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

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.