July  2014, 19(5): 1249-1278. doi: 10.3934/dcdsb.2014.19.1249

Phase transition and diffusion among socially interacting self-propelled agents

1. 

Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-7058, United States

2. 

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received  July 2012 Revised  October 2012 Published  April 2014

We consider a hydrodynamic model of swarming behavior derived from the kinetic description of a particle system combining a noisy Cucker-Smale consensus force and self-propulsion. In the large self-propulsive force limit, we provide evidence of a phase transition from disordered to ordered motion which manifests itself as a change of type of the limit model (from hyperbolic to diffusive) at the crossing of a critical noise intensity. In the hyperbolic regime, the resulting model, referred to as the `Self-Organized Hydrodynamics (SOH)', consists of a system of compressible Euler equations with a speed constraint. We show that the range of SOH models obtained by this limit is restricted. To waive this restriction, we compute the Navier-Stokes diffusive corrections to the hydrodynamic model. Adding these diffusive corrections, the limit of a large propulsive force yields unrestricted SOH models and offers an alternative to the derivation of the SOH using kinetic models with speed constraints.
Citation: Alethea B. T. Barbaro, Pierre Degond. Phase transition and diffusion among socially interacting self-propelled agents. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1249-1278. doi: 10.3934/dcdsb.2014.19.1249
References:
[1]

I. Aoki, A simulation study on the schooling mechanism in fish, Bulletin of the Japan Society of Scientific Fisheries, 48 (1982), 1081-1088. doi: 10.2331/suisan.48.1081.

[2]

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

[3]

A. B. T. Barbaro, J. A. Cañizo, J. A. Carrillo and P. Degond, Phase transitions in a Cucker-Smale model with self-propulsion, in preparation.

[4]

A. B. T. Barbaro, K. Taylor, P. F. Trethewey, L. Youseff and B. Birnir, Discrete and continuous models of the dynamics of pelagic fish: Applications to the capelin, Mathematics and Computers in Simulation, 79 (2009), 3397-3414. doi: 10.1016/j.matcom.2008.11.018.

[5]

A. Barbaro, B. Einarsson, B. Birnir, S. Sigurdsson, H. Valdimarsson, O. K. Pálsson, S. Sveinbjornsson and Th. Sigurdsson, Modelling and simulations of the migration of pelagic fish, ICES Journal of Marine Science, 66 (2009), 826-838. doi: 10.1093/icesjms/fsp067.

[6]

Ch. Becco, N. Vandewalle, J. Delcourt and P. Poncin, Experimental evidences of a structural and dynamical transition in fish schools, Physica A, 367 (2006), 487-793. doi: 10.1016/j.physa.2005.11.041.

[7]

E. Bertin, M. Droz and G. Grégoire, Boltzmann and hydrodynamic description for self-propelled particles, Phys. Rev. E, 74 (2006), 022101. doi: 10.1103/PhysRevE.74.022101.

[8]

E. Bertin, M. Droz and G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, J. Phys. A: Math. Theor., 42 (2009), 445001. doi: 10.1088/1751-8113/42/44/445001.

[9]

A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009.

[10]

M. Bostan and J. A. Carrillo, Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming, Math. Models Methods Appl. Sci., 23 (2013), 2353-2393. doi: 10.1142/S0218202513500346.

[11]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702.

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F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett., 25 (2012), 339-343. doi: 10.1016/j.aml.2011.09.011.

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H. Brézis, Analyse Fonctionnelle. Thèorie et Applications, Masson, Paris, 1983.

[14]

J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller and S. J. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406. doi: 10.1126/science.1125142.

[15]

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.

[16]

J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Models Methods Appl. Sci., 20 (2010), 1533-1552. doi: 10.1142/S0218202510004684.

[17]

Y-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Physica D, 232 (2007), 33-47. doi: 10.1016/j.physd.2007.05.007.

[18]

I. D. Couzin and N. R. Franks., Self-organized lane formation and optimized traffic flow in army ants, Proceedings of the Royal Society B: Biological Sciences, 270 (2003), 139-146. doi: 10.1098/rspb.2002.2210.

[19]

I. D. Couzin and J. Krause, Self-organization and collective behavior in vertebrates, Advance in the Study of Behavior, 32 (2003), 1-74. doi: 10.1016/S0065-3454(03)01001-5.

[20]

I. D. Couzin, J. Krause, R. James, G. D. Ruxton and N. R. Franks, Collective Memory and Spatial Sorting in Animal Groups, J. theor. Biol., 218 (2002), 1-11. doi: 10.1006/jtbi.2002.3065.

[21]

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.

[22]

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

[23]

P. Degond, Macroscopic limits of the Boltzmann equation: A review, in Modeling and Computational Methods for Kinetic Equations, (eds. P. Degond, L. Pareschi, and G. Russo), Birkhäuser Boston, (2004), 3-57.

[24]

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

[25]

P. Degond, J-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114. doi: 10.4310/MAA.2013.v20.n2.a1.

[26]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Mathematical Models and Methods in Applied Sciences, 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[27]

P. Degond, L. Navoret, R. Bon and D. Sanchez, Congestion in a macroscopic model of self-driven particles modeling gregariousness, J. Stat. Phys., 138 (2010), 85-125. doi: 10.1007/s10955-009-9879-x.

[28]

P. Degond and T. Yang, Diffusion in a continuum model of self-propelled particles with alignment interaction, Mathematical Models and Methods in Applied Sciences, 20 (2010), 1459-1490. doi: 10.1142/S0218202510004659.

[29]

J. Deseigne, O. Dauchot and H. Chaté, Collective motion of vibrated polar disks, Phys. Rev. Lett., 105 (2010), 098001. doi: 10.1103/PhysRevLett.105.098001.

[30]

M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302. doi: 10.1103/PhysRevLett.96.104302.

[31]

R. Eftimie, G. de Vries and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, Proc. Natl. Acad. Sci. USA, 104 (2007), 6974-6979. doi: 10.1073/pnas.0611483104.

[32]

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

[33]

A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci., 22 (2012), 1250011-1250051. doi: 10.1142/S021820251250011X.

[34]

A. Frouvelle and J.-G. Liu, Dynamics in a kinetic model of oriented particles with phase transition, SIAM J. Math Anal, 44 (2012), 791-826. doi: 10.1137/110823912.

[35]

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

[36]

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

[37]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[38]

C. K. Hemelrijk and H. Hildenbrandt, Some causes of the variable shape of flocks of birds, PLOS ONE, 6 (2011), e22479. doi: 10.1371/journal.pone.0022479.

[39]

Y. Katz, K. Tunstrom, C. C. Ioannou, C. Huepe and I. D. Couzin, Inferring the structure and dynamics of interactions in schooling fish, Proc. Nat. Acad. Sci., 108 (2011), 18720-18725. doi: 10.1073/pnas.1107583108.

[40]

R. Lukeman, Y.-X. Li and L. Edelstein-Keshet, Inferring individual rules from collective behavior, Proc. Nat. Acad. Sci. USA, 107 (2010), 12576-12580. doi: 10.1073/pnas.1001763107.

[41]

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

[42]

A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation, J. Math. Biol., 47 (2003), 353-389. doi: 10.1007/s00285-003-0209-7.

[43]

S. Mishra, A. Baskaran and C. Marchetti, Fluctuations and pattern formation in self-propelled particles, Phys. Rev. E, 81 (2010), 061916. doi: 10.1103/PhysRevE.81.061916.

[44]

S. Motsch and L. Navoret, Numerical simulations of a nonconservative hyperbolic system with geometric constraints describing swarming behavior, Multiscale Modeling and Simulation, 9 (2011), 1253-1275. doi: 10.1137/100794067.

[45]

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.

[46]

V. I. Ratushnaya, D. Bedeaux, V. L. Kulinskii and A. V. Zvelindovsky, Collective behavior of self-propelling particles with kinematic constraints: the relation between the discrete and the continuous description, Physica A, 381 (2007), 39-46. doi: 10.1016/j.physa.2007.03.045.

[47]

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

[48]

N. J. Suematsu, S. Nakata, A. Awazu and H. Nishimori, Collective behavior of inanimate boats, Phys. Rev. E, 81 (2010), 056210. doi: 10.1103/PhysRevE.81.056210.

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[50]

J. Toner, Y. Tu and S. Ramaswamy, Hydrodynamics and phases of flocks, Annals of Physics, 318 (2005), 170-244. doi: 10.1016/j.aop.2005.04.011.

[51]

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.

[52]

C. M. Topaz, A. 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.

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T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, 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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show all references

References:
[1]

I. Aoki, A simulation study on the schooling mechanism in fish, Bulletin of the Japan Society of Scientific Fisheries, 48 (1982), 1081-1088. doi: 10.2331/suisan.48.1081.

[2]

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

[3]

A. B. T. Barbaro, J. A. Cañizo, J. A. Carrillo and P. Degond, Phase transitions in a Cucker-Smale model with self-propulsion, in preparation.

[4]

A. B. T. Barbaro, K. Taylor, P. F. Trethewey, L. Youseff and B. Birnir, Discrete and continuous models of the dynamics of pelagic fish: Applications to the capelin, Mathematics and Computers in Simulation, 79 (2009), 3397-3414. doi: 10.1016/j.matcom.2008.11.018.

[5]

A. Barbaro, B. Einarsson, B. Birnir, S. Sigurdsson, H. Valdimarsson, O. K. Pálsson, S. Sveinbjornsson and Th. Sigurdsson, Modelling and simulations of the migration of pelagic fish, ICES Journal of Marine Science, 66 (2009), 826-838. doi: 10.1093/icesjms/fsp067.

[6]

Ch. Becco, N. Vandewalle, J. Delcourt and P. Poncin, Experimental evidences of a structural and dynamical transition in fish schools, Physica A, 367 (2006), 487-793. doi: 10.1016/j.physa.2005.11.041.

[7]

E. Bertin, M. Droz and G. Grégoire, Boltzmann and hydrodynamic description for self-propelled particles, Phys. Rev. E, 74 (2006), 022101. doi: 10.1103/PhysRevE.74.022101.

[8]

E. Bertin, M. Droz and G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, J. Phys. A: Math. Theor., 42 (2009), 445001. doi: 10.1088/1751-8113/42/44/445001.

[9]

A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009.

[10]

M. Bostan and J. A. Carrillo, Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming, Math. Models Methods Appl. Sci., 23 (2013), 2353-2393. doi: 10.1142/S0218202513500346.

[11]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702.

[12]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett., 25 (2012), 339-343. doi: 10.1016/j.aml.2011.09.011.

[13]

H. Brézis, Analyse Fonctionnelle. Thèorie et Applications, Masson, Paris, 1983.

[14]

J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller and S. J. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406. doi: 10.1126/science.1125142.

[15]

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.

[16]

J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Models Methods Appl. Sci., 20 (2010), 1533-1552. doi: 10.1142/S0218202510004684.

[17]

Y-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Physica D, 232 (2007), 33-47. doi: 10.1016/j.physd.2007.05.007.

[18]

I. D. Couzin and N. R. Franks., Self-organized lane formation and optimized traffic flow in army ants, Proceedings of the Royal Society B: Biological Sciences, 270 (2003), 139-146. doi: 10.1098/rspb.2002.2210.

[19]

I. D. Couzin and J. Krause, Self-organization and collective behavior in vertebrates, Advance in the Study of Behavior, 32 (2003), 1-74. doi: 10.1016/S0065-3454(03)01001-5.

[20]

I. D. Couzin, J. Krause, R. James, G. D. Ruxton and N. R. Franks, Collective Memory and Spatial Sorting in Animal Groups, J. theor. Biol., 218 (2002), 1-11. doi: 10.1006/jtbi.2002.3065.

[21]

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.

[22]

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

[23]

P. Degond, Macroscopic limits of the Boltzmann equation: A review, in Modeling and Computational Methods for Kinetic Equations, (eds. P. Degond, L. Pareschi, and G. Russo), Birkhäuser Boston, (2004), 3-57.

[24]

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

[25]

P. Degond, J-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114. doi: 10.4310/MAA.2013.v20.n2.a1.

[26]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Mathematical Models and Methods in Applied Sciences, 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[27]

P. Degond, L. Navoret, R. Bon and D. Sanchez, Congestion in a macroscopic model of self-driven particles modeling gregariousness, J. Stat. Phys., 138 (2010), 85-125. doi: 10.1007/s10955-009-9879-x.

[28]

P. Degond and T. Yang, Diffusion in a continuum model of self-propelled particles with alignment interaction, Mathematical Models and Methods in Applied Sciences, 20 (2010), 1459-1490. doi: 10.1142/S0218202510004659.

[29]

J. Deseigne, O. Dauchot and H. Chaté, Collective motion of vibrated polar disks, Phys. Rev. Lett., 105 (2010), 098001. doi: 10.1103/PhysRevLett.105.098001.

[30]

M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302. doi: 10.1103/PhysRevLett.96.104302.

[31]

R. Eftimie, G. de Vries and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, Proc. Natl. Acad. Sci. USA, 104 (2007), 6974-6979. doi: 10.1073/pnas.0611483104.

[32]

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

[33]

A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci., 22 (2012), 1250011-1250051. doi: 10.1142/S021820251250011X.

[34]

A. Frouvelle and J.-G. Liu, Dynamics in a kinetic model of oriented particles with phase transition, SIAM J. Math Anal, 44 (2012), 791-826. doi: 10.1137/110823912.

[35]

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

[36]

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

[37]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[38]

C. K. Hemelrijk and H. Hildenbrandt, Some causes of the variable shape of flocks of birds, PLOS ONE, 6 (2011), e22479. doi: 10.1371/journal.pone.0022479.

[39]

Y. Katz, K. Tunstrom, C. C. Ioannou, C. Huepe and I. D. Couzin, Inferring the structure and dynamics of interactions in schooling fish, Proc. Nat. Acad. Sci., 108 (2011), 18720-18725. doi: 10.1073/pnas.1107583108.

[40]

R. Lukeman, Y.-X. Li and L. Edelstein-Keshet, Inferring individual rules from collective behavior, Proc. Nat. Acad. Sci. USA, 107 (2010), 12576-12580. doi: 10.1073/pnas.1001763107.

[41]

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

[42]

A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation, J. Math. Biol., 47 (2003), 353-389. doi: 10.1007/s00285-003-0209-7.

[43]

S. Mishra, A. Baskaran and C. Marchetti, Fluctuations and pattern formation in self-propelled particles, Phys. Rev. E, 81 (2010), 061916. doi: 10.1103/PhysRevE.81.061916.

[44]

S. Motsch and L. Navoret, Numerical simulations of a nonconservative hyperbolic system with geometric constraints describing swarming behavior, Multiscale Modeling and Simulation, 9 (2011), 1253-1275. doi: 10.1137/100794067.

[45]

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.

[46]

V. I. Ratushnaya, D. Bedeaux, V. L. Kulinskii and A. V. Zvelindovsky, Collective behavior of self-propelling particles with kinematic constraints: the relation between the discrete and the continuous description, Physica A, 381 (2007), 39-46. doi: 10.1016/j.physa.2007.03.045.

[47]

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

[48]

N. J. Suematsu, S. Nakata, A. Awazu and H. Nishimori, Collective behavior of inanimate boats, Phys. Rev. E, 81 (2010), 056210. doi: 10.1103/PhysRevE.81.056210.

[49]

J. Toner and Y. Tu, Flocks, Long-range order in a two-dimensional dynamical XY model: how birds fly together, Phys. Rev. Lett., 75 (1995), 4326-4329. doi: 10.1103/PhysRevLett.75.4326.

[50]

J. Toner, Y. Tu and S. Ramaswamy, Hydrodynamics and phases of flocks, Annals of Physics, 318 (2005), 170-244. doi: 10.1016/j.aop.2005.04.011.

[51]

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.

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