American Institute of Mathematical Sciences

Advanced
September  2014, 34(9): 3683-3702. doi: 10.3934/dcds.2014.34.3683

Effectual leadership in flocks with hierarchy and individual preference

Zhuchun Li 1,

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received  July 2013 Revised  December 2013 Published  March 2014

We study the emergent flocking behavior in a group of interacting particles following a Cucker-Smale type model with hierarchical leadership and individual effects. In this model, each individual adjusts its velocity to match that of its leaders and in meantime has a preferred acceleration if there is no local velocity consensus. We give some sufficient conditions on the range of parameters and initial configurations to have an effectual leadership, i.e., to guarantee the asymptotic convergence to a velocity agreement with the leader. To do this, we explore a special matrix norm for the flocking under hierarchical leadership.
Keywords: individual effects., hierarchical leadership, Cucker-Smale flocking.
Mathematics Subject Classification: Primary: 92D25, 92D50; Secondary: 91D3.
Citation: Zhuchun Li. Effectual leadership in flocks with hierarchy and individual preference. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3683-3702. doi: 10.3934/dcds.2014.34.3683
References:
[1]

S. Ahn, H.-O. Bae, S.-Y. Ha, Y. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628. doi: 10.1142/S0218202513500176.

[2]

S. Ahn, H. Choi, S.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Comm. Math. Sci., 10 (2012), 625-643. doi: 10.4310/CMS.2012.v10.n2.a10.

[3]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301. doi: 10.1063/1.3496895.

[4]

H. Bae, Y.-P. Choi, S.-Y. Ha and M. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177. doi: 10.1088/0951-7715/25/4/1155.

[5]

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Nat. Acad. Sci., 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.

[6]

N. Bellomo, H. Beresticky, F. Brezzi and J. P. Nadal, Mathematics and complexity in life and human sciences, Math. Models Methods Appl. Sci., 20 (2010), 1391-1395. doi: 10.1142/S0218202510004702.

[7]

N. Bellomo and F. Brezzi, Mathematics and complexity of multi-particle systems, Math. Models Methods Appl. Sci., 22 (2012), 1103001. doi: 10.1142/S0218202511030011.

[8]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236. doi: 10.1137/090757290.

[9]

I. D. Couzin, J. Krause, N. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516. doi: 10.1038/nature03236.

[10]

F. Cucker and J.-G. Dong, On the critical exponent for flocks under hierarchical leadership, Math. Models Methods in Appl. Sci., 19 (2009), 1391-1404. doi: 10.1142/S0218202509003851.

[11]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243. doi: 10.1109/TAC.2010.2042355.

[12]

F. Cucker and C. Huepe, Flocking with informed agents, MathS in Action, 1 (2008), 1-25. doi: 10.5802/msia.1.

[13]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002.

[14]

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

[15]

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

[16]

F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math., 71 (2010), 1307-1316. doi: 10.1137/100785910.

[17]

R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145. doi: 10.1007/s00220-010-1110-z.

[18]

M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via the Povzner-Boltzmann equation, Phys. D, 240 (2011), 21-31. doi: 10.1016/j.physd.2010.08.003.

[19]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Comm. Math. Sci., 7 (2009), 453-469. doi: 10.4310/CMS.2009.v7.n2.a9.

[20]

S.-Y. Ha and Z. Li, Complete synchronization of Kuramoto oscillators with hierarchical leadership, Comm. Math. Sci., 12 (2014), 485-508. doi: 10.4310/CMS.2014.v12.n3.a5.

[21]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean field limit, Comm. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.

[22]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Mod., 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[23]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.

[24]

A. Jadbabaie, J. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.

[25]

N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74. doi: 10.1109/JPROC.2006.887295.

[26]

Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, submitted to Quart. Appl. Math.

[27]

Z. Li, S.-Y. Ha and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 2013. doi: 10.1142/S0218202514500043.

[28]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174. doi: 10.1137/100791774.

[29]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9.

[30]

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

[31]

J. Park, H. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control, 55 (2010), 2617-2623. doi: 10.1109/TAC.2010.2061070.

[32]

L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guid. Control Dynam., 32 (2009), 527-537. doi: 10.2514/1.36269.

[33]

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

[34]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174. doi: 10.1137/S0036139903437424.

[35]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

show all references

References:
[1]

S. Ahn, H.-O. Bae, S.-Y. Ha, Y. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628. doi: 10.1142/S0218202513500176.

[2]

S. Ahn, H. Choi, S.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Comm. Math. Sci., 10 (2012), 625-643. doi: 10.4310/CMS.2012.v10.n2.a10.

[3]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301. doi: 10.1063/1.3496895.

[4]

H. Bae, Y.-P. Choi, S.-Y. Ha and M. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177. doi: 10.1088/0951-7715/25/4/1155.

[5]

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Nat. Acad. Sci., 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.

[6]

N. Bellomo, H. Beresticky, F. Brezzi and J. P. Nadal, Mathematics and complexity in life and human sciences, Math. Models Methods Appl. Sci., 20 (2010), 1391-1395. doi: 10.1142/S0218202510004702.

[7]

N. Bellomo and F. Brezzi, Mathematics and complexity of multi-particle systems, Math. Models Methods Appl. Sci., 22 (2012), 1103001. doi: 10.1142/S0218202511030011.

[8]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236. doi: 10.1137/090757290.

[9]

I. D. Couzin, J. Krause, N. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516. doi: 10.1038/nature03236.

[10]

F. Cucker and J.-G. Dong, On the critical exponent for flocks under hierarchical leadership, Math. Models Methods in Appl. Sci., 19 (2009), 1391-1404. doi: 10.1142/S0218202509003851.

[11]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control, 55 (2010), 1238-1243. doi: 10.1109/TAC.2010.2042355.

[12]

F. Cucker and C. Huepe, Flocking with informed agents, MathS in Action, 1 (2008), 1-25. doi: 10.5802/msia.1.

[13]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002.

[14]

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

[15]

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

[16]

F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math., 71 (2010), 1307-1316. doi: 10.1137/100785910.

[17]

R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145. doi: 10.1007/s00220-010-1110-z.

[18]

M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via the Povzner-Boltzmann equation, Phys. D, 240 (2011), 21-31. doi: 10.1016/j.physd.2010.08.003.

[19]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Comm. Math. Sci., 7 (2009), 453-469. doi: 10.4310/CMS.2009.v7.n2.a9.

[20]

S.-Y. Ha and Z. Li, Complete synchronization of Kuramoto oscillators with hierarchical leadership, Comm. Math. Sci., 12 (2014), 485-508. doi: 10.4310/CMS.2014.v12.n3.a5.

[21]

S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean field limit, Comm. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.

[22]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Mod., 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[23]

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 2013.

[24]

A. Jadbabaie, J. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.

[25]

N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74. doi: 10.1109/JPROC.2006.887295.

[26]

Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, submitted to Quart. Appl. Math.

[27]

Z. Li, S.-Y. Ha and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 2013. doi: 10.1142/S0218202514500043.

[28]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174. doi: 10.1137/100791774.

[29]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9.

[30]

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

[31]

J. Park, H. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control, 55 (2010), 2617-2623. doi: 10.1109/TAC.2010.2061070.

[32]

L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guid. Control Dynam., 32 (2009), 527-537. doi: 10.2514/1.36269.

[33]

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

[34]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174. doi: 10.1137/S0036139903437424.

[35]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

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