Mathematical Study on a Dynamical Predator-Prey Model with Constant Prey Harvesting and Proportional Harvesting in Predator

Abstract

A dynamical predator-prey model with constant prey harvesting, proportional harvesting in predator has been studied. The square root functional response also has been incorporated in the system to describe the prey herd behaviour, assuming the average handling time is zero. The existence and the local stability of equilibria of the system have been discussed. It is examined that, two types of bifurcation occur in the system. The two types of bifurcations have been analyzed, and it has been found by analyzing the saddle-node bifurcation that, there is a maximum sustainable yield. It is observed that if harvesting rate is greater than the maximum sustainable yield, the prey population abolish from the system and then extinction of the predator population happen. But if harvesting rate is lesser than the maximum sustainable yield, the extinction of the prey population can not be possible. By analyzing the Hopf bifurcation, it is obtained that, there exists an unstable limit cycle around the interior equilibrium point. Several numerical simulations are performed to check the results.