Existence of hopf bifurcation in a delay partial dependent predator-prey model with allelopathic effect

In this paper, the dynamics of a partial dependent predator-prey model with allelopathic effect and delay is studied. The model has four equilibrium points, i.e., the extinction of prey point, the extinction of predator point, the extinction of both predator and prey point, and the coexistent equilibrium. It is shown that the stability properties of the first three equilibrium points are not affected by the time delay; i.e., the extinction of predator point and the extinction of both predator and prey point are unstable, while the extinction of prey point is conditionally stable. However, the coexistent equilibrium may exhibit a Hopf bifurcation driven by time delay.