American Institute of Mathematical Sciences

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June  2017, 22(4): 1295-1327. doi: 10.3934/dcdsb.2017063

A continuum model for nematic alignment of self-propelled particles

Pierre Degond 1,, , Angelika Manhart 2, and  Hui Yu 3,

1. 

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

Faculty of Mathematics, University of Vienna, Austria, Current affiliation: Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
3. 

Université de Toulouse; UPS, INSA, UT1, UTM, CNRS; Institut de Mathématiques de Toulouse, UMR 5219, France, and Department of Mathematics, Imperial College London, United Kingdom, Current affiliation: Inst. für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, 52056, Germany

* Corresponding author: Pierre Degond

Received  May 2016 Revised  October 2016 Published  February 2017

A continuum model for a population of self-propelled particles interacting through nematic alignment is derived from an individual-based model. The methodology consists of introducing a hydrodynamic scaling of the corresponding mean field kinetic equation. The resulting perturbation problem is solved thanks to the concept of generalized collision invariants. It yields a hyperbolic but non-conservative system of equations for the nematic mean direction of the flow and the densities of particles flowing parallel or anti-parallel to this mean direction. Diffusive terms are introduced under a weakly non-local interaction assumption and the diffusion coefficient is proven to be positive. An application to the modeling of myxobacteria is outlined.

Keywords: Self-propelled particles, nematic alignment, hydrodynamic limit, generalized collision invariant, diffusion correction, weakly non-local interaction, myxobacteria.
Mathematics Subject Classification: 35L60, 35K55, 35Q70, 82C05, 82C22, 82C70, 92D50.
Citation: Pierre Degond, Angelika Manhart, Hui Yu. A continuum model for nematic alignment of self-propelled particles. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1295-1327. doi: 10.3934/dcdsb.2017063
References:
[1]

I. Aoki, A simulation study on the schooling mechanism in fish, Bull. Jpn. Soc. Sci. Fish., 48 (1982), 1081-1088.  doi: 10.2331/suisan.48.1081.  Google Scholar

[2]

J. P. Arcede and E. A. Cabral, An equivalent definition for the backwards Itô integral, Thai J. Math., 9 (2011), 619-630.   Google Scholar

[3]

A. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1249-1278.  doi: 10.3934/dcdsb.2014.19.1249.  Google Scholar

[4]

A. Baskaran and M. Marchetti, Hydrodynamics of self-propelled hard rods, Phys. Rev. E, 77 (2008), 011920, 9pp.  doi: 10.1103/PhysRevE.77.011920.  Google Scholar

[5]

A. Baskaran and C. M. Marchetti, Nonequilibrium statistical mechanics of self-propelled hard rods, J. Stat. Mech.: Theory Exp., 2010 (2010), P04019.  doi: 10.1088/1742-5468/2010/04/P04019.  Google Scholar

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E. Ben-Jacob, I. Cohen and H. Levine, Cooperative self-organization of microorganisms, Adv. in Phys., 49 (2000), 395-554.  doi: 10.1080/000187300405228.  Google Scholar

[7]

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.  Google Scholar

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U. Börner, A. Deutsch, H. Reichenbach and M. Bär, Rippling patterns in aggregates of myxobacteria arise from cell-cell collisions, Phys. Rev. Lett., 89 (2002), 078101.   Google Scholar

[9]

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

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J. Carrillo, M. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378.  doi: 10.3934/krm.2009.2.363.  Google Scholar

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A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini and M. Viale, Scale-free correlations in starling flocks, Proc. Natl. Acad. Sci. USA, 107 (2010), 11865-11870.  doi: 10.1073/pnas.1005766107.  Google Scholar

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H. Chaté, F. Ginelli, G. Grégoire and F. Raynaud, Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E, 77 (2008), 046113.   Google Scholar

[13]

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

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

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P. Degond, F. Delebecque and D. Peurichard, Continuum model for linked fibers with alignment interactions, Math. Models Methods Appl. Sci., 26 (2016), 269-318.  doi: 10.1142/S0218202516400030.  Google Scholar

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P. Degond, G. Dimarco and T. B. N. Mac, Hydrodynamics of the Kuramoto-Vicsek model of rotating self-propelled particles, Math. Models Methods Appl. Sci., 24 (2014), 277-325.  doi: 10.1142/S0218202513400095.  Google Scholar

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P. Degond, G. Dimarco, T. B. N. Mac and N. Wang, Macroscopic models of collective motion with repulsion, Commun. Math. Sci., 13 (2015), 1615-1638.  doi: 10.4310/CMS.2015.v13.n6.a12.  Google Scholar

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P. Degond, A. 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

[20]

P. Degond, A. Frouvelle and J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Arch. Ration. Mech. Anal., 216 (2015), 63-115.  doi: 10.1007/s00205-014-0800-7.  Google Scholar

[21]

P. Degond and J.-G. Liu, Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation, Math. Models Methods Appl. Sci., 22 (2012), 114001, 18pp.  doi: 10.1142/S021820251140001X.  Google Scholar

[22]

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.  Google Scholar

[23]

P. Degond, J.-G. Liu and C. Ringhofer, Evolution of wealth in a nonconservative economy driven by local Nash equilibria, Philos. Trans. A, 372 (2014), 20130394, 15pp.  doi: 10.1098/rsta.2013.0394.  Google Scholar

[24]

P. Degond, A. Manhart and H. Yu, An age-structured continuum model for myxobacteria, In preparation, 372 (2017).   Google Scholar

[25]

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

[26]

P. Degond and L. Navoret, A multi-layer model for self-propelled disks interacting through alignment and volume exclusion, Math. Models Methods Appl. Sci., 25 (2015), 2439-2475.  doi: 10.1142/S021820251540014X.  Google Scholar

[27]

P. Degond and H. Yu, Self-organized hydrodynamics in an annular domain: Modal analysis and nonlinear effects, Math. Models Methods Appl. Sci., 25 (2015), 495-519.  doi: 10.1142/S0218202515400047.  Google Scholar

[28]

P. Dhar, T. Fischer, Y. Wang, T. Mallouk, W. Paxton and A. Sen, Autonomously moving nanorods at a viscous interface, Nano Lett., 6 (2006), 66-72.  doi: 10.1021/nl052027s.  Google Scholar

[29]

M. Dworkin, Recent advances in the social and developmental biology of the myxobacteria, Microbiol. Rev., 60 (1996), 70-102.   Google Scholar

[30]

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, 40pp.  doi: 10.1142/S021820251250011X.  Google Scholar

[31]

F. Ginelli, F. Peruani, M. Bär and H. Chaté, Large-scale collective properties of self-propelled rods, Phys. Rev. Lett., 104 (2010), 184502.  doi: 10.1103/PhysRevLett.104.184502.  Google Scholar

[32]

D. Helbing, A. Johansson and H. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E, 75 (2007), 046109.  doi: 10.1103/PhysRevE.75.046109.  Google Scholar

[33]

O. Igoshin, A. Mogilner, R. Welch, D. Kaiser and G. Oster, Pattern formation and traveling waves in myxobacteria: Theory and modeling, Proc. Natl. Acad. Sci. USA, 98 (2001), 14913-14918.  doi: 10.1073/pnas.221579598.  Google Scholar

[34]

O. Igoshin, R. Welch, D. Kaiser and G. Oster, Waves and aggregation patterns in myxobacteria, Proc. Natl. Acad. Sci. USA, 101 (2004), 4256-4261.  doi: 10.1073/pnas.0400704101.  Google Scholar

[35]

D. Kaiser, Coupling cell movement to multicellular development in myxobacteria, Nat. Rev. Microbiol., 1 (2003), 45-54.  doi: 10.1038/nrmicro733.  Google Scholar

[36]

P. Lançon, G. Batrouni, L. Lobry and N. Ostrowsky, Drift without flux: Brownian walker with a space-dependent diffusion coefficient, Europhys. Lett., 54 (2001), 28.   Google Scholar

[37]

P. Lançon, G. Batrouni, L. Lobry and N. Ostrowsky, Brownian walker in a confined geometry leading to a space-dependent diffusion coefficient, Physica A, 304 (2002), 65.   Google Scholar

[38]

A. Lau and T. Lubensky, State-dependent diffusion: Thermodynamic consistency and its path integral formulation, Phys. Rev. E, 76 (2007), 011123, 17pp.  doi: 10.1103/PhysRevE.76.011123.  Google Scholar

[39]

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

[40]

M. Moussaïd, N. Perozo, S. Garnier, D. Helbing and G. Theraulaz, The walking behaviour of pedestrian social groups and its impact on crowd dynamics, PLoS ONE, 5 (2010), 1-7.   Google Scholar

[41]

N. Jiang, L. Xiong and T.-F. Zhang, Hydrodynamic Limits of the Kinetic Self-Organized Models, SIAM J. Math. Anal., 48 (2016), 3383-3411.  doi: 10.1137/15M1035665.  Google Scholar

[42]

V. Narayan, S. Ramaswamy and N. Menon, Long-lived giant number fluctuations in a swarming granular nematic, Science, 317 (2007), 105-108.  doi: 10.1126/science.1140414.  Google Scholar

[43]

S. Ngo, F. Ginelli and H. Chaté, Competing ferromagnetic and nematic alignment in self-propelled polar particles, Phys. Rev. E, 86 (2012), 050101(R).  doi: 10.1103/PhysRevE.86.050101.  Google Scholar

[44] J. Parrish and W. Hamner, Animal Groups in Three Dimensions: How Species Aggregate, Cambridge University Press, 1997.  doi: 10.1017/CBO9780511601156.  Google Scholar
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F. Peruani, F. Ginelli, M. Bär and H. Chaté, Polar vs. apolar alignment in systems of polar self-propelled particles, J. Phys. Conf. Ser., 297 (2011), 012014.  doi: 10.1088/1742-6596/297/1/012014.  Google Scholar

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C. Reynold, Flocks, herds, and schools: A distributed behavioral model, SIGGRAPH Comput. Graph., 21 (1987), 25-34.  doi: 10.1145/37401.37406.  Google Scholar

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A. Sokolov, I. Aranson, J. Kessler and R. Goldstein, Concentration dependence of the collective dynamics of swimming bacteria, Phys. Rev. Lett., 98 (2007), 158102.  doi: 10.1103/PhysRevLett.98.158102.  Google Scholar

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J. Toner and Y. Tu, Long-range order in a two-dimensional dynamical xy model: How birds fly together, Phys. Rev. Lett., 75 (1995), 4326-4329.   Google Scholar

[49]

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.  Google Scholar

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R. Welch and D. Kaiser, Cell behavior in traveling wave patterns of myxobacteria, Proc. Natl. Acad. Sci. USA, 98 (2001), 14907-14912.  doi: 10.1073/pnas.261574598.  Google Scholar

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show all references

References:
[1]

I. Aoki, A simulation study on the schooling mechanism in fish, Bull. Jpn. Soc. Sci. Fish., 48 (1982), 1081-1088.  doi: 10.2331/suisan.48.1081.  Google Scholar

[2]

J. P. Arcede and E. A. Cabral, An equivalent definition for the backwards Itô integral, Thai J. Math., 9 (2011), 619-630.   Google Scholar

[3]

A. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1249-1278.  doi: 10.3934/dcdsb.2014.19.1249.  Google Scholar

[4]

A. Baskaran and M. Marchetti, Hydrodynamics of self-propelled hard rods, Phys. Rev. E, 77 (2008), 011920, 9pp.  doi: 10.1103/PhysRevE.77.011920.  Google Scholar

[5]

A. Baskaran and C. M. Marchetti, Nonequilibrium statistical mechanics of self-propelled hard rods, J. Stat. Mech.: Theory Exp., 2010 (2010), P04019.  doi: 10.1088/1742-5468/2010/04/P04019.  Google Scholar

[6]

E. Ben-Jacob, I. Cohen and H. Levine, Cooperative self-organization of microorganisms, Adv. in Phys., 49 (2000), 395-554.  doi: 10.1080/000187300405228.  Google Scholar

[7]

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.  Google Scholar

[8]

U. Börner, A. Deutsch, H. Reichenbach and M. Bär, Rippling patterns in aggregates of myxobacteria arise from cell-cell collisions, Phys. Rev. Lett., 89 (2002), 078101.   Google Scholar

[9]

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

[10]

J. Carrillo, M. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378.  doi: 10.3934/krm.2009.2.363.  Google Scholar

[11]

A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini and M. Viale, Scale-free correlations in starling flocks, Proc. Natl. Acad. Sci. USA, 107 (2010), 11865-11870.  doi: 10.1073/pnas.1005766107.  Google Scholar

[12]

H. Chaté, F. Ginelli, G. Grégoire and F. Raynaud, Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E, 77 (2008), 046113.   Google Scholar

[13]

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

[14]

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

[15] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2 edition, Oxford University Press, United Kingdom, 1993.   Google Scholar
[16]

P. Degond, F. Delebecque and D. Peurichard, Continuum model for linked fibers with alignment interactions, Math. Models Methods Appl. Sci., 26 (2016), 269-318.  doi: 10.1142/S0218202516400030.  Google Scholar

[17]

P. Degond, G. Dimarco and T. B. N. Mac, Hydrodynamics of the Kuramoto-Vicsek model of rotating self-propelled particles, Math. Models Methods Appl. Sci., 24 (2014), 277-325.  doi: 10.1142/S0218202513400095.  Google Scholar

[18]

P. Degond, G. Dimarco, T. B. N. Mac and N. Wang, Macroscopic models of collective motion with repulsion, Commun. Math. Sci., 13 (2015), 1615-1638.  doi: 10.4310/CMS.2015.v13.n6.a12.  Google Scholar

[19]

P. Degond, A. 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

[20]

P. Degond, A. Frouvelle and J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Arch. Ration. Mech. Anal., 216 (2015), 63-115.  doi: 10.1007/s00205-014-0800-7.  Google Scholar

[21]

P. Degond and J.-G. Liu, Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation, Math. Models Methods Appl. Sci., 22 (2012), 114001, 18pp.  doi: 10.1142/S021820251140001X.  Google Scholar

[22]

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.  Google Scholar

[23]

P. Degond, J.-G. Liu and C. Ringhofer, Evolution of wealth in a nonconservative economy driven by local Nash equilibria, Philos. Trans. A, 372 (2014), 20130394, 15pp.  doi: 10.1098/rsta.2013.0394.  Google Scholar

[24]

P. Degond, A. Manhart and H. Yu, An age-structured continuum model for myxobacteria, In preparation, 372 (2017).   Google Scholar

[25]

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

[26]

P. Degond and L. Navoret, A multi-layer model for self-propelled disks interacting through alignment and volume exclusion, Math. Models Methods Appl. Sci., 25 (2015), 2439-2475.  doi: 10.1142/S021820251540014X.  Google Scholar

[27]

P. Degond and H. Yu, Self-organized hydrodynamics in an annular domain: Modal analysis and nonlinear effects, Math. Models Methods Appl. Sci., 25 (2015), 495-519.  doi: 10.1142/S0218202515400047.  Google Scholar

[28]

P. Dhar, T. Fischer, Y. Wang, T. Mallouk, W. Paxton and A. Sen, Autonomously moving nanorods at a viscous interface, Nano Lett., 6 (2006), 66-72.  doi: 10.1021/nl052027s.  Google Scholar

[29]

M. Dworkin, Recent advances in the social and developmental biology of the myxobacteria, Microbiol. Rev., 60 (1996), 70-102.   Google Scholar

[30]

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, 40pp.  doi: 10.1142/S021820251250011X.  Google Scholar

[31]

F. Ginelli, F. Peruani, M. Bär and H. Chaté, Large-scale collective properties of self-propelled rods, Phys. Rev. Lett., 104 (2010), 184502.  doi: 10.1103/PhysRevLett.104.184502.  Google Scholar

[32]

D. Helbing, A. Johansson and H. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E, 75 (2007), 046109.  doi: 10.1103/PhysRevE.75.046109.  Google Scholar

[33]

O. Igoshin, A. Mogilner, R. Welch, D. Kaiser and G. Oster, Pattern formation and traveling waves in myxobacteria: Theory and modeling, Proc. Natl. Acad. Sci. USA, 98 (2001), 14913-14918.  doi: 10.1073/pnas.221579598.  Google Scholar

[34]

O. Igoshin, R. Welch, D. Kaiser and G. Oster, Waves and aggregation patterns in myxobacteria, Proc. Natl. Acad. Sci. USA, 101 (2004), 4256-4261.  doi: 10.1073/pnas.0400704101.  Google Scholar

[35]

D. Kaiser, Coupling cell movement to multicellular development in myxobacteria, Nat. Rev. Microbiol., 1 (2003), 45-54.  doi: 10.1038/nrmicro733.  Google Scholar

[36]

P. Lançon, G. Batrouni, L. Lobry and N. Ostrowsky, Drift without flux: Brownian walker with a space-dependent diffusion coefficient, Europhys. Lett., 54 (2001), 28.   Google Scholar

[37]

P. Lançon, G. Batrouni, L. Lobry and N. Ostrowsky, Brownian walker in a confined geometry leading to a space-dependent diffusion coefficient, Physica A, 304 (2002), 65.   Google Scholar

[38]

A. Lau and T. Lubensky, State-dependent diffusion: Thermodynamic consistency and its path integral formulation, Phys. Rev. E, 76 (2007), 011123, 17pp.  doi: 10.1103/PhysRevE.76.011123.  Google Scholar

[39]

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

[40]

M. Moussaïd, N. Perozo, S. Garnier, D. Helbing and G. Theraulaz, The walking behaviour of pedestrian social groups and its impact on crowd dynamics, PLoS ONE, 5 (2010), 1-7.   Google Scholar

[41]

N. Jiang, L. Xiong and T.-F. Zhang, Hydrodynamic Limits of the Kinetic Self-Organized Models, SIAM J. Math. Anal., 48 (2016), 3383-3411.  doi: 10.1137/15M1035665.  Google Scholar

[42]

V. Narayan, S. Ramaswamy and N. Menon, Long-lived giant number fluctuations in a swarming granular nematic, Science, 317 (2007), 105-108.  doi: 10.1126/science.1140414.  Google Scholar

[43]

S. Ngo, F. Ginelli and H. Chaté, Competing ferromagnetic and nematic alignment in self-propelled polar particles, Phys. Rev. E, 86 (2012), 050101(R).  doi: 10.1103/PhysRevE.86.050101.  Google Scholar

[44] J. Parrish and W. Hamner, Animal Groups in Three Dimensions: How Species Aggregate, Cambridge University Press, 1997.  doi: 10.1017/CBO9780511601156.  Google Scholar
[45]

F. Peruani, F. Ginelli, M. Bär and H. Chaté, Polar vs. apolar alignment in systems of polar self-propelled particles, J. Phys. Conf. Ser., 297 (2011), 012014.  doi: 10.1088/1742-6596/297/1/012014.  Google Scholar

[46]

C. Reynold, Flocks, herds, and schools: A distributed behavioral model, SIGGRAPH Comput. Graph., 21 (1987), 25-34.  doi: 10.1145/37401.37406.  Google Scholar

[47]

A. Sokolov, I. Aranson, J. Kessler and R. Goldstein, Concentration dependence of the collective dynamics of swimming bacteria, Phys. Rev. Lett., 98 (2007), 158102.  doi: 10.1103/PhysRevLett.98.158102.  Google Scholar

[48]

J. Toner and Y. Tu, Long-range order in a two-dimensional dynamical xy model: How birds fly together, Phys. Rev. Lett., 75 (1995), 4326-4329.   Google Scholar

[49]

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Figure 1.  $M_0(\theta)$ for $\kappa=0.5, 2,10$ (red-dotted, black-solid, blue-dashed).
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Figure 2.  $g(\theta)$ for $\kappa=0.5, 2,10$ (red-dotted, black-solid, blue-dashed).
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Figure 3.  Local dynamics for $\lambda(\rho)$ given by 82. The arrows mark the flow field in the $(\rho_+,\rho_-)$ plane. The red-dotted and green-dashed lines show the values for which $\lambda(\rho_+)\rho_--\lambda(\rho_-)\rho_+=0$. The blue-solid line shows the threshold values $\rho_++\rho_-=2\sqrt{\frac{\lambda_0}{\lambda_1}}$.
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