Summary: We consider a predator-prey system with distributed time delay where the predator dynamics is logistic with the carrying capacity proportional to prey population. In [Chaos Solitons Fractals 37, No. 1, 87--99 (2008; Zbl 1152.34059
); ibid. 42, No. 3, 1474--1484 (2009; Zbl 1198.34149
)], we studied the impact of the discrete time delay on the stability of the model; however, in this paper, we investigate the effect of the distributed delay for the same model. By choosing the delay time $\tau $ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time $\tau $ passes some critical values. Using normal form theory and the center manifold theorem, we establish the direction and the stability of Hopf bifurcation. Some numerical simulations justifying the theoretical analysis are also presented.