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

TL;DR

This paper investigates how the shape and size of a region influence population survival or extinction in a moving-boundary Fisher-Stefan model, revealing that geometry, not just size, determines outcomes.

Contribution

It extends the understanding of the Fisher-Stefan model to two-dimensional regions, showing that geometry affects survival-extinction thresholds beyond simple length measures.

Findings

Region geometry influences survival or extinction outcomes.

Critical area alone is insufficient to predict survival; shape matters.

Numerical simulations confirm the importance of geometry in two-dimensional settings.

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.