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With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of a delayed ratio-dependent predator-prey system with stage structure for predator. The approach involves some new technique of priori estimate. For the system without delay, by constructing a suitable Lyapunov function, some sufficient conditions which guarantee the existence of a unique global attractive positive periodic solution are obtained. Those results have further applications in population dynamics.