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

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. The boundary evolves analogously to the classical Stefan problem, such that boundary speed is proportional to the local population density gradient. By combining the Fisher-KPP and classical Stefan theory, the Fisher-Stefan model alleviates two limitations of the Fisher–KPP equation for biological populations. Unlike the Fisher-KPP equation, solutions to the Fisher–Stefan model have compact support, explicitly defining the region occupied by the population. Furthermore, the Fisher-Stefan model admits travelling wave solutions with non-negative density for all wave speeds, and can thus model both population invasion and population recession. 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.