doi: 10.3934/krm.2021035
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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 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, doi: 10.3934/krm.2021035
References:
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S. BernardiR. Eftimie and K. J. Painter, Leadership through influence: What mechanisms allow leaders to steer a swarm?, Bull. Math. Biol., 83 (2021), 1-33.  doi: 10.1007/s11538-021-00901-8.  Google Scholar

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L. J. N. BrentD. W. FranksE. A. FosterK. C. BalcombM. A. Cant and D. P. Croft, Ecological knowledge, leadership, and the evolution of menopause in killer whales, Current Biology, 25 (2015), 746-750.  doi: 10.1016/j.cub.2015.01.037.  Google Scholar

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J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, (2010), 297–336. doi: 10.1007/978-0-8176-4946-3_12.  Google Scholar

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I. D. CouzinJ. KrauseN. R. Franks and S. A. Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516.  doi: 10.1038/nature03236.  Google Scholar

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F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

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F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

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P. DegondA. Frouvelle and J.-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., 23 (2013), 427-456.  doi: 10.1007/s00332-012-9157-y.  Google Scholar

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P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

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[18]

G. Estrada-Rodriguez and H. Gimperlein, Interacting particles with lévy strategies: Limits of transport equations for swarm robotic systems, SIAM J. Appl. Math., 80 (2020), 476-498.  doi: 10.1137/18M1205327.  Google Scholar

[19]

U. GreggersC. SchoeningJ. Degen and R. Menzel, Scouts behave as streakers in honeybee swarms, Naturwissenschaften, 100 (2013), 805-809.  doi: 10.1007/s00114-013-1077-7.  Google Scholar

[20]

A. HaegerK. WolfM. M. Zegers and P. Friedl, Collective cell migration: Guidance principles and hierarchies, Trends in Cell Biology, 25 (2015), 556-566.  doi: 10.1016/j.tcb.2015.06.003.  Google Scholar

[21]

A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, 48 (2003), 29 pp. doi: 10.1109/TAC.2003.812781.  Google Scholar

[22]

N. C. MakrisP. RatilalS. JagannathanZ. GongM. AndrewsI. BertsatosO. R. GodøR. W. Nero and J. M. Jech, Critical population density triggers rapid formation of vast oceanic fish shoals, Science, 323 (2009), 1734-1737.  doi: 10.1126/science.1169441.  Google Scholar

[23]

R. Mayor and S. Etienne-Manneville, The front and rear of collective cell migration, Nature Reviews Molecular Cell Biology, 17 (2016), 97-109.  doi: 10.1038/nrm.2015.14.  Google Scholar

[24]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[25]

T. MuellerR. B. O'HaraS. J. ConverseR. P. Urbanek and W. F. Fagan, Social learning of migratory performance, Science, 341 (2013), 999-1002.  doi: 10.1126/science.1237139.  Google Scholar

[26]

M. NagyZ. AkosD. Biro and T. Vicsek, Hierarchical group dynamics in pigeon flocks, Nature, 464 (2010), 890-893.  doi: 10.1038/nature08891.  Google Scholar

[27]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

[28]

K. PainterJ. BloomfieldJ. Sherratt and A. Gerisch, A nonlocal model for contact attraction and repulsion in heterogeneous cell populations, Bulletin of Mathematical Biology, 77 (2015), 1132-1165.  doi: 10.1007/s11538-015-0080-x.  Google Scholar

[29]

R. Rainey, Radar observations of locust swarms, Science, 157 (1967), 98-99.  doi: 10.1126/science.157.3784.98.  Google Scholar

[30]

S. G. Reebs, Can a minority of informed leaders determine the foraging movements of a fish shoal?, Animal Behaviour, 59 (2000), 403-409.  doi: 10.1006/anbe.1999.1314.  Google Scholar

[31] T. D. Seeley, Honeybee Democracy, Princeton University Press, 2010.  doi: 10.1515/9781400835959.  Google Scholar
[32]

J. Shen, Cucker–smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.  Google Scholar

[33]

R. SkafG. B. Popov and J. Roffey, The desert locust: An international challenge, Philosophical Transactions of the Royal Society of London. B, Biological Sciences, 328 (1990), 525-538.  doi: 10.1098/rstb.1990.0125.  Google Scholar

[34]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[35]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Bull. Math. Biol., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

show all references

References:
[1]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.  Google Scholar

[2]

M. BeekmanR. L. Fathke and T. D. Seeley, How does an informed minority of scouts guide a honeybee swarm as it flies to its new home?, Animal Behaviour, 71 (2006), 161-171.  doi: 10.1016/j.anbehav.2005.04.009.  Google Scholar

[3]

N. Bellomo, Modeling Complex Living Systems: A Kinetic Theory and Stochastic Game Approach, Springer Science & Business Media, 2008.  Google Scholar

[4]

A. M. BerdahlA. B. KaoA. FlackP. A. WestleyE. A. CodlingI. D. CouzinA. I. Dell and D. Biro, Collective animal navigation and migratory culture: From theoretical models to empirical evidence, Philosophical Transactions of the Royal Society B: Biological Sciences, 373 (2018), 20170009.  doi: 10.1098/rstb.2017.0009.  Google Scholar

[5]

S. BernardiA. Colombi and M. Scianna, A particle model analysing the behavioural rules underlying the collective flight of a bee swarm towards the new nest, J. Biol. Dyn., 12 (2018), 632-662.  doi: 10.1080/17513758.2018.1501105.  Google Scholar

[6]

S. BernardiR. Eftimie and K. J. Painter, Leadership through influence: What mechanisms allow leaders to steer a swarm?, Bull. Math. Biol., 83 (2021), 1-33.  doi: 10.1007/s11538-021-00901-8.  Google Scholar

[7]

L. J. N. BrentD. W. FranksE. A. FosterK. C. BalcombM. A. Cant and D. P. Croft, Ecological knowledge, leadership, and the evolution of menopause in killer whales, Current Biology, 25 (2015), 746-750.  doi: 10.1016/j.cub.2015.01.037.  Google Scholar

[8]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, (2010), 297–336. doi: 10.1007/978-0-8176-4946-3_12.  Google Scholar

[9]

I. D. CouzinJ. KrauseN. R. Franks and S. A. Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516.  doi: 10.1038/nature03236.  Google Scholar

[10]

I. D. CouzinJ. KrauseR. JamesG. D. Ruxton and N. R. Franks, Collective memory and spatial sorting in animal groups, J. Theoret. Biol., 218 (2002), 1-11.  doi: 10.1006/jtbi.2002.3065.  Google Scholar

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[12]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

[13]

P. DegondA. Frouvelle and J.-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., 23 (2013), 427-456.  doi: 10.1007/s00332-012-9157-y.  Google Scholar

[14]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[15]

G. Dimarco and S. Motsch, Self-alignment driven by jump processes: Macroscopic limit and numerical investigation, Math. Models Methods Appl. Sci., 26 (2016), 1385-1410.  doi: 10.1142/S0218202516500330.  Google Scholar

[16]

R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review, J. Math. Biol., 65 (2012), 35-75.  doi: 10.1007/s00285-011-0452-2.  Google Scholar

[17]

R. EftimieG. De VriesM. Lewis and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol., 69 (2007), 1537-1565.  doi: 10.1007/s11538-006-9175-8.  Google Scholar

[18]

G. Estrada-Rodriguez and H. Gimperlein, Interacting particles with lévy strategies: Limits of transport equations for swarm robotic systems, SIAM J. Appl. Math., 80 (2020), 476-498.  doi: 10.1137/18M1205327.  Google Scholar

[19]

U. GreggersC. SchoeningJ. Degen and R. Menzel, Scouts behave as streakers in honeybee swarms, Naturwissenschaften, 100 (2013), 805-809.  doi: 10.1007/s00114-013-1077-7.  Google Scholar

[20]

A. HaegerK. WolfM. M. Zegers and P. Friedl, Collective cell migration: Guidance principles and hierarchies, Trends in Cell Biology, 25 (2015), 556-566.  doi: 10.1016/j.tcb.2015.06.003.  Google Scholar

[21]

A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, 48 (2003), 29 pp. doi: 10.1109/TAC.2003.812781.  Google Scholar

[22]

N. C. MakrisP. RatilalS. JagannathanZ. GongM. AndrewsI. BertsatosO. R. GodøR. W. Nero and J. M. Jech, Critical population density triggers rapid formation of vast oceanic fish shoals, Science, 323 (2009), 1734-1737.  doi: 10.1126/science.1169441.  Google Scholar

[23]

R. Mayor and S. Etienne-Manneville, The front and rear of collective cell migration, Nature Reviews Molecular Cell Biology, 17 (2016), 97-109.  doi: 10.1038/nrm.2015.14.  Google Scholar

[24]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[25]

T. MuellerR. B. O'HaraS. J. ConverseR. P. Urbanek and W. F. Fagan, Social learning of migratory performance, Science, 341 (2013), 999-1002.  doi: 10.1126/science.1237139.  Google Scholar

[26]

M. NagyZ. AkosD. Biro and T. Vicsek, Hierarchical group dynamics in pigeon flocks, Nature, 464 (2010), 890-893.  doi: 10.1038/nature08891.  Google Scholar

[27]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

[28]

K. PainterJ. BloomfieldJ. Sherratt and A. Gerisch, A nonlocal model for contact attraction and repulsion in heterogeneous cell populations, Bulletin of Mathematical Biology, 77 (2015), 1132-1165.  doi: 10.1007/s11538-015-0080-x.  Google Scholar

[29]

R. Rainey, Radar observations of locust swarms, Science, 157 (1967), 98-99.  doi: 10.1126/science.157.3784.98.  Google Scholar

[30]

S. G. Reebs, Can a minority of informed leaders determine the foraging movements of a fish shoal?, Animal Behaviour, 59 (2000), 403-409.  doi: 10.1006/anbe.1999.1314.  Google Scholar

[31] T. D. Seeley, Honeybee Democracy, Princeton University Press, 2010.  doi: 10.1515/9781400835959.  Google Scholar
[32]

J. Shen, Cucker–smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.  doi: 10.1137/060673254.  Google Scholar

[33]

R. SkafG. B. Popov and J. Roffey, The desert locust: An international challenge, Philosophical Transactions of the Royal Society of London. B, Biological Sciences, 328 (1990), 525-538.  doi: 10.1098/rstb.1990.0125.  Google Scholar

[34]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[35]

T. VicsekA. CzirókE. Ben-JacobI. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Bull. Math. Biol., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

Figure 1.  Illustration of switching between streakers and passive leaders
Figure 2.  Illustration of different swarm shapes
Figure 3.  Evolution of follower and leader populations in model example
Figure 4.  Percentage of active and passive leaders as a function of time
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