# American Institute of Mathematical Sciences

March  2016, 9(1): 131-164. doi: 10.3934/krm.2016.9.131

## 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

Received  May 2015 Revised  July 2015 Published  October 2015

The present paper deals with the modeling of formation and destruction of swarms using a nonlinear Boltzmann--like equation. We introduce a new model that contains parameters characterizing the attractiveness or repulsiveness of individuals. The model can represent both gregarious and solitarious behaviors. In the latter case we provide a mathematical analysis in the space homogeneous case. Moreover we identify relevant hydrodynamic limits on a formal way. We introduce some preliminary results in the case of gregarious behavior and we indicate open problems for further research. Finally, we provide numerical simulations to illustrate the ability of the model to represent formation or destruction of swarms.
Citation: Martin Parisot, Mirosław Lachowicz. A kinetic model for the formation of swarms with nonlinear interactions. Kinetic and Related Models, 2016, 9 (1) : 131-164. doi: 10.3934/krm.2016.9.131
##### 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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##### 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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