How far does small chemotactic interaction perturb the Fisher-KPP dynamics?

This paper deals with nonnegative solutions of the Neumann initial-boundary value problem for the fully parabolic chemotaxis-growth system $(u_{\varepsilon})_t$ $=\Delta u_{\varepsilon} - \varepsilon \nabla \cdot ( u_\varepsilon \nabla v_\varepsilon) + \mu u_\varepsilon(1 - u_\varepsilon)$, $(v_{\varepsilon})_t=\Delta v_\varepsilon -v_\varepsilon+u_\varepsilon,$ with positive small parameter $\varepsilon>0$ in a bounded convex domain $\Omega\subset\mathbb{R}^n$ ($n\geq 1$) with smooth boundary. The solutions converge to the solution $u$ to the Fisher-KPP equation as $\varepsilon\to 0$. It is shown that for all $\mu>0$ and any suitably regular nonnegative initial data $(u_{init},v_{init})$ there are some constants $\varepsilon_0>0$ and $C>0$ such that $$\sup_{t>0}\|u_\varepsilon(\cdot,t)-u(\cdot,t)\|_{L^\infty(\Omega)} \leq C\varepsilon \quad for\ all\ \varepsilon\in(0,\varepsilon_0).$$