Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity

Abstract

This paper deals with the two-species chemotaxis-competition system \begin{document}$\begin{equation*} \begin{cases} u_t = d_1Δ u - \nabla · (u χ_1(w)\nabla w) +μ_1 u(1-u-a_1 v)&{\rm in} Ω × (0, ∞), v_t = d_2Δ v - \nabla · (v χ_2(w)\nabla w) +μ_2 v(1-a_2u-v)&{\rm in} Ω × (0, ∞), w_t = d_3Δ w + α u + β v - γ w&{\rm in} Ω × (0, ∞), \end{cases} \end{equation*}$ \end{document} where \begin{document}$Ω$\end{document} is a bounded domain in \begin{document}$\mathbb{R}^n$\end{document} with smooth boundary \begin{document}$\partial Ω$\end{document} , \begin{document}$n≥ 2$\end{document} ; \begin{document}$χ_i$\end{document} are functions satisfying some conditions. About this problem, Bai-Winkler [ 1 ] first obtained asymptotic stability in (1) under some conditions in the case that \begin{document}$a_1, a_2∈ (0, 1)$\end{document} . 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 \begin{document}$a_1, a_2∈ (0, 1)$\end{document} .