Self-organization on Riemannian manifolds

TL;DR

This paper studies an aggregation model on Riemannian manifolds, analyzing how populations self-organize into equilibria influenced by nonlocal interactions, with specific focus on the sphere and hyperbolic space.

Contribution

It introduces a framework for constructing interaction potentials on Riemannian manifolds that produce constant-support equilibria, with analytical and numerical analysis on specific geometries.

Findings

Equilibria can be characterized analytically on the sphere and hyperbolic space.

Numerical simulations confirm the long-time stability of certain equilibria.

Different interaction potentials lead to diverse equilibrium configurations.

Abstract

We consider an aggregation model that consists of an active transport equation for the macroscopic population density, where the velocity has a nonlocal functional dependence on the density, modelled via an interaction potential. We set up the model on general Riemannian manifolds and provide a framework for constructing interaction potentials which lead to equilibria that are constant on their supports. We consider such potentials for two specific cases (the two-dimensional sphere and the two-dimensional hyperbolic space) and investigate analytically and numerically the long-time behaviour and equilibrium solutions of the aggregation model on these manifolds. Equilibria obtained numerically with other interaction potentials are also presented.