A systematic study of autonomous and nonautonomous predator-prey models for the combined effects of fear, refuge, cooperation and harvesting.

In the present study, we investigate the roles of fear, refuge and hunting cooperation on the dynamics of a predator-prey system, where the predator population is subject to harvesting at a nonlinear rate. We also focus on the effects of seasonal forcing by letting some of the model parameters to vary with time. We rigorously analyze the autonomous and nonautonomous models mathematically as well as numerically. Our simulation results show that the birth rate of prey and the fear of predators causing decline in it, and harvesting of predators first destabilize and then stabilize the system around the coexistence of prey and predator; if the birth rate of prey is very low, both prey and predator populations extinct from the ecosystem, and for a range of this parameter, only the prey population survive. The fear of predators responsible for increase in the intraspecific competition among the prey species and the refuge behavior of prey have tendency to stabilize the system, whereas the cooperative behavior of predators during the hunting time destroys stability in the ecosystem. Numerical investigations of the seasonally forced model showcase the appearances of periodic solution, higher periodic solutions, bursting patterns and chaotic dynamics.