An autonomous and nonautonomous predator–prey model with fear, refuge, and nonlinear harvesting: Backward, Bogdanov–Takens, transcritical bifurcations, and optimal control

Abstract

In recent research, fear, refuge, and harvesting have been highlighted, but their combined impact must also be explored. We investigate the combined effects of these three ecologically important factors in the context of a prey–predator system using Holling type II as an interaction term. By using the parametric conditions, we can determine the existence of biologically feasible equilibria with their local and global stability. We investigate transcritical, Saddle‐node, Hopf, backward, Bogdanov–Takens bifurcations about different equilibria theoretically and numerically. We observe that the system is stable for lower and higher values of catchability effect and harvesting parameters. And the system shows stable behavior for a high value of fear and refuge parameters. Our model is extended by considering fear, refuge, and harvesting as time‐dependent parameters. Nonautonomous systems exhibit periodic solutions when the corresponding autonomous system is stable. Further, the nonautonomous system shows complex dynamics such as higher periods, chaos and bursting patterns whenever the associated autonomous system goes through limit cycle oscillations.