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

• For the weak competition case, if ${\mu }_1,{\mu }_2$ are sufficiently large, then the solution $(u_1,u_2,v_1,v_2)$ converges to $\begin{array}{rl}& \left(\frac{1-a_1}{1-a_1{a}_2},\frac{1-a_2}{1-a_1{a}_2},\frac{{\alpha }_{11}(1-a_1)+{\alpha }_{12}(1-a_2)}{{\lambda }_1(1-a_1{a}_2)},\right.\\ & \left.\frac{{\alpha }_{21}(1-a_1)+{\alpha }_{22}(1-a_2)}{{\lambda }_2(1-a_1{a}_2)}\right)\end{array}$ exponentially as $t\to \mathrm{\infty }$.

• For the strong-weak competition case, if ${\mu }_2$ is sufficiently large, then the solution $(u_1,u_2,v_1,v_2)$ converges to $(0,1,{\alpha }_{12}/{\lambda }_1,{\alpha }_{22}/{\lambda }_2)$ with exponential decay when $a_1>1$, and with algebraic decay when $a_1=1$.

Keywords:

• Chemotaxis system
• Lotka–Volterra
• competitive kinetics
• global boundedness
• asymptotic stability

2010 Mathematics Subject Classifications:

• 35B35
• 35B34
• 35K55
• 92C17

Acknowledgments

This work is supported in part by the Fundamental Research Funds for the Central Universities under grant XDJK2020C054, 106112016CDJXZ238826 and 2019CDJCYJ001, NSFC under grants 11771062 and 11971082, the Postdoctoral Program for Innovative Talent Support of Chongqing, Chongqing Key Laboratory of Analytic Mathematics and Applications.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported in part by the Fundamental Research Funds for the Central Universities [grant numbers XDJK2020C054, 106112016CDJXZ238826 and 2019CDJCYJ001], the National Natural Science Foundation of China (NSFC) [grant numbers 11771062 and 11971082] and the Postdoctoral Program for Innovative Talent Support of Chongqing, Chongqing Key Laboratory of Analytic Mathematics and Applications.