In this paper, we discuss a predator-prey model with two different delays in the chemostat. The consumption terms take the Monod type response functions. By analyzing the characteristic equations of delay differential equations, we obtain the sufficient conditions of stability of boundary equilibria. About the effect of delay on stability of the positive equilibrium, using stability switch theory and bifurcation theory, we give the sufficient conditions of appearance of stability switches and Hopf bifurcations in the delay equations as a result of changing the delay. Finally, numerical simulations are carried out to illustrate the theoretical results.