Coexistence states for a modified Leslie-Gower type predator-prey model with diffusion

This paper is concerned with a modified Leslie-Gower predator-prey model with general functional response under homogeneous Robin boundary conditions. We establish the existence of coexistence states by the fixed index theory on positive cones. As an example, we apply the obtained results to this model with Holling-type II functional response. Our results show that the intrinsic growth rates and the principle eigenvalues of the corresponding elliptic problems with respect to the Robin boundary conditions play more important roles than other parameters for the existence of positive solutions.