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

We investigate pattern formation in a two-dimensional (2D) Fisher–Stefan model, which involves solving the Fisher–KPP equation on compactly-supported region with moving boundary. The boundary evolves analogously to classical one-phase Stefan problem, such that speed is proportional local population density gradient. By combining and theory, model alleviates two limitations of for biological populations. Unlike equation, solutions have compact support, explicitly defining occupied by population. Furthermore, admits travelling wave non-negative all speeds c∈(−∞,∞), can thus both invasion recession. In this work, we whether 2D predicts formation, analysing linear stability planar sinusoidal transverse perturbations. Planar fronts are linearly stable. Similarly, demonstrate invading (c>0) stable perturbations numbers. However, our analysis demonstrates receding (c<0) unstable This analogous solidification problem. Introducing surface tension regularisation stabilises short-wavelength perturbations, giving rise range modes most number. supplement level-set numerical corroborate theoretical results. Overall, front instability suggests new mechanism