Stability analysis on a type of steady state for the SKT competition model with large cross diffusion

This paper is concerned with the existence and the stability of steady state solutions for the SKT biological competition model with cross-diffusion. By applying the higher order expansion and some detailed spectral analysis to the limiting system as the cross diffusion rate tends to infinite, it is proved that the nontrivial positive steady states with some special bifurcating structure are unstable. Further, the existence and the instability of the corresponding nontrivial positive steady states for the original cross-diffusion system are proved by applying perturbation argument. Finally, we show the global existence of solutions for the limiting system and present some numerical simulation on the large time behavior of the solution with more general initial data.