of prey refuge. However, an increase in the level of fear stabilizes the system by eliminating oscillatory solutions and also decreases the predator density at a coexistence steady state. The extinction of predators is only possible if the level of prey refuge is extremely high. A multiple Hopf bifurcation is observed by choosing handling time as a bifurcation parameter. Also, we can observe that an increasing level of handling time destabilizes the system via multiple stability switches.