The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion

This paper concerns with the existence and stability properties of non-constant positive steady states in one dimensional space for the following competition system with cross diffusion (Equation presented) by Lyapunov-Schmidt method, we obtain the existence and the detailed structure of a type of small nontrivial positive steady states to the shadow system of (1) as ρ12 → ∞ and when d2 is near a₂/π², which also verifies some related existence results obtained earlier in [11] by a different method. Then, based on the detailed structure of the steady states, we further establish the stability of the small nontrivial positive steady states for the shadow system by spectral analysis. Finally, we prove the existence and stability of the corresponding nontrivial positive steady states for the original cross diffusion system (1) when ρ12 is large enough and d2 is near a ₂/π².