Fear effect in a predator-prey model with additional food, prey refuge and harvesting on super predator

Abstract

The shape of the interaction between three species: prey, predator, and super predator has been considered in this work as a new predator-prey model. In this article, the fear of predator in prey species has been introduced. Also, we have considered that the predation of both prey and predator occurs according to the Beddington-DeAngelis type functional response. The predator is thought to be acting as a shelter for the super predator species. It is anticipated that a constant amount of increased food will be delivered to super predators due to their dread of predators and their refuge behavior, and that predators will benefit from this excess food. The non-negativity and boundedness of solutions of the prosed model have been examined. The system's local and global stability at the interior equilibrium points has been explored. The existence condition of Hopf bifurcation of the system with respect to fear f of predator has been studied. From the analysis of the model, it is found that the fear f of predator may be responsible for the system's stability. It's also been discovered that providing extra food to the super predator may help to keep the system in check. Further, it is found that the refuge of predator to the super predator has the ability to stabilize the predator-prey system. Finally, some intriguing numerical simulation findings were provided in order to study the model's dynamics.