# American Institute of Mathematical Sciences

February  2020, 13(2): 269-278. doi: 10.3934/dcdss.2020015

## Improvement of conditions for 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  May 2017 Published  January 2019

This paper deals with the two-species chemotaxis-competition system
 $\begin{equation*} \begin{cases} u_t = d_1Δ u - \nabla · (u χ_1(w)\nabla w) +μ_1 u(1-u-a_1 v)&{\rm in} \ Ω × (0, ∞), \\ v_t = d_2Δ v - \nabla · (v χ_2(w)\nabla w) +μ_2 v(1-a_2u-v)&{\rm in} \ Ω × (0, ∞), \\ w_t = d_3Δ w + α u + β v - γ w&{\rm in} \ Ω × (0, ∞), \end{cases} \end{equation*}$
where
 $Ω$
is a bounded domain in
 $\mathbb{R}^n$
with smooth boundary
 $\partial Ω$
,
 $n≥ 2$
;
 $χ_i$
are functions satisfying some conditions. About this problem, Bai-Winkler [1] first obtained asymptotic stability in (1) under some conditions in the case that
 $a_1, a_2∈ (0, 1)$
. Recently, the conditions assumed in [1] were improved ([6]); however, there is a gap between the conditions assumed in [1] and [6]. The purpose of this work is to improve the conditions assumed in the previous works for asymptotic behavior in the case that
 $a_1, a_2∈ (0, 1)$
.
Citation: Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 269-278. doi: 10.3934/dcdss.2020015
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