Stability and Bifurcation Analysis in a Predator–Prey System with Constant Harvesting and Prey Group Defense

Abstract

In this paper, we study a predator–prey system that the prey population gathers in herds to defend its predator and both are harvested by constant rate. The defensive strategy of the gathered prey makes the individuals at the border of the herd mostly suffer from the attacks of the predators. This behavior can be described by a modified Holling-type II functional response in mathematics. Notably, we consider harvesting under two cases: prey harvesting only and predator harvesting only. We investigate the existence of equilibria for both cases, and then find there exists the maximum sustainable yield for two cases to guarantee predator and prey to coexist. Moreover, both species can coexist under some conditions and initial values through investigation of stability of the interior equilibrium in the given system. These results demonstrate that, when hunting the prey or predator for economic interest, harvesting rate must be chosen at a suitable value (not merely less than the maximum sustainable yield) to maintain the coexistence of the predator and prey as well as ecological balance. Finally, we analyze the saddle-node bifurcation and Hopf bifurcation, and determine the direction of Hopf bifurcation by calculating the first Lyapunov number for both cases. In particular, Bogdanov–Takens bifurcation occurs only in the given system with predator harvesting.