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December  2021, 14(6): 981-1002. doi: 10.3934/krm.2021035

## Macroscopic descriptions of follower-leader systems

 1 Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 2 Laboratoire Jacques-Louis Lions, Sorbonne-Université, 4, pl. Jussieu, F-75005 Paris, France 3 Maxwell Institute for Mathematical Sciences and Department of Mathematics, Heriot-–Watt University, Edinburgh, EH14 4AS, United Kingdom 4 Institute for Mathematics, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany 5 Interuniversity Department of Regional and Urban Studies and Planning, Politecnico di Torino, Torino, 10125, Italy

* Corresponding author: Heiko Gimperlein

Received  March 2020 Revised  October 2021 Published  December 2021 Early access  November 2021

Fund Project: G. E. R. was supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh

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.

Citation: Sara Bernardi, Gissell Estrada-Rodriguez, Heiko Gimperlein, Kevin J. Painter. Macroscopic descriptions of follower-leader systems. Kinetic & Related Models, 2021, 14 (6) : 981-1002. doi: 10.3934/krm.2021035
##### References:

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##### References:
Illustration of switching between streakers and passive leaders
Illustration of different swarm shapes
Evolution of follower and leader populations in model example
Percentage of active and passive leaders as a function of time
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