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### Open Access Journals

DCDS-B

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

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.

DCDS

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

## Year of publication

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