Travelling front solutions in a spatially heterogeneous reaction-diffusion system

TL;DR

This paper studies travelling front solutions in a reaction-diffusion system with spatial heterogeneity, constructing solutions and deriving a delay-differential equation for front position without relying on spectral analysis.

Contribution

It extends Fenichel theory to non-compact cases and derives a novel delay-differential equation for front dynamics in heterogeneous media.

Findings

Existence of bi-stability with stable heterogeneous states

Construction of stationary and travelling front solutions

Derivation and validation of a front position delay equation

Abstract

We investigate a two-component reaction-diffusion system with a slow-fast structure and spatially varying coefficients $f_1$ and $f_2$ appearing in the slow equation. Under mild boundedness and regularity conditions on $f_1$ and $f_2$ the system is shown to exhibit bi-stability in the form of two stable stationary heterogeneous background states. These background states can be connected by stationary and travelling front solutions. Travelling fronts feature an interface that moves with a non-uniform speed through the motionless spatially varying background states it connects. We construct both the background states and stationary fronts using an extension of Fenichel theory to the non-compact case. Additionally, we establish the existence of travelling front solutions and derive a leading-order expression for the dynamic position of the moving interface through a time-dependent spatial dynamics approach. This expression takes the form of a delay-differential equation, and its accuracy is validated through numerical simulations. A key contribution of our work lies in the general treatment of $f_1$ and $f_2$, which are neither (necessarily) asymptotically small nor restricted to specific forms such as periodic or localized structures. Furthermore, our derivation of the front position formula circumvents the traditional reliance on spectral analysis, enabling us to describe front dynamics beyond bifurcations from stationary fronts. This approach has the potential to be extended to other settings in which spectral properties at onset preclude conventional reduction techniques.