The effect of geometry on survival and extinction in a moving-boundary problem motivated by the Fisher–KPP equation

Abstract

The Fisher–Stefan model involves solving the Fisher–KPP equation on a domain whose boundary evolves according to a Stefan-like condition. The Fisher–Stefan model alleviates two practical limitations of the standard Fisher–KPP model when applied to biological invasion. First, unlike the Fisher–KPP equation, solutions to the Fisher–Stefan model have compact support, enabling one to define the interface between occupied and unoccupied regions unambiguously. Second, the Fisher–Stefan model admits solutions for which the population becomes extinct, which is not possible in the Fisher–KPP equation. Previous research showed that population survival or extinction in the Fisher–Stefan model depends on a critical length in one-dimensional Cartesian or radially-symmetric geometry. However, the survival and extinction behaviour for general two-dimensional regions remains unexplored. We combine analysis and level-set numerical simulations of the Fisher–Stefan model to investigate the survival–extinction conditions for rectangular-shaped initial conditions. We show that it is insufficient to generalise the critical length conditions to critical area in two-dimensions. Instead, knowledge of the region geometry is required to determine whether a population will survive or become extinct.