The positive solutions are discussed for a class of the Holling-Tanner prey-predator ecological model with diffusion and cross-diffusion. The biological implication of cross-diffusion means that the prey species exercise a self-defense mechanism to protect themselves from the attack of the predator.By means of maximum principle and Harnack inequality, the prior estimate to the positive solutions of the model is given. Furthermore,by using the integral property, the non-existence of the non-constant positive solutions is considered, and it is proved that the model has no non-constant positive solution when the diffusion coefficient d1 and d2 are larger than the special positive constants and cross-diffusion coefficient d3 is bounded. Lastly the degree theory is utilized for discussing the existence of the non-constant positive solutions, therefore, we will obtain that the model has at least one non-constant positive solution if the algebraic multiplicity of the positive eigenvalue of the linearized operator of the model is odd and cross-diffusion coefficient d3 is not less than some given positive constant.

cross-diffusion;Holling-Tanner prey-predator model;positive solution;existence