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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

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