Global asymptotic stability of constant equilibrium point in attraction-repulsion chemotaxis model with logistic source term

Author(s): Abdelhakam Hassan Mohammed1,2, Ali. B. B. Almurad1
1Department of Mathematics and Computer, College of Education, Alsalam University, Alfula, Sudan.
2College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, P.R. China.
Copyright © Abdelhakam Hassan Mohammed, Ali. B. B. Almurad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with nonnegative solutions of the Neumann initial-boundary value problem for an attraction-repulsion chemotaxis model with logistic source term of Eq. (1) in bounded convex domains \(\Omega\subset\mathbb{R}^{n},~ n\geq1\), with smooth boundary. It is shown that if the ratio \(\frac{\mu}{\chi \alpha-\xi \gamma}\) is sufficiently large, then the unique nontrivial spatially homogeneous equilibrium given by \((u_{1},u_{2},u_{3})=(1,~\frac{\alpha}{\beta},~\frac{\gamma}{\eta})\) is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data \((u_{10},u_{20},u_{30})\) such that \(u_{10}\not\equiv0\), the above problem possesses uniquely determined global classical solution \((u_{1},u_{2},u_{3})\) with \((u_{1},u_{2},u_{3})|_{t=0}=(u_{10},u_{20},u_{30})\) which satisfies \(\left\|u_{1}(\cdot,t)-1\right\|_{L^{\infty}(\Omega)}\rightarrow{0},~~
\left\|u_{2}(\cdot,t)-\frac{\alpha}{\beta}\right\|_{L^{\infty}(\Omega)}\rightarrow{0},\left\|u_{3}(\cdot,t)-\frac{\gamma}{\eta}\right\|_{L^{\infty}(\Omega)}\rightarrow{0}\,,\) \(\mathrm{as}~t\rightarrow{\infty}\).

Keywords: Keller-Segel model; Logistic source; Chemotaxis; Attraction-Repulsion; Asymptotic Stability.