# Sorting Phenomena in a Mathematical Model For Two Mutually Attracting/Repelling Species

## TL;DR

This paper investigates a mathematical model for two interacting species with attraction and repulsion, demonstrating phase separation and spatial sorting, with proofs of stationary solutions' existence, shape, and conditions for uniqueness.

## Contribution

It extends existing models to include different intra- and inter-species attractions, proving phase separation and stationary solution properties using advanced mathematical techniques.

## Key findings

- Support of stationary solutions has zero Lebesgue measure.
- Support of the sum of densities is simply connected.
- Existence and shape of segregated solutions are established.

## 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.