Front stability for a moving-boundary model for biological invasion and recession
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Abstract: The Fisher-Stefan model is a modification to the Fisher-KPP equation, a common prototype model in mathematical biology. The Fisher-Stefan model involves solving the Fisher-KPP equation on a compactly-supported region with a moving boundary that evolves analogously to the classical one-phase Stefan condition. Unlike the Fisher-KPP equation, the Fisher-Stefan model admits solutions with compact support, and has travelling wave solutions with non-negative density for any wave speed. Despite these advantages, there remains much to study about the Fisher-Stefan and related moving-boundary models. In this work, we use linear stability analysis and numerical solutions using the level-set method to investigate whether planar Fisher-Stefan fronts are stable to transverse perturbations. We found that invading planar fronts are linearly stable (like solutions to the Fisher-KPP equation), but receding planar fronts are unstable. Instability in receding fronts suggests a mechanism for pattern formation in receding biological populations.