The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross-diffusion

This paper is concerned with the existence and stability of traveling waves with transition layers for a quasi-linear competition system with cross diffusion,which was first proposed by Shegesada,Kawasaki and Teramoto.When one of the random diffusion rates is small and the cross-diffusion rate is not small,by the geometric singular perturbation method,the existence of traveling waves with transition layers is obtained.Further,by the detailed spectral analysis and topological index method,the traveling waves with transition layers are proved to be locally exponentially stable with shift.