Title: Sorting Phenomena in a Mathematical Model For Two Mutually Attracting/Repelling Species,
Abstract: Macroscopic models for systems involving diffusion, short-range repulsion,
and long-range attraction have been studied extensively in the last decades. In
this paper we extend the analysis to a system for two species interacting with
each other according to different inner- and intra-species attractions. Under
suitable conditions on this self- and crosswise attraction an interesting
effect can be observed, namely phase separation into neighbouring regions, each
of which contains only one of the species. We prove that the intersection of
the support of the stationary solutions of the continuum model for the two
species has zero Lebesgue measure, while the support of the sum of the two
densities is simply connected.
Preliminary results indicate the existence of phase separation, i.e. spatial
sorting of the different species. A detailed analysis in one spatial dimension
follows. The existence and shape of segregated stationary solutions is shown
via the Krein-Rutman theorem. Moreover, for small repulsion/nonlinear
diffusion, also uniqueness of these stationary states is proved.