Macroscopic descriptions of follower-leader systems

Sara Bernardi; Gissell Estrada-Rodriguez; Heiko Gimperlein; Kevin J. Painter

doi:10.3934/krm.2021035

Article Contents

2021, Volume 14, Issue 6: 981-1002. Doi: 10.3934/krm.2021035

This issue Previous Article The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions Next Article Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics

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
^* Corresponding author: Heiko Gimperlein

Received: March 2020

Revised: October 2021

Early access: November 2021

Published: December 2021

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

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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.

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Mathematics Subject Classification: Primary: 92D25; Secondary: 92D50, 35Q92, 82C22.

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Figure 1. Illustration of switching between streakers and passive leaders

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Figure 2. Illustration of different swarm shapes

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Figure 3. Evolution of follower and leader populations in model example

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Figure 4. Percentage of active and passive leaders as a function of time

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