Mathematical analysis of the global dynamics of an eco-epidemiological model with time delay

In this paper, an eco-epidemiological predator-prey model with time delay representing the gestation period of the predator is investigated. In the model, it is assumed that the predator population suffers a transmissible disease by contact. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free equilibrium, the prey-infected predator equilibrium and the endemic-coexistence equilibrium are established. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global asymptotic stability of the predator-extinction equilibrium, the disease-free equilibrium, the prey-infected predator equilibrium and the endemic-coexistence equilibrium of the model.