Survival, extinction, and interface stability in a two--phase moving boundary model of biological invasion

TL;DR

This paper develops a two-phase moving boundary model for biological invasion, analyzing survival, extinction, and interface stability of competing populations, revealing differences from single-population models and emphasizing the importance of multi-population interactions.

Contribution

It introduces a novel two-phase Stefan boundary model for biological invasion and explores its properties through numerical simulations, highlighting differences from traditional single-population models.

Findings

Survival and extinction depend on initial conditions and parameters.

The moving front's stability varies with perturbations, affecting invasion outcomes.

Two-phase interactions lead to different dynamics than single-population models.

Abstract

We consider a moving boundary mathematical model of biological invasion. The model describes the spatiotemporal evolution of two adjacent populations: each population undergoes linear diffusion and logistic growth, and the boundary between the two populations evolves according to a two--phase Stefan condition. This mathematical model describes situations where one population invades into regions occupied by the other population, such as the spreading of a malignant tumour into surrounding tissues. Full time--dependent numerical solutions are obtained using a level--set numerical method. We use these numerical solutions to explore several properties of the model including: (i) survival and extinction of one population initially surrounded by the other; and (ii) linear stability of the moving front boundary in the context of a travelling wave solution subjected to transverse perturbations. Overall, we show that many features of the well--studied one--phase single population analogue of this model can be very different in the more realistic two--phase setting. These results are important because realistic examples of biological invasion involve interactions between multiple populations and so great care should be taken when extrapolating predictions from a one--phase single population model to cases for which multiple populations are present. Open source Julia--based software is available on GitHub to replicate all results in this study.