Boundedness and 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}$\left\{ {\begin{array}{*{20}{l}}{{u_t} = {d_1}\Delta u - \nabla \cdot (u{\chi _1}(w)\nabla w) + {\mu _1}u(1 - u - {a_1}v)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\\{{v_t} = {d_2}\Delta v - \nabla \cdot (v{\chi _2}(w)\nabla w) + {\mu _2}v(1 - {a_2}u - v)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\\{{w_t} = {d_3}\Delta w + h(u,v,w)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\end{array}} \right.$ \end{document} where \begin{document}$\Omega$\end{document} is a bounded domain in \begin{document}$\mathbb{R}^n$\end{document} with smooth boundary \begin{document}$\partial \Omega$\end{document} , \begin{document}$n\in \mathbb{N}$\end{document} ; \begin{document}$h$\end{document} , \begin{document}$\chi_i$\end{document} are functions satisfying some conditions. In the case that \begin{document}$\chi_i(w)=\chi_i$\end{document} , Bai–Winkler [ 1 ] proved asymptotic behavior of solutions to the above system under some conditions which roughly mean largeness of \begin{document}$\mu_1, \mu_2$\end{document} . 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 \begin{document}$\mu_1, \mu_2$\end{document} are extended.