Dynamics in a parabolic-elliptic two-species population competition model with cross-diffusion for one species

This work deals with a simplified parabolic-elliptic population competition model with cross-diffusion for one species. We first claim that this model has at least a non-trivial global smooth solution, and then we detect explicit parameter ranges under which the coexistence steady state and the two semi-trivial steady states are globally and asymptotically stable.