Generic configurations in 2D strongly competing systems

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.