# Macroscopic descriptions of follower-leader systems

## TL;DR

This paper derives and analyzes macroscopic models for follower-leader swarms, emphasizing the impact of interaction kernels on swarm behavior and the role of leaders in initiating movement.

## Contribution

It provides a clear derivation of macroscopic equations for fixed follower-leader systems and analyzes how interaction kernels influence swarm dynamics over different time scales.

## Key findings

- Interaction kernel critically affects transient swarm behavior
- Number of leaders needed depends on interaction kernel properties
- Long-term swarm movement can be initiated with minimal leaders

## Abstract

The fundamental derivation of macroscopic model equations to describe swarms based on microscopic movement laws and mathematical analyses into their self-organisation capabilities remains a challenge from the perspective of both modelling and analysis. In this paper we clarify relevant continuous macroscopic model equations that describe follower-leader interactions for a swarm where these two populations are fixed. We study the behaviour of the swarm over long and short time scales to shed light on the number of leaders needed to initiate swarm movement, according to the homogeneous or inhomogeneous nature of the interaction (alignment) kernel. The results indicate the crucial role played by the interaction kernel to model transient behaviour.