Modelling and analysis of a delayed predator–prey model with disease in the predator

In this paper, we study a predator–prey model with a transmissible disease spreading in the predator population and a time delay representing the gestation period of the predator. 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 coexistence 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-extinction equilibrium and the disease-free equilibrium and the global attractiveness of the coexistence equilibrium of the system, respectively. Numerical simulations are carried out to support the theoretical analysis.