Dynamical behavior for an eco-epidemiological model with discrete and distributed delay

In this paper, the dynamical behavior of an eco-epidemiological model with discrete and distributed delay is studied. Sufficient conditions for the local as-ymptotical stability of the nonnegative equilibria are obtained. We prove that there exists a threshold value of the feedback time delay tau beyond which the positive equilibrium bifurcates towards a periodic solution. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, the direction and the periodic of bifurcating period solutions are derived. Numerical sim-ulations are carried out to explain the mathematical conclusions.