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

A continuum model for nematic alignment of self-propelled particles

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.

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:
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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.

[2]

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

[3]

A. Barbaro, 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.

[4]

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

[5]

A. Baskaran, 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.

[6]

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

[7]

E. Bertin, M. Droz, 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.

[8]

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

[9]

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

[10]

J. Carrillo, M. D'Orsogna, 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.

[11]

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

[12]

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

[13]

Y. Chuang, M. D'Orsogna, D. Marthaler, A. Bertozzi, 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.

[14]

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

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

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

[17]

P. Degond, G. Dimarco, 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.

[18]

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

[19]

P. Degond, A. Frouvelle, 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.

[20]

P. Degond, A. Frouvelle, 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.

[21]

P. Degond, 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.

[22]

P. Degond, J.-G. Liu, S. Motsch, 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.

[23]

P. Degond, J.-G. Liu, 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.

[24]

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

[25]

P. Degond, S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[26]

P. Degond, 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.

[27]

P. Degond, 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.

[28]

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

[29]

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

[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.

[31]

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

[32]

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

[33]

O. Igoshin, A. Mogilner, R. Welch, D. Kaiser, 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.

[34]

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

[35]

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

[36]

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

[37]

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

[38]

A. Lau, 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.

[39]

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

[40]

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

[41]

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

[42]

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

[43]

S. Ngo, F. Ginelli, 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.

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

F. Peruani, F. Ginelli, M. Bär, 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.

[46]

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

[47]

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

[48]

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

[49]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, 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.

[50]

T. Vicsek, A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004.

[51]

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

[52]

Y. Wu, A. Kaiser, Y. Jiang, M. Alber, Periodic reversal of direction allows myxobacteria to swarm, Proc. Natl. Acad. Sci. USA, 106 (2008), 1222-1227. doi: 10.1073/pnas.0811662106.

[53]

H.-P. Zhang, A. Be'er, E.-L. Florin, H. Swinney, Collective motion and density fluctuations in bacterial colonies, Proc. Natl. Acad. Sci. USA, 107 (2010), 13626-13630. doi: 10.1073/pnas.1001651107.

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.

[2]

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

[3]

A. Barbaro, 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.

[4]

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

[5]

A. Baskaran, 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.

[6]

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

[7]

E. Bertin, M. Droz, 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.

[8]

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

[9]

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

[10]

J. Carrillo, M. D'Orsogna, 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.

[11]

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

[12]

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

[13]

Y. Chuang, M. D'Orsogna, D. Marthaler, A. Bertozzi, 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.

[14]

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

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

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

[17]

P. Degond, G. Dimarco, 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.

[18]

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

[19]

P. Degond, A. Frouvelle, 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.

[20]

P. Degond, A. Frouvelle, 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.

[21]

P. Degond, 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.

[22]

P. Degond, J.-G. Liu, S. Motsch, 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.

[23]

P. Degond, J.-G. Liu, 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.

[24]

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

[25]

P. Degond, S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[26]

P. Degond, 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.

[27]

P. Degond, 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.

[28]

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

[29]

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

[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.

[31]

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

[32]

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

[33]

O. Igoshin, A. Mogilner, R. Welch, D. Kaiser, 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.

[34]

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

[35]

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

[36]

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

[37]

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

[38]

A. Lau, 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.

[39]

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

[40]

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

[41]

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

[42]

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

[43]

S. Ngo, F. Ginelli, 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.

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

F. Peruani, F. Ginelli, M. Bär, 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.

[46]

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

[47]

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

[48]

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

[49]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, 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.

[50]

T. Vicsek, A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004.

[51]

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

[52]

Y. Wu, A. Kaiser, Y. Jiang, M. Alber, Periodic reversal of direction allows myxobacteria to swarm, Proc. Natl. Acad. Sci. USA, 106 (2008), 1222-1227. doi: 10.1073/pnas.0811662106.

[53]

H.-P. Zhang, A. Be'er, E.-L. Florin, H. Swinney, Collective motion and density fluctuations in bacterial colonies, Proc. Natl. Acad. Sci. USA, 107 (2010), 13626-13630. doi: 10.1073/pnas.1001651107.

Figure 1.  $M_0(\theta)$ for $\kappa=0.5, 2,10$ (red-dotted, black-solid, blue-dashed).
Figure 2.  $g(\theta)$ for $\kappa=0.5, 2,10$ (red-dotted, black-solid, blue-dashed).
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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José A. Carrillo, M. R. D’Orsogna, V. Panferov. Double milling in self-propelled swarms from kinetic theory. Kinetic & Related Models, 2009, 2 (2) : 363-378. doi: 10.3934/krm.2009.2.363

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