Pattern formation and front stability for a moving-boundary model of biological invasion and recession

TL;DR

This paper analyzes the stability of moving boundaries in a 2D Fisher--Stefan model, revealing conditions under which biological invasion fronts are stable or unstable, and suggesting a new pattern formation mechanism.

Contribution

It provides a linear stability analysis of the 2D Fisher--Stefan model, highlighting the stability of invading fronts and instability of receding fronts, with surface tension effects.

Findings

Invading fronts are linearly stable to perturbations.

Receding fronts are linearly unstable for all wave numbers.

Surface tension stabilizes short-wavelength receding fronts.

Abstract

We investigate pattern formation in a two-dimensional (2D) Fisher--Stefan model, which involves solving the Fisher--KPP equation on a compactly-supported region with a moving boundary. By combining the Fisher--KPP and classical Stefan theory, the Fisher--Stefan model alleviates two limitations of the Fisher--KPP equation for biological populations. In this work, we investigate whether the 2D Fisher--Stefan model predicts pattern formation, by analysing the linear stability of planar travelling wave solutions to sinusoidal transverse perturbations. Planar fronts of the Fisher--KPP equation are linearly stable. Similarly, we demonstrate that invading planar fronts ($c > 0$) of the Fisher--Stefan model are linearly stable to perturbations of all wave numbers. However, our analysis demonstrates that receding planar fronts ($c < 0$) of the Fisher--Stefan model are linearly unstable for all wave numbers. This is analogous to unstable solutions for planar solidification in the classical Stefan problem. Introducing a surface tension regularisation stabilises receding fronts for short-wavelength perturbations, giving rise to a range of unstable modes and a most unstable wave number. We supplement linear stability analysis with level-set numerical solutions that corroborate theoretical results. Overall, front instability in the Fisher--Stefan model suggests a new mechanism for pattern formation in receding biological populations.