Stability and Local Hopf Bifurcation for a Predator-Prey Model with Delay

A predator-prey system with disease in the predator is investigated, where the discrete delay is regarded as a parameter. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By analyzing the associated characteristic equation, it is found that Hopf bifurcation occurs when crosses some critical values. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions are derived.