Abstract This paper investigates the dynamic behavior analysis on the prey-predator model with ratio-dependent Monod-Haldane response function under the homogeneous Dirichlet boundary conditions, which is used to simulate a class of biological system. Firstly, the sufficient and necessary conditions on existence and non-existence of coexistence states of this model are discussed by comparison principle and fixed point index theory. Secondly, taking a as a main bifurcation parameter, the structure of global bifurcation curve on positive solutions is established by using global bifurcation theorem and properties of principal eigenvalue. Finally, the stability of coexistence states is obtained by the eigenvalue perturbation theory; the multiplicity of coexistence states is investigated when a satisfies some condition by the fixed point index theory.