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.
Categories: math.AP