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A kinetic model for the formation of swarms with nonlinear interactions
1. | INRIA, ANGE Project-Team, Rocquencourt, F-78153 Le Chesnay Cedex, France |
2. | Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa |
References:
[1] |
L. Arlotti, A. Deutsch and M. Lachowicz, A discrete Boltzmann-type model of swarming, Math. Comput. Modelling, 41 (2005), 1193-1201.
doi: 10.1016/j.mcm.2005.05.011. |
[2] |
J. Banasiak and M. Lachowicz, On a macroscopic limit of a kinetic model of alignment, Math. Models Methods Appl. Sci., 23 (2013), 2647-2670.
doi: 10.1142/S0218202513500425. |
[3] |
E. Ben-Naim, Opinion dynamics: Rise and fall of political parties, EPL (Europhysics Letters), 69 (2005), p671.
doi: 10.1209/epl/i2004-10421-1. |
[4] |
L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation, M2AN Math. Model. Numer. Anal., 43 (2009), 507-522.
doi: 10.1051/m2an/2009004. |
[5] |
J. A. Carrillo, M. R. 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. |
[6] |
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. |
[7] |
J. A. Carrillo, S. Martin and V. Panferov, A new interaction potential for swarming models, Phys. D, 260 (2013), 112-126.
doi: 10.1016/j.physd.2013.02.004. |
[8] |
F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[9] |
P. Daskalopoulos and M. Del Pino, On nonlinear parabolic equations of very fast diffusion, Arch. Rational Mech. Anal., 137 (1997), 363-380.
doi: 10.1007/s002050050033. |
[10] |
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. |
[11] |
B. Després, F. Lagoutière, E. Labourasse and I. Marmajou, An antidissipative transport scheme on unstructured meshes for multicomponent flows, Int. J. Finite Vol., 7 (2010), 36pp. |
[12] |
L. Edelstein-Keshet, J. Watmough and D. Grunbaum, Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts, J. Math. Biol., 36 (1998), 515-549.
doi: 10.1007/s002850050112. |
[13] |
R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review, J. Math. Biol., 65 (2012), 35-75.
doi: 10.1007/s00285-011-0452-2. |
[14] |
R. Erban and J. Haskovec, From individual to collective behaviour of coupled velocity jump processes: a locust example, Kinet. Relat. Models, 5 (2012), 817-842.
doi: 10.3934/krm.2012.5.817. |
[15] |
E. Frénod and O. Sire, An explanatory model to validate the way water activity rules periodic terrace generation in proteus mirabilis swarm, J. Math. Biol., 59 (2009), 439-466.
doi: 10.1007/s00285-008-0235-6. |
[16] |
E. Geigant and M. Stoll, Bifurcation analysis of an orientational aggregation model, J. Math. Biol., 46 (2003), 537-563.
doi: 10.1007/s00285-002-0187-1. |
[17] |
M. Greenwood and R. Chapman, Differences in numbers of sensilla on the antennae of solitarious and gregarious locusta migratoria l.(orthoptera: Acrididae), International Journal of Insect Morphology and Embryology, 13 (1984), 295-301.
doi: 10.1016/0020-7322(84)90004-7. |
[18] |
D. Grünbaum, K. Chan, E. Tobin and M. T. Nishizaki, Non-linear advection-diffusion equations approximate swarming but not schooling populations, Math. Biosci., 214 (2008), 38-48.
doi: 10.1016/j.mbs.2008.06.002. |
[19] |
S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.
doi: 10.3934/krm.2008.1.415. |
[20] |
S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454 (electronic).
doi: 10.1137/S1064827598334599. |
[21] |
K. Kang, B. Perthame, A. Stevens and J. J. L. Velázquez, Corrigendum to "An integro-differential equation model for alignment and orientational aggregation'' [J. Differential Equations 246 (4) (2009) 1387-1421] [mr2488690], J. Differential Equations, 252 (2012), 5125-5128.
doi: 10.1016/j.jde.2008.11.006. |
[22] |
S. Kaniel and M. Shinbrot, The Boltzmann equation. I. Uniqueness and local existence, Comm. Math. Phys., 58 (1978), 65-84. |
[23] |
R. Mach and F. Schweitzer, Modeling vortex swarming in daphnia, Bull. Math. Biol., 69 (2007), 539-562.
doi: 10.1007/s11538-006-9135-3. |
[24] |
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. |
[25] |
A. H. Øien, Daphnicle dynamics based on kinetic theory: An analogue-modelling of swarming and behaviour of daphnia, Bull. Math. Biol., 66 (2004), 1-46.
doi: 10.1016/S0092-8240(03)00065-X. |
[26] |
H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[27] |
L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013. |
[28] |
F. Peruani, T. Klauss, A. Deutsch and A. Voss-Boehme, Traffic jams, gliders, and bands in the quest for collective motion of self-propelled particles, Phys. Rev. Lett., 106 (2011), 128101.
doi: 10.1103/PhysRevLett.106.128101. |
[29] |
F. Peruani, J. Starruß, V. Jakovljevic, L. Søgaard-Andersen, A. Deutsch and M. Bär, Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria, Phys. Rev. Lett., 108 (2012), 098102.
doi: 10.1103/PhysRevLett.108.098102. |
[30] |
P.-A. Raviart and J.-M. Thomas, Introduction à L'analyse Numérique des Équations Aux Dérivées Partielles, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983. |
[31] |
F. Salvarani and G. Toscani, The diffusive limit of Carleman-type models in the range of very fast diffusion equations, J. Evol. Equ., 9 (2009), 67-80.
doi: 10.1007/s00028-009-0005-y. |
[32] |
F. Salvarani and J. L. Vázquez, The diffusive limit for Carleman-type kinetic models, Nonlinearity, 18 (2005), 1223-1248.
doi: 10.1088/0951-7715/18/3/015. |
[33] |
S. J. Simpson, D. Raubenheimer, S. T. Behmer, A. Whitworth and G. A. Wright, A comparison of nutritional regulation in solitarious- and gregarious-phase nymphs of the desert locust schistocerca gregaria, Journal of Experimental Biology, 205 (2002), 121-129. |
[34] |
S. J. Simpson, A. R. McCaffery and B. F. Hägele, A behavioural analysis of phase change in the desert locust, Biological Reviews, 74 (1999), 461-480. |
[35] |
G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.
doi: 10.4310/CMS.2006.v4.n3.a1. |
[36] |
J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, vol. 33 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2006, Equations of porous medium type.
doi: 10.1093/acprof:oso/9780199202973.001.0001. |
[37] |
Y. Wu, Y. Jiang, D. Kaiser and M. Alber, Social interactions in myxobacterial swarming, PLoS Comput. Biol., 3 (2007), 2546-2558.
doi: 10.1371/journal.pcbi.0030253. |
[38] |
T. I. Zohdi, Mechanistic modeling of swarms, Comput. Methods Appl. Mech. Engrg., 198 (2009), 2039-2051.
doi: 10.1016/j.cma.2008.12.029. |
show all references
References:
[1] |
L. Arlotti, A. Deutsch and M. Lachowicz, A discrete Boltzmann-type model of swarming, Math. Comput. Modelling, 41 (2005), 1193-1201.
doi: 10.1016/j.mcm.2005.05.011. |
[2] |
J. Banasiak and M. Lachowicz, On a macroscopic limit of a kinetic model of alignment, Math. Models Methods Appl. Sci., 23 (2013), 2647-2670.
doi: 10.1142/S0218202513500425. |
[3] |
E. Ben-Naim, Opinion dynamics: Rise and fall of political parties, EPL (Europhysics Letters), 69 (2005), p671.
doi: 10.1209/epl/i2004-10421-1. |
[4] |
L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation, M2AN Math. Model. Numer. Anal., 43 (2009), 507-522.
doi: 10.1051/m2an/2009004. |
[5] |
J. A. Carrillo, M. R. 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. |
[6] |
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. |
[7] |
J. A. Carrillo, S. Martin and V. Panferov, A new interaction potential for swarming models, Phys. D, 260 (2013), 112-126.
doi: 10.1016/j.physd.2013.02.004. |
[8] |
F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[9] |
P. Daskalopoulos and M. Del Pino, On nonlinear parabolic equations of very fast diffusion, Arch. Rational Mech. Anal., 137 (1997), 363-380.
doi: 10.1007/s002050050033. |
[10] |
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. |
[11] |
B. Després, F. Lagoutière, E. Labourasse and I. Marmajou, An antidissipative transport scheme on unstructured meshes for multicomponent flows, Int. J. Finite Vol., 7 (2010), 36pp. |
[12] |
L. Edelstein-Keshet, J. Watmough and D. Grunbaum, Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts, J. Math. Biol., 36 (1998), 515-549.
doi: 10.1007/s002850050112. |
[13] |
R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review, J. Math. Biol., 65 (2012), 35-75.
doi: 10.1007/s00285-011-0452-2. |
[14] |
R. Erban and J. Haskovec, From individual to collective behaviour of coupled velocity jump processes: a locust example, Kinet. Relat. Models, 5 (2012), 817-842.
doi: 10.3934/krm.2012.5.817. |
[15] |
E. Frénod and O. Sire, An explanatory model to validate the way water activity rules periodic terrace generation in proteus mirabilis swarm, J. Math. Biol., 59 (2009), 439-466.
doi: 10.1007/s00285-008-0235-6. |
[16] |
E. Geigant and M. Stoll, Bifurcation analysis of an orientational aggregation model, J. Math. Biol., 46 (2003), 537-563.
doi: 10.1007/s00285-002-0187-1. |
[17] |
M. Greenwood and R. Chapman, Differences in numbers of sensilla on the antennae of solitarious and gregarious locusta migratoria l.(orthoptera: Acrididae), International Journal of Insect Morphology and Embryology, 13 (1984), 295-301.
doi: 10.1016/0020-7322(84)90004-7. |
[18] |
D. Grünbaum, K. Chan, E. Tobin and M. T. Nishizaki, Non-linear advection-diffusion equations approximate swarming but not schooling populations, Math. Biosci., 214 (2008), 38-48.
doi: 10.1016/j.mbs.2008.06.002. |
[19] |
S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435.
doi: 10.3934/krm.2008.1.415. |
[20] |
S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), 441-454 (electronic).
doi: 10.1137/S1064827598334599. |
[21] |
K. Kang, B. Perthame, A. Stevens and J. J. L. Velázquez, Corrigendum to "An integro-differential equation model for alignment and orientational aggregation'' [J. Differential Equations 246 (4) (2009) 1387-1421] [mr2488690], J. Differential Equations, 252 (2012), 5125-5128.
doi: 10.1016/j.jde.2008.11.006. |
[22] |
S. Kaniel and M. Shinbrot, The Boltzmann equation. I. Uniqueness and local existence, Comm. Math. Phys., 58 (1978), 65-84. |
[23] |
R. Mach and F. Schweitzer, Modeling vortex swarming in daphnia, Bull. Math. Biol., 69 (2007), 539-562.
doi: 10.1007/s11538-006-9135-3. |
[24] |
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. |
[25] |
A. H. Øien, Daphnicle dynamics based on kinetic theory: An analogue-modelling of swarming and behaviour of daphnia, Bull. Math. Biol., 66 (2004), 1-46.
doi: 10.1016/S0092-8240(03)00065-X. |
[26] |
H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[27] |
L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013. |
[28] |
F. Peruani, T. Klauss, A. Deutsch and A. Voss-Boehme, Traffic jams, gliders, and bands in the quest for collective motion of self-propelled particles, Phys. Rev. Lett., 106 (2011), 128101.
doi: 10.1103/PhysRevLett.106.128101. |
[29] |
F. Peruani, J. Starruß, V. Jakovljevic, L. Søgaard-Andersen, A. Deutsch and M. Bär, Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria, Phys. Rev. Lett., 108 (2012), 098102.
doi: 10.1103/PhysRevLett.108.098102. |
[30] |
P.-A. Raviart and J.-M. Thomas, Introduction à L'analyse Numérique des Équations Aux Dérivées Partielles, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983. |
[31] |
F. Salvarani and G. Toscani, The diffusive limit of Carleman-type models in the range of very fast diffusion equations, J. Evol. Equ., 9 (2009), 67-80.
doi: 10.1007/s00028-009-0005-y. |
[32] |
F. Salvarani and J. L. Vázquez, The diffusive limit for Carleman-type kinetic models, Nonlinearity, 18 (2005), 1223-1248.
doi: 10.1088/0951-7715/18/3/015. |
[33] |
S. J. Simpson, D. Raubenheimer, S. T. Behmer, A. Whitworth and G. A. Wright, A comparison of nutritional regulation in solitarious- and gregarious-phase nymphs of the desert locust schistocerca gregaria, Journal of Experimental Biology, 205 (2002), 121-129. |
[34] |
S. J. Simpson, A. R. McCaffery and B. F. Hägele, A behavioural analysis of phase change in the desert locust, Biological Reviews, 74 (1999), 461-480. |
[35] |
G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.
doi: 10.4310/CMS.2006.v4.n3.a1. |
[36] |
J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, vol. 33 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2006, Equations of porous medium type.
doi: 10.1093/acprof:oso/9780199202973.001.0001. |
[37] |
Y. Wu, Y. Jiang, D. Kaiser and M. Alber, Social interactions in myxobacterial swarming, PLoS Comput. Biol., 3 (2007), 2546-2558.
doi: 10.1371/journal.pcbi.0030253. |
[38] |
T. I. Zohdi, Mechanistic modeling of swarms, Comput. Methods Appl. Mech. Engrg., 198 (2009), 2039-2051.
doi: 10.1016/j.cma.2008.12.029. |
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