The Evolution of Travelling Waves in a KPP Reaction-Diffusion Model with Cut-off Reaction Rate. I. Permanent Form Travelling Waves

TL;DR

This paper investigates the existence and properties of permanent form travelling wave solutions in a KPP reaction-diffusion model with a discontinuous reaction rate cut-off, analyzing asymptotic behaviors for small and large cut-off values.

Contribution

It proves the existence of unique travelling waves with monotone speeds for each cut-off value and extends asymptotic analysis to new regimes, supported by numerical validation.

Findings

Existence of unique travelling waves for all cut-off values

Asymptotic behaviors characterized for small and large cut-off limits

Numerical results confirm analytical asymptotic predictions

Abstract

We consider Kolmogorov--Petrovskii--Piscounov (KPP) type models in the presence of a discontinuous cut-off in reaction rate at concentration $u=u_c$. In Part I we examine permanent form travelling wave solutions (a companion paper, Part II, is devoted to their evolution in the large time limit). For each fixed cut-off value $0<u_c<1$, we prove the existence of a unique permanent form travelling wave with a continuous and monotone decreasing propagation speed $v^*(u_c)$. We extend previous asymptotic results in the limit of small $u_c$ and present new asymptotic results in the limit of large $u_c$ which are respectively obtained via the systematic use of matched and regular asymptotic expansions. The asymptotic results are confirmed against numerical results obtained for the particular case of a cut-off Fisher reaction function.