Asymptotics and periodic dynamics in a negative chemotaxis system with cell lethality

TL;DR

This paper analyzes a negative chemotaxis PDE system with cell death, showing solutions tend to a steady state or periodic behavior depending on the external chemical supply function, with convergence results linking PDE and ODE models.

Contribution

It establishes conditions under which solutions of a chemotaxis PDE system converge to solutions of an associated ODE system, including cases with periodic external forcing.

Findings

Solutions approach the ODE system's solutions as time progresses.

Convergence occurs when the external function f(x,t) stabilizes spatially.

Periodic external forcing can induce periodic solutions in the PDE system.

Abstract

This work studies the following system of parabolic partial differential equations \begin{equation*} \begin{cases} \displaystyle \frac{\partial u}{\partial t} = D\Delta u + \chi \nabla \cdot(u \nabla v) + ru(1-u) - u v, \quad & x \in \Omega, ~t > 0, \\ \displaystyle \frac{\partial v}{\partial t} = \Delta v + a u -v+ f(x,t), \quad & x \in \Omega, ~t > 0, \end{cases} \end{equation*} modeling the negative chemotaxis interactions between a biological species and a lethal chemical substance that is supplied according to the known function $f(x,t)$. \\\\ It is shown that if $f$ converges to a spatially homogeneous function $\tilde{f}$ in a certain sense, then the solution $(u,v)$ satisfies $$ ||u-\tilde{u}||_{L^2(\Omega)} + ||v-\tilde{v}||_{L^2(\Omega)} \to 0 \quad \text{as } t \to \infty, $$ where $(\tilde{u},\tilde{v})$ is the solution to the associated ODE system \begin{equation*} \begin{cases} \displaystyle \frac{d \tilde{u}}{dt~} = r \tilde{u} (1 - \tilde{u}) - \tilde{u}\tilde{v}, \quad & t>0,\\ \displaystyle \frac{d \tilde{v}}{dt~} = a\tilde{u} - \tilde{v} + \tilde{f},\quad & t>0. \end{cases} \end{equation*} Some final remarks are given for the case in which $\tilde{f}$ is a time periodic function, and under which hypotheses do $(\tilde{u},\tilde{v})$ inherit this periodicity.