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

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.