# American Institute of Mathematical Sciences

August  2017, 22(6): 2301-2319. doi: 10.3934/dcdsb.2017097

## Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity

 Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received  June 2016 Revised  November 2016 Published  March 2017

This paper deals with the two-species chemotaxis-competition system
 $\left\{ {\begin{array}{*{20}{l}}{{u_t} = {d_1}\Delta u - \nabla \cdot (u{\chi _1}(w)\nabla w) + {\mu _1}u(1 - u - {a_1}v)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\\{{v_t} = {d_2}\Delta v - \nabla \cdot (v{\chi _2}(w)\nabla w) + {\mu _2}v(1 - {a_2}u - v)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\\{{w_t} = {d_3}\Delta w + h(u,v,w)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\end{array}} \right.$
where
 $\Omega$
is a bounded domain in
 $\mathbb{R}^n$
with smooth boundary
 $\partial \Omega$
,
 $n\in \mathbb{N}$
;
 $h$
,
 $\chi_i$
are functions satisfying some conditions. In the case that
 $\chi_i(w)=\chi_i$
, Bai–Winkler [1] proved asymptotic behavior of solutions to the above system under some conditions which roughly mean largeness of
 $\mu_1, \mu_2$
. The main purpose of this paper is to extend the previous method for obtaining asymptotic stability. As a result, the present paper improves the conditions assumed in [1], i.e., the ranges of
 $\mu_1, \mu_2$
are extended.
Citation: Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097
##### References:

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##### References:
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