Bifurcation analysis in a predator–prey system with time delay

In this paper, a predator–prey system with a discrete delay and a distributed delay is investigated. We first consider the stability of the positive equilibrium and the existence of local Hopf bifurcations. In succession, using the normal form theory and center manifold argument, we derive the explicit formulas determining stability, direction and other properties of bifurcating periodic solutions. Finally, several numerical simulations for supporting the theoretical analysis are also given.