Front selection in reaction-diffusion systems via diffusive normal forms

TL;DR

This paper proves that invasion speeds in reaction-diffusion systems are determined by spectral stability, extending the marginal stability conjecture to multi-component systems and analyzing diffusive dynamics in the invasion front.

Contribution

It provides a full proof of the marginal stability conjecture for multi-component reaction-diffusion systems and introduces coordinate transformations to analyze diffusive dynamics in the leading edge.

Findings

Propagation speeds are determined by spectral stability conditions.

Coordinate transformations reveal diffusive behavior in the invasion front.

Established front selection through nonlinear matching and sharp linear estimates.

Abstract

We show that propagation speeds in invasion processes modeled by reaction-diffusion systems are determined by marginal spectral stability conditions, as predicted by the marginal stability conjecture. This conjecture was recently settled in scalar equations; here we give a full proof for the multi-component case. The main new difficulty lies in precisely characterizing diffusive dynamics in the leading edge of invasion fronts. To overcome this, we introduce coordinate transformations which allow us to recognize a leading order diffusive equation relying only on an assumption of generic marginal pointwise stability. We are then able to use self-similar variables to give a detailed description of diffusive dynamics in the leading edge, which we match with a traveling invasion front in the wake. We then establish front selection by controlling these matching errors in a nonlinear iteration scheme, relying on sharp estimates on the linearization about the invasion front. We briefly discuss applications to parametrically forced amplitude equations, competitive Lotka-Volterra systems, and a tumor growth model.