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A three stage-structured prey-predator model with digestion delay and density dependent delay for the predator is investigated. The stability of the equilibrium point and the Hopf bifurcation of the system by choosing time delay as a bifurcation parameter in five cases are considered, and the conditions for the positive equilibrium occurring local Hopf bifurcation are given in each case. Numerical results show that delayed system considered has not only periodic oscillation, stability switches but also chaotic oscillation, even unbounded oscillation. Finally, delays induced Hopf bifurcation, stability switches, complicated dynamic behaviors of the system are discussed in detail.