Addendum to ‘Non-cooperative Fisher–KPP systems: traveling waves and long-time behavior’

Abstract

The solution to a problem left open in the original paper [1] and solved afterwards is presented. Additionally, a few conflicting notations and a typo are corrected.

A Liouville-Type Result for Non-cooperative Fisher–KPP Systems and Nonlocal Equations in Cylinders

“…(This scalar equation can actually be understood as a KPP system of dimension 1.) This similarity mainly concerns the behavior close to u = 0 and it leads to several classical results: a sharp persistence-extinction criterion [24,25], the existence of traveling waves for all speeds larger than or equal to a linearly determined minimal wave speed c [24,28,30], the equality between this minimal wave speed and the asymptotic speed of spreading for initially compactly supported solutions of the Cauchy problem [5,24] and an exponential equivalent of the profile at the leading edge [23,30]. However, away from u = 0 and in particular in the wake of a traveling wave solution p (x − ct), the picture is more complicated.…”

“…The system (1) is a particular example of non-cooperative KPP systems. The first author studied these systems in [23][24][25][26]. The second author studied them with collaborators in [3,28] and gave an epidemiological interpretation in [29].…”