Asymptotics and periodic dynamics in a negative chemotaxis system with cell lethality
Federico Herrero-Hervás, Mihaela Negreanu
TL;DR
The paper investigates a parabolic Keller–Segel-type system with negative chemotaxis, cell lethality, and an external source $f(x,t)$, proving that when $f$ converges to a spatially homogeneous $\tilde{f}(t)$ in a specified sense, the PDE solution $(u,v)$ converges in $L^2(\Omega)$ to the solution $(\tilde{u},\tilde{v})$ of the associated ODE system. The analysis combines global existence and boundedness results with a robust set of a priori estimates, including a mass lower bound, a decomposition of the chemical equation, and Lyapunov-type functionals, to establish time-integrable fluctuations and ultimately $L^2$-convergence. The main result extends prior work by handling the fully coupled parabolic system (with $a>0$) and providing a rigorous link to the reduced ODE dynamics under time-varying forcing. The work also discusses the periodic forcing scenario, showing that periodicity can be inherited by the ODE limit under certain conditions (notably when $a=0$), and it highlights open questions for the general periodically forced, strongly coupled case, with implications for understanding regime shifts and pattern formation under seasonal or cyclic external inputs.
Abstract
This work studies the following system of parabolic partial differential equations \begin{equation*} \begin{cases} \displaystyle \frac{\partial u}{\partial t} = DΔu + χ\nabla \cdot(u \nabla v) + ru(1-u) - u v, \quad & x \in Ω, ~t > 0, \\ \displaystyle \frac{\partial v}{\partial t} = Δv + a u -v+ f(x,t), \quad & x \in Ω, ~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 $$ ||u-\tilde{u}||_{L^2(Ω)} + ||v-\tilde{v}||_{L^2(Ω)} \to 0 \quad \text{as } t \to \infty, $$ 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.