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The Fisher-Stefan model involves solving the Fisher-KPP equation on a domainwhose boundary evolves according to a Stefan-like condition. The Fisher-Stefanmodel alleviates two practical limitations of the standard Fisher-KPP modelwhen applied to biological invasion. First, unlike the Fisher-KPP equation,solutions to the Fisher-Stefan model have compact support, enabling one todefine the interface between occupied and unoccupied regions unambiguously.Second, the Fisher-Stefan model admits solutions for which the populationbecomes extinct, which is not possible in the Fisher-KPP equation. Previousresearch showed that population survival or extinction in the Fisher-Stefanmodel depends on a critical length in one-dimensional Cartesian orradially-symmetric geometry. However, the survival and extinction behaviour forgeneral two-dimensional regions remains unexplored. We combine analysis andlevel-set numerical simulations of the Fisher-Stefan model to investigate thesurvival-extinction conditions for rectangular-shaped initial conditions. Weshow that it is insufficient to generalise the critical length conditions tocritical area in two-dimensions. Instead, knowledge of the region geometry isrequired to determine whether a population will survive or become extinct.

The Effect of Geometry on Survival and Extinction in a Moving-Boundary Problem Motivated by the Fisher-KPP Equation