This paper is conced with the existence and stability of steady state solutions for the SKT biological competition model with cross-diffusion.By applying the detailed spectral analysis and in virtue of the bifurcating direction to the limiting system as the cross diffusion rate tends to infinity,it is proved the stability/instability of the nontrivial positive steady states with some special bifurcating structure.Further,the existence and stability/instability of the corresponding nontrivial positive steady states for the original cross-diffusion system are proved by applying perturbation argument.