Generic configurations in 2D strongly competing systems

TL;DR

This paper investigates the typical structure of domain partitions in a multi-species segregation model, showing that generically only three species meet at boundary junctions, using harmonic maps and complex analysis tools.

Contribution

It demonstrates that in multi-species segregation models, the domain partitioning typically involves only triple junctions, advancing understanding of free boundary problems in planar systems.

Findings

Domain partitions usually have only triple junctions.

Multiple species can coexist locally, but with limited boundary complexity.

Harmonic maps and Hopf differentials are key analytical tools.

Abstract

We study a problem modelling segregation of an arbitrary number of competing species in planar domains. The solutions give rise to a well known free boundary problem with the domain partitioning itself into subdomains occupied by different species. In principle, several of them can coexist in a neighborhood of any point. However, we show that {\it generically} the domain partitions into subdomains with only triple junctions, meaning that at most three populations meet at the free boundary. Our main tools are the use of the formalism of harmonic maps into singular spaces and the introduction of a complex structure via the Hopf differential.