In this paper, an eco-epidemiological model with time delay is studied. The local stability of the four equilibria, the existence of stability switches about the predation- free equilibrium and the coexistence equilibrium are discussed. It is found that Hopf bifurcations occur when the delay passes through some critical values. Formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Some numerical simulations are carried out to illustrate the theoretical results.

In this paper, an eco-epidemiological model with time delay is studied. The local stability of the four equilibria, the existence of stability switches about the predation- free equilibrium and the coexistence equilibrium are discussed. It is found that Hopf bifurcations occur when the delay passes through some critical values. Formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Some numerical simulations are carried out to illustrate the theoretical results.