Front selection in reaction-diffusion systems via diffusive normal forms

Montie Avery

Published 2022-11-21Version 1

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. Using appropriate rescalings and a functional analytic approach to regularize singular perturbations, we show that our assumptions hold in general reaction-diffusion systems when the nonlinearity undergoes a transcritical, saddle-node, or supercritical pitchfork bifurcation, demonstrating that our results capture universal aspects of the onset of instability in spatially extended systems. We discuss further applications to parametrically forced amplitude equations, competitive Lotka-Volterra systems, and a tumor growth model.