Stability and Hopf bifurcation analysis on a predator–prey model with discrete and distributed delays

A predator–prey model with discrete and distributed delays is investigated, where the discrete delay ττ 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 ττ crosses some critical value. 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.