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

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.