• Previous Article
    Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics
  • KRM Home
  • This Issue
  • Next Article
    The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions
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 and Related Models, 2021, 14 (6) : 981-1002. doi: 10.3934/krm.2021035
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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[29]

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

[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.

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

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

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[29]

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

[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.

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

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

[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.

[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.

[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.

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
[1]

Lukas Neumann, Christian Schmeiser. A kinetic reaction model: Decay to equilibrium and macroscopic limit. Kinetic and Related Models, 2016, 9 (3) : 571-585. doi: 10.3934/krm.2016007

[2]

S.-I. Ei, M. Mimura, M. Nagayama. Interacting spots in reaction diffusion systems. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 31-62. doi: 10.3934/dcds.2006.14.31

[3]

María del Mar González, Regis Monneau. Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1255-1286. doi: 10.3934/dcds.2012.32.1255

[4]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic and Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011

[5]

David Cowan. Rigid particle systems and their billiard models. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 111-130. doi: 10.3934/dcds.2008.22.111

[6]

Xihong Yan. An augmented Lagrangian-based parallel splitting method for a one-leader-two-follower game. Journal of Industrial and Management Optimization, 2016, 12 (3) : 879-890. doi: 10.3934/jimo.2016.12.879

[7]

N. Bellomo, A. Bellouquid. From a class of kinetic models to the macroscopic equations for multicellular systems in biology. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 59-80. doi: 10.3934/dcdsb.2004.4.59

[8]

Felipe Cucker, Jiu-Gang Dong. A conditional, collision-avoiding, model for swarming. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1009-1020. doi: 10.3934/dcds.2014.34.1009

[9]

Eliot Fried. New insights into the classical mechanics of particle systems. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1469-1504. doi: 10.3934/dcds.2010.28.1469

[10]

Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic and Related Models, 2021, 14 (3) : 541-570. doi: 10.3934/krm.2021015

[11]

Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258

[12]

Fuke Wu, George Yin, Zhuo Jin. Kolmogorov-type systems with regime-switching jump diffusion perturbations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2293-2319. doi: 10.3934/dcdsb.2016048

[13]

M. S. Mahmoud, P. Shi, Y. Shi. $H_\infty$ and robust control of interconnected systems with Markovian jump parameters. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 365-384. doi: 10.3934/dcdsb.2005.5.365

[14]

Yanqing Liu, Yanyan Yin, Kok Lay Teo, Song Wang, Fei Liu. Probabilistic control of Markov jump systems by scenario optimization approach. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1447-1453. doi: 10.3934/jimo.2018103

[15]

Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control and Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019

[16]

Steffen Härting, Anna Marciniak-Czochra, Izumi Takagi. Stable patterns with jump discontinuity in systems with Turing instability and hysteresis. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 757-800. doi: 10.3934/dcds.2017032

[17]

Elena Rossi. A justification of a LWR model based on a follow the leader description. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 579-591. doi: 10.3934/dcdss.2014.7.579

[18]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic and Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[19]

Helge Holden, Nils Henrik Risebro. The continuum limit of Follow-the-Leader models — a short proof. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 715-722. doi: 10.3934/dcds.2018031

[20]

Nicolas Forcadel, Cyril Imbert, Régis Monneau. Homogenization of some particle systems with two-body interactions and of the dislocation dynamics. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 785-826. doi: 10.3934/dcds.2009.23.785

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (136)
  • HTML views (148)
  • Cited by (0)

[Back to Top]