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We investigate pattern formation in a two-dimensional (2D) Fisher--Stefanmodel, which involves solving the Fisher--KPP equation on a compactly-supportedregion with a moving boundary. By combining the Fisher--KPP and classicalStefan theory, the Fisher--Stefan model alleviates two limitations of theFisher--KPP equation for biological populations. In this work, we investigatewhether the 2D Fisher--Stefan model predicts pattern formation, by analysingthe linear stability of planar travelling wave solutions to sinusoidaltransverse perturbations. Planar fronts of the Fisher--KPP equation arelinearly stable. Similarly, we demonstrate that invading planar fronts ($c >0$) of the Fisher--Stefan model are linearly stable to perturbations of allwave numbers. However, our analysis demonstrates that receding planar fronts($c < 0$) of the Fisher--Stefan model are linearly unstable for all wavenumbers. This is analogous to unstable solutions for planar solidification inthe classical Stefan problem. Introducing a surface tension regularisationstabilises receding fronts for short-wavelength perturbations, giving rise to arange of unstable modes and a most unstable wave number. We supplement linearstability analysis with level-set numerical solutions that corroboratetheoretical results. Overall, front instability in the Fisher--Stefan modelsuggests a new mechanism for pattern formation in receding biologicalpopulations.

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