Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system

This paper deals with the two-species chemotaxis system $\left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \nabla \cdot (u{\chi _1}(w)\nabla w) + {\mu _1}u(1 - u)}&{{\text{in}}\;\Omega \times (0,\infty ),} \\ {{v_t} = \Delta v - \nabla \cdot (v{\chi _2}(w)\nabla w) + {\mu _2}v(1 - v)}&{{\text{in}}\;\Omega \times (0,\infty ),} \\ {{w_t} = d\Delta w + h(u,v,w)}&{{\text{in}}\;\Omega \times (0,\infty ),} \end{array}} \right.$ 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. Global existence and asymptotic stability of solutions to the above system were established under some conditions [11]. The main purpose of the present paper is to improve smallness conditions for chemotactic effect deriving asymptotic stability and to give the convergence rate in stabilization which cannot be attained in the previous work.