Study Of An Eco-epidemiological Model With Disease In The Predator

Recently, epidemiological models have received much attention from scientists. These epidemic models were formulated to describe the spread of the disease. In the natural world, species does not exist alone. While species spreads the disease, it also competes with the other species for space or food, or is predated by other species. Therefore, it is more of bio-logical significance to consider the effect of interacting species when we study the dynamical behaviors of epidemiological models.In this paper, we study a predator-prey models with a transmissible disease spread-ing in the predator population and a time delay representing the gestation period of the predator. Only the susceptible predators have ability to capture prey with Holling type-II response function, and the infected predator are unable to catch prey due to a high infection. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free equilibrium and the co-existence equilibrium are established, respectively. By means of Lyapunov functionals and LaSalle’s invariance principle, sufficient conditions are derived for the global stability of the predator-extinct equilibrium, the disease-free equilibrium and the coexistence equilibrium of the system, respectively.