Determining the optimal coefficient of the spatially periodic Fisher-KPP equation that minimizes the spreading speed

TL;DR

This paper finds the explicit form of the function that minimizes the spreading speed in a spatially periodic Fisher-KPP equation, providing a novel example of calculating minimal wave speeds in such models.

Contribution

It introduces a method to explicitly determine the optimal periodic growth rate function that minimizes the spreading speed in the Fisher-KPP equation.

Findings

Explicit form of the minimizing function r(x) obtained.

First calculable example of minimal speed in spatially periodic Fisher-KPP.

Provides insights into controlling propagation speed in heterogeneous environments.

Abstract

This paper is concerned with the spatially periodic Fisher-KPP equation $u_t=(d(x)u_x)_x+(r(x)-u)u$, $x\in \mathbb{R}$, where $d(x)$ and $r(x)$ are periodic functions with period $L>0$. We assume that $r(x)$ has positive mean and $d(x)>0$. It is known that there exists a positive number $c^*_d(r)$, called the minimal wave speed, such that a periodic traveling wave solution with average speed $c$ exists if and only if $c \geq c^*_d(r)$. In the one-dimensional case, the minimal speed $c^*_d(r)$ coincides with the ``spreading speed'', that is, the asymptotic speed of the propagating front of a solution with compactly supported initial data. In this paper, we study the minimizing problem for the minimal speed $c^*_d(r)$ by varying $r(x)$ under a certain constraint, while $d(x)$ arbitrarily. We have been able to obtain an explicit form of the minimizing function $r(x)$. Our result provides the first calculable example of the minimal speed for spatially periodic Fisher-KPP equations as far as the author knows.