Global asymptotic stability in a parabolic–elliptic chemotaxis system with competitive kinetics and loop

Abstract

This paper deals with the initial-boundary value problem for the two-species chemotaxis-competition system with two signals $\left\{\begin{array}{l}{\mathrm{\partial }}_t{u}_1=\mathrm{\Delta }{u}_1-{\chi }_{11}\mathrm{\nabla }\cdot \left(u_1\mathrm{\nabla }{v}_1\right)-{\chi }_{12}\mathrm{\nabla }\cdot \left(u_1\mathrm{\nabla }{v}_2\right)+{\mu }_1{u}_1(1-u_1-a_1{u}_2),\\ {\mathrm{\partial }}_t{u}_2=\mathrm{\Delta }{u}_2-{\chi }_{21}\mathrm{\nabla }\cdot \left(u_2\mathrm{\nabla }{v}_1\right)-{\chi }_{22}\mathrm{\nabla }\cdot \left(u_2\mathrm{\nabla }{v}_2\right)+{\mu }_2{u}_2(1-u_2-a_2{u}_1),\\ 0=\mathrm{\Delta }{v}_1-{\lambda }_1{v}_1+{\alpha }_{11}{u}_1+{\alpha }_{12}{u}_2,\\ 0=\mathrm{\Delta }{v}_2-{\lambda }_2{v}_2+{\alpha }_{21}{u}_1+{\alpha }_{22}{u}_2,\end{array}\right.$ under the homogeneous Neumann boundary condition, where $x\in \mathrm{\Omega },t>0$, ${\chi }_{ij}>0$, ${\mu }_i>0$, $a_i>0$, ${\alpha }_{ij}>0$, ${\lambda }_i>0$ $(i,j=1,2)$, and $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 2)$ is a smooth bounded domain. If ${\chi }_{11}/{\mu }_1$, ${\chi }_{12}/{\mu }_1$, ${\chi }_{21}/{\mu }_2$ and ${\chi }_{22}/{\mu }_2$ are sufficiently small, then the system possesses a globally bounded classical solution for any suitably regular initial data $u_{10},u_{20}$. Furthermore, by constructing some appropriate functionals, it is shown that