Swarm equilibria in domains with boundaries

TL;DR

This paper analyzes equilibrium solutions in bounded domains for a first-order aggregation model, revealing that while only connected states are stable, the dynamics often favor the formation of unstable disconnected equilibria.

Contribution

It introduces a detailed analysis of both connected and disconnected equilibria, highlighting their stability properties and dynamical tendencies in bounded domains.

Findings

Only connected equilibria are stable.

Disconnected equilibria are not energy minimizers.

Dynamics tend to favor formation of unstable disconnected states.

Abstract

We study equilibria in domains with boundaries for a first-order aggregation model that includes social interactions and exogenous forces. Such equilibrium solutions can be connected or disconnected, the latter consisting in a delta concentration on the boundary and a free swarm component in the interior of the domain. Equilibria are stationary points of an energy functional, and stable configurations are local minimizers of this functional. We find a one-parameter family of disconnected equilibrium configurations which are not energy minimizers; the only stable equilibria are the connected states. Nevertheless, we demonstrate that in certain cases the dynamical evolution, along the gradient flow of the energy functional, tends to overwhelmingly favour the formation of (unstable) disconnected equilibria.