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A delayed predator-prey system with a constant rate immigration is considered. Local and global stability of the equilibria are studied, a fixed point bifurcation appears near the boundary equilibrium and Hopf bifurcation occurs near the positive equilibrium when the time delay passes some critical values. We also show the existence of the global Hopf bifurcation, and the properties of the fixed point bifurcation and the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem.