Instability of spiky steady states for S-K-T biological competing model with cross-diffusion

This paper is concerned with the stability/instability of a type of positive spiky steady states for a quasi-linear cross-diffusion system describing two species competition. By detailed spectral analysis, it is proved that the spiky steady states for the related shadow system are linearly unstable when the diffusion rate is small enough and the positive spiky steady states for the original cross-diffusion system are unstable when the cross diffusion rate for one species is large enough and the diffusion rate of another species is small.