Stability and Hopf Bifurcation of a Delayed Prey-Predator Model with Disease in the Predator

Dealing with the epidemiological prey-predator is very important for us to understand the dynamical characteristics of population models. The existing literature has shown that disease introduction into the predator group can destabilize the established prey-predator communities. In this paper, we establish a new delayed SIS epidemiological prey-predator model with the assumptions that the disease is transmitted among the predator species only and different type of predators have different functional responses, viz. the infected predator consumes the prey according to Holling type-II functional response and the susceptible predator consumes the prey following the law of mass action. The positivity of solutions, the existence of various equilibrium points, the stability and bifurcation at those equilibrium points are investigated at length. Using the incubation period as bifurcation parameter, it is observed that a Hopf bifurcation may occur around the equilibrium points when the parameter passes through some critical values. We also discuss the direction and stability of the Hopf bifurcation around the interior equilibrium point. Simulations are arranged to show the correctness and effectiveness of these theoretical results.