An extension of the Wright's 3/2-theorem for the KPP-Fisher delayed equation

TL;DR

This paper extends Wright's 3/2-stability theorem to the delayed KPP-Fisher equation, establishing conditions for the existence of positive traveling fronts with delays and non-monotonicity.

Contribution

It provides a short proof of an extended stability theorem for the delayed KPP-Fisher equation, broadening the understanding of traveling front solutions.

Findings

Conditions $ au \,\leq\, 3/2$ and $c \geq 2$ ensure positive traveling fronts.

Traveling fronts are not necessarily monotone under these conditions.

The proof simplifies and extends previous stability results.

Abstract

We present a short proof of the following natural extension of the famous Wright's 3/2-stability theorem: the conditions $\tau \leq 3/2, \ c \geq 2$ imply the presence of the positive traveling fronts (not necessarily monotone) $u = \phi(x\cdot \nu+ct), \ |\nu| =1,$ in the delayed KPP-Fisher equation $u_t(t,x) = \Delta u(t,x) + u(t,x)(1-u(t-\tau,x))$, $u\geq 0,$ $x \in \R^m.$