Bifurcation of a delayed Gause predator-prey model with Michaelis-Menten type harvesting

Abstract

In this paper, a Gause predator-prey model with gestation delay and Michaelis-Menten type harvesting of prey is proposed and analyzed by considering Holling type III functional response. We first consider the local stability of the interior equilibrium by investigating the corresponding characteristic equation. In succession, we derive some sufficient conditions on the occurrence of the stability switches of the positive steady state by taking the gestation delay as a bifurcation parameter. It is shown that the delay can induce instability and small amplitude oscillations of population densities via Hopf bifurcations. Furthermore, the stability and direction of the Hopf bifurcations are determined by employing the center manifold argument. Finally, computer simulations are performed to illustrate our analytical findings, and the biological implications of our analytical findings are also discussed.

Keywords: Bifurcation; Delay; Michaelis-Menten harvesting; Oscillation; Predator-prey; Stability.