DCDS-B
The stability of bifurcating steady states of several classes of chemotaxis systems
Qian Xu
This paper concerns with the stability of bifurcating steady states obtained in [13] of several chemotaxis systems. By spectral analysis and the principle of the linearized stability, we prove that the bifurcating steady states are stable when the parameters satisfy some certain conditions.
keywords: Stability spectral analysis bifurcating steady states chemotaxis systems. expansion
DCDS
The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion
Yaping Wu Qian Xu
This paper is concerned with the existence of large positive spiky steady states for S-K-T competition systems with cross-diffusion. Firstly by detailed integral and perturbation estimates, the existence and detailed fast-slow structure of a class of spiky steady states are obtained for the corresponding shadow system, which also verify and extend some existence results on spiky steady states obtained in [10] by different method of proof. Further by applying special perturbation method, we prove the existence of large positive spiky steady states for the original competition systems with large cross-diffusion rate.
keywords: shadow system singular perturbation. existence spiky steady state cross diffusion
DCDS
The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion
Wei-Ming Ni Yaping Wu Qian Xu
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 $$\left\{ \begin{array}{ll} u_t=[(d_{1}+\rho_{12}v)u]_{xx}+u(a_{1}-b_{1}u-c_{1}v),&x\in(0,1), t>0, \\ v_t= d_{2}v_{xx}+v(a_{2}-b_{2}u-c_{2}v),& x\in(0,1),t>0,                    (1) \\ u_{x}=v_{x}=0, &x=0,1, t>0. \end{array}\right. $$ First, 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 $\rho_{12}\to \infty$ and when $d_2$ is near $a_2/\pi^2$, 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 $\rho_{12}$ is large enough and $d_2$ is near $a_2/\pi^2$.
keywords: steady states cross diffusion spectral analysis shadow system. Existence stability

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