Criteria for the (in)stability of planar interfaces in singularly perturbed 2-component reaction-diffusion equations

TL;DR

This paper develops stability criteria for planar interfaces in singularly perturbed 2-component reaction-diffusion systems, analyzing their stability and pattern formation, especially in dryland ecosystem models.

Contribution

It introduces two new 2D stability criteria for slow-fast-slow reaction-diffusion fronts, linking system nonlinearities to interface stability and pattern formation.

Findings

Analytically proved instability of certain dryland interfaces.

Numerically observed fingering patterns in vegetation models.

Established criteria applicable to various reaction-diffusion systems.

Abstract

We consider a class of singularly perturbed 2-component reaction-diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow-fast-slow, interfaces between stable homogeneous rest states. In many example systems, such as models of desertification fronts in dryland ecosystems, such fronts can exhibit an instability by which the interface destabilizes into fingering patterns. Motivated by the appearance of such patterns, we propose two versions of a 2D stability criterion for (transversal) long wavelength perturbations along the interface of these traveling slow-fast-slow fronts. The fronts are constructed using geometric singular perturbation techniques by connecting slow orbits on two distinct normally hyperbolic slow manifolds through a heteroclinic orbit in the fast problem. The associated stability criteria are expressed in terms of the nonlinearities of the system and the slow-fast-slow structure of the fronts. We illustrate and further elaborate the general set-up by explicitly working out the existence and transversal (in)stability of traveling fronts in a number of example systems/models. We analytically establish the instability of invading bare soil/vegetation interfaces against transversal long wavelength perturbations in several dryland ecosystem models and numerically recover fingering vegetation patterns counter-invading an invading desertification front.