By means of spectral analysis and bifurcation theory, the coexistence state of the predator-prey model between two species with non-monotonic functional response under Robin boundary condition is investi- gated. The existence of the local bifurcation solutions from semi-trivial solutions to this system is obtained through the use of c as a bifurcation parametersand the local bifurcation can be extended to global bifurcation. Moreover,the stability of the local bifurcation solutions is presented based on the methods of linear eigenvalue perturbation theory.