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

TL;DR

This paper investigates how small chemotactic interactions affect the Fisher-KPP equation, demonstrating that solutions of the chemotaxis-growth system converge uniformly to Fisher-KPP solutions as the chemotactic parameter approaches zero.

Contribution

It establishes quantitative convergence rates of chemotaxis-growth solutions to Fisher-KPP solutions in bounded domains as chemotactic effects become negligible.

Findings

Solutions converge uniformly with order epsilon as chemotactic parameter tends to zero.

Convergence holds for all positive growth rates and regular initial data.

Provides explicit bounds on the difference between chemotaxis and Fisher-KPP solutions.

Abstract

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 \[ \sup_{t>0}\|u_\varepsilon(\cdot,t)-u(\cdot,t)\|_{L^\infty(\Omega)} \leq C\varepsilon \quad for\ all\ \varepsilon\in(0,\varepsilon_0). \]