Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response

We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response. The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response. Here, a positive solution corresponds to a coexistence state of the model. Firstly, we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane. In addition, we present the uniqueness of positive solutions in one dimension case. Secondly, we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system, and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.