Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistence

TL;DR

This paper investigates a two-species chemotaxis-competition system with singular sensitivity, establishing conditions for global existence, boundedness, and persistence of solutions, which advances understanding of complex biological interactions modeled mathematically.

Contribution

It is the first to analyze a two-species chemotaxis-competition system with singular sensitivity and Lotka-Volterra kinetics, proving global existence and persistence under certain conditions.

Findings

Global classical solutions exist under certain parameter conditions.

Solutions are bounded and persist over time.

The system exhibits stable long-term behavior with bounded population densities.

Abstract

This paper is concerned with 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. This is the first work on two-species chemotaxis-competition system with singular sensitivity and Lotka-Volterra competitive kinetics. Among others, we prove 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 $(u(t,x;u_0,v_0),v(t,x;u_0,v_0),w(t,x;u_0,v_0))$ with $u(0,x;u_0,v_0)=u_0(x)$ and $v(0,x;u_0,v_0)=v_0(x)$ provided that $\min\{a_1,a_2\}$ is large relative to $\chi_1,\chi_2$ and $u_0+v_0$ is not small. Moreover, under the same condition, we prove that \begin{equation*} \limsup_{t\to\infty} \|u(t,\cdot;u_0,v_0)+v(t,\cdot;u_0,v_0)\|_\infty\le M^*, \end{equation*} and \begin{equation*} \liminf_{t\to\infty} \inf_{x\in\Omega}(u(t,x,u_0,v_0)+v(t,x;u_0,v_0))\ge m^*, \end{equation*} for some positive constants $M^*,m^*$ independent of $u_0,v_0$, the latter is referred to as combined pointwise persistence.