Slowly oscillating wavefronts of the KPP-Fisher delayed equation

TL;DR

This paper investigates semi-wavefront solutions of the delayed KPP-Fisher equation, establishing conditions for their monotonicity, existence, and asymptotic behavior, and linking wavefront existence to Wright's stability conjecture.

Contribution

It provides a complete characterization of semi-wavefronts, proves their wavefront nature under certain conditions, and connects wavefront existence to a famous stability conjecture.

Findings

Semi-wavefronts are either monotone or slowly oscillating.

Wavefronts exist for c ≥ 2 and τ ≤ 1, and are monotone in this case.

The existence of wavefronts relates to Wright's global stability conjecture.

Abstract

This paper concerns the semi-wavefronts (i.e. bounded solutions $u=\phi(x \nu +ct) >0,$ $ |\nu|=1, $ satisfying $\phi(-\infty)=0$) to the delayed KPP-Fisher equation $$u_t(t,x) = \Delta u(t,x) + u(t,x)(1-u(t-\tau,x)), \ u \geq 0,\ x \in \R^m. \eqno(*)$$ First, we show that each semi-wavefront should be either monotone or slowly oscillating. Then a complete solution to the problem of existence of semi-wavefronts is provided. We prove next that the semi-wavefronts are in fact wavefronts (i.e. additionally $\phi(+\infty)=1$) if $c \geq 2$ and $\tau \leq 1$; our proof uses dynamical properties of some auxiliary one-dimensional map with the negative Schwarzian. The analysis of the fronts' asymptotic expansions at infinity is another key ingredient of our approach. It allows to indicate the maximal domain ${\mathcal D}_n$ of $(\tau,c)$ where the existence of non-monotone wavefronts can be expected. Here we show that the problem of wavefront's existence is closely related to the Wright's global stability conjecture.