# Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics

Source: [Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics](<https://www.sota2.com/research/paper/68fa5ba61e209131ce9ca359>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-27T05:31:24.286Z

Authors: Halil Ibrahim Kurt, Wenxian Shen

Published: 2024-04-04T02:12:36.000Z

## Abstract

The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, \\begin{equation} \\begin{cases} u\_t=\\Delta u-\\chi\_1 \\nabla\\cdot (\\frac{u}{w} \\nabla w)+u(a\_1-b\_1u-c\_1v) ,\\quad &x\\in \\Omega\\cr v\_t=\\Delta v-\\chi\_2 \\nabla\\cdot (\\frac{v}{w} \\nabla w)+v(a\_2-b\_2v-c\_2u),\\quad &x\\in \\Omega\\cr 0=\\Delta w-\\mu w +\\nu u+ \\lambda v,\\quad &x\\in \\Omega \\cr \\frac{\\partial u}{\\partial n}=\\frac{\\partial v}{\\partial n}=\\frac{\\partial w}{\\partial n}=0,\\quad &x\\in\\partial\\Omega, \\end{cases} \\end{equation} where $\\Omega \\subset \\mathbb{R}^N$ is a bounded smooth domain, and $\\chi\_i,a\_i, b\_i, c\_i$ ($i=1,2$) and $\\mu,\\, \\nu, \\, \\lambda$ are positive constants. In \[25\], we proved that for any given nonnegative initial data $u\_0,v\_0\\in C^0(\\bar\\Omega)$ with $u\_0+v\_0\\not \\equiv 0$, (0.1) has a unique globally defined classical solution provided that $\\min\\{a\_1,a\_2\\}$ is large relative to $\\chi\_1,\\chi\_2$, and $u\_0+v\_0$ is not small in the case that $(\\chi\_1-\\chi\_2)^2\\le \\max\\{4\\chi\_1,4\\chi\_2\\}$ and $u\_0+v\_0$ is neither small nor big in the case that $(\\chi\_1-\\chi\_2)^2\>\\max\\{4\\chi\_1,4\\chi\_2\\}$. In this paper, we proved that (0.1) has a unique positive constant solution $(u^\*,v^\*,w^\*)$, where $$ u^\*=\\frac{a\_1b\_2-c\_1a\_2}{b\_1b\_2-c\_1c\_2},\\quad v^\*=\\frac{b\_1a\_2-a\_1c\_2}{b\_1b\_2-c\_1c\_2}, \\quad w^\*=\\frac{\\nu}{\\mu}u^\*+\\frac{\\lambda}{\\mu} v^\*. $$ We obtain some explicit conditions on $\\chi\_1,\\chi\_2$ which ensure that the positive constant solution $(u^\*,v^\*,w^\*)$ is globally stable in the sense that for any given nonnegative initial data $u\_0,v\_0\\in C^0(\\bar\\Omega)$ with $u\_0\\not \\equiv 0$ and $v\_0\\not \\equiv 0$, $$ \\lim\_{t\\to\\infty}\\Big(\\\|u(t,\\cdot;u\_0,v\_0)-u^\*\\\|\_\\infty +\\\|v(t,\\cdot;u\_0,v\_0)-v^\*\\\|\_\\infty+\\\|w(t,\\cdot;u\_0,v\_0)-w^\*\\\|\_\\infty\\Big)=0. $$

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2404.03158>)

[PDF](<https://arxiv.org/pdf/2404.03158.pdf>)