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 & 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. doi: 10.1016/j.mcm.2005.05.011. Google Scholar

[2]

J. Banasiak and M. Lachowicz, On a macroscopic limit of a kinetic model of alignment,, Math. Models Methods Appl. Sci., 23 (2013), 2647. doi: 10.1142/S0218202513500425. Google Scholar

[3]

E. Ben-Naim, Opinion dynamics: Rise and fall of political parties,, EPL (Europhysics Letters), 69 (2005). doi: 10.1209/epl/i2004-10421-1. Google Scholar

[4]

L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation,, M2AN Math. Model. Numer. Anal., 43 (2009), 507. doi: 10.1051/m2an/2009004. Google Scholar

[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. doi: 10.3934/krm.2009.2.363. Google Scholar

[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. doi: 10.1137/090757290. Google Scholar

[7]

J. A. Carrillo, S. Martin and V. Panferov, A new interaction potential for swarming models,, Phys. D, 260 (2013), 112. doi: 10.1016/j.physd.2013.02.004. Google Scholar

[8]

F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852. doi: 10.1109/TAC.2007.895842. Google Scholar

[9]

P. Daskalopoulos and M. Del Pino, On nonlinear parabolic equations of very fast diffusion,, Arch. Rational Mech. Anal., 137 (1997), 363. doi: 10.1007/s002050050033. Google Scholar

[10]

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

[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). Google Scholar

[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. doi: 10.1007/s002850050112. Google Scholar

[13]

R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review,, J. Math. Biol., 65 (2012), 35. doi: 10.1007/s00285-011-0452-2. Google Scholar

[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. doi: 10.3934/krm.2012.5.817. Google Scholar

[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. doi: 10.1007/s00285-008-0235-6. Google Scholar

[16]

E. Geigant and M. Stoll, Bifurcation analysis of an orientational aggregation model,, J. Math. Biol., 46 (2003), 537. doi: 10.1007/s00285-002-0187-1. Google Scholar

[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. doi: 10.1016/0020-7322(84)90004-7. Google Scholar

[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. doi: 10.1016/j.mbs.2008.06.002. Google Scholar

[19]

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

[20]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations,, SIAM J. Sci. Comput., 21 (1999), 441. doi: 10.1137/S1064827598334599. Google Scholar

[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. doi: 10.1016/j.jde.2008.11.006. Google Scholar

[22]

S. Kaniel and M. Shinbrot, The Boltzmann equation. I. Uniqueness and local existence,, Comm. Math. Phys., 58 (1978), 65. Google Scholar

[23]

R. Mach and F. Schweitzer, Modeling vortex swarming in daphnia,, Bull. Math. Biol., 69 (2007), 539. doi: 10.1007/s11538-006-9135-3. Google Scholar

[24]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior,, J. Stat. Phys., 144 (2011), 923. doi: 10.1007/s10955-011-0285-9. Google Scholar

[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. doi: 10.1016/S0092-8240(03)00065-X. Google Scholar

[26]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[27]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods,, Oxford University Press, (2013). Google Scholar

[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). doi: 10.1103/PhysRevLett.106.128101. Google Scholar

[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). doi: 10.1103/PhysRevLett.108.098102. Google Scholar

[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], (1983). Google Scholar

[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. doi: 10.1007/s00028-009-0005-y. Google Scholar

[32]

F. Salvarani and J. L. Vázquez, The diffusive limit for Carleman-type kinetic models,, Nonlinearity, 18 (2005), 1223. doi: 10.1088/0951-7715/18/3/015. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

G. Toscani, Kinetic models of opinion formation,, Commun. Math. Sci., 4 (2006), 481. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[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, (2006). doi: 10.1093/acprof:oso/9780199202973.001.0001. Google Scholar

[37]

Y. Wu, Y. Jiang, D. Kaiser and M. Alber, Social interactions in myxobacterial swarming,, PLoS Comput. Biol., 3 (2007), 2546. doi: 10.1371/journal.pcbi.0030253. Google Scholar

[38]

T. I. Zohdi, Mechanistic modeling of swarms,, Comput. Methods Appl. Mech. Engrg., 198 (2009), 2039. doi: 10.1016/j.cma.2008.12.029. Google Scholar

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. doi: 10.1016/j.mcm.2005.05.011. Google Scholar

[2]

J. Banasiak and M. Lachowicz, On a macroscopic limit of a kinetic model of alignment,, Math. Models Methods Appl. Sci., 23 (2013), 2647. doi: 10.1142/S0218202513500425. Google Scholar

[3]

E. Ben-Naim, Opinion dynamics: Rise and fall of political parties,, EPL (Europhysics Letters), 69 (2005). doi: 10.1209/epl/i2004-10421-1. Google Scholar

[4]

L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation,, M2AN Math. Model. Numer. Anal., 43 (2009), 507. doi: 10.1051/m2an/2009004. Google Scholar

[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. doi: 10.3934/krm.2009.2.363. Google Scholar

[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. doi: 10.1137/090757290. Google Scholar

[7]

J. A. Carrillo, S. Martin and V. Panferov, A new interaction potential for swarming models,, Phys. D, 260 (2013), 112. doi: 10.1016/j.physd.2013.02.004. Google Scholar

[8]

F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852. doi: 10.1109/TAC.2007.895842. Google Scholar

[9]

P. Daskalopoulos and M. Del Pino, On nonlinear parabolic equations of very fast diffusion,, Arch. Rational Mech. Anal., 137 (1997), 363. doi: 10.1007/s002050050033. Google Scholar

[10]

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

[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). Google Scholar

[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. doi: 10.1007/s002850050112. Google Scholar

[13]

R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review,, J. Math. Biol., 65 (2012), 35. doi: 10.1007/s00285-011-0452-2. Google Scholar

[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. doi: 10.3934/krm.2012.5.817. Google Scholar

[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. doi: 10.1007/s00285-008-0235-6. Google Scholar

[16]

E. Geigant and M. Stoll, Bifurcation analysis of an orientational aggregation model,, J. Math. Biol., 46 (2003), 537. doi: 10.1007/s00285-002-0187-1. Google Scholar

[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. doi: 10.1016/0020-7322(84)90004-7. Google Scholar

[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. doi: 10.1016/j.mbs.2008.06.002. Google Scholar

[19]

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

[20]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations,, SIAM J. Sci. Comput., 21 (1999), 441. doi: 10.1137/S1064827598334599. Google Scholar

[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. doi: 10.1016/j.jde.2008.11.006. Google Scholar

[22]

S. Kaniel and M. Shinbrot, The Boltzmann equation. I. Uniqueness and local existence,, Comm. Math. Phys., 58 (1978), 65. Google Scholar

[23]

R. Mach and F. Schweitzer, Modeling vortex swarming in daphnia,, Bull. Math. Biol., 69 (2007), 539. doi: 10.1007/s11538-006-9135-3. Google Scholar

[24]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior,, J. Stat. Phys., 144 (2011), 923. doi: 10.1007/s10955-011-0285-9. Google Scholar

[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. doi: 10.1016/S0092-8240(03)00065-X. Google Scholar

[26]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[27]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods,, Oxford University Press, (2013). Google Scholar

[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). doi: 10.1103/PhysRevLett.106.128101. Google Scholar

[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). doi: 10.1103/PhysRevLett.108.098102. Google Scholar

[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], (1983). Google Scholar

[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. doi: 10.1007/s00028-009-0005-y. Google Scholar

[32]

F. Salvarani and J. L. Vázquez, The diffusive limit for Carleman-type kinetic models,, Nonlinearity, 18 (2005), 1223. doi: 10.1088/0951-7715/18/3/015. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

G. Toscani, Kinetic models of opinion formation,, Commun. Math. Sci., 4 (2006), 481. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[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, (2006). doi: 10.1093/acprof:oso/9780199202973.001.0001. Google Scholar

[37]

Y. Wu, Y. Jiang, D. Kaiser and M. Alber, Social interactions in myxobacterial swarming,, PLoS Comput. Biol., 3 (2007), 2546. doi: 10.1371/journal.pcbi.0030253. Google Scholar

[38]

T. I. Zohdi, Mechanistic modeling of swarms,, Comput. Methods Appl. Mech. Engrg., 198 (2009), 2039. doi: 10.1016/j.cma.2008.12.029. Google Scholar

[1]

Nicolas Crouseilles, Giacomo Dimarco, Mohammed Lemou. Asymptotic preserving and time diminishing schemes for rarefied gas dynamic. Kinetic & Related Models, 2017, 10 (3) : 643-668. doi: 10.3934/krm.2017026

[2]

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

[3]

Jeffrey R. Haack, Cory D. Hauck. Oscillatory behavior of Asymptotic-Preserving splitting methods for a linear model of diffusive relaxation. Kinetic & Related Models, 2008, 1 (4) : 573-590. doi: 10.3934/krm.2008.1.573

[4]

Nikolaos Halidias. Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 153-160. doi: 10.3934/dcdsb.2015.20.153

[5]

Laurent Boudin, Francesco Salvarani. The quasi-invariant limit for a kinetic model of sociological collective behavior. Kinetic & Related Models, 2009, 2 (3) : 433-449. doi: 10.3934/krm.2009.2.433

[6]

Alina Chertock, Changhui Tan, Bokai Yan. An asymptotic preserving scheme for kinetic models with singular limit. Kinetic & Related Models, 2018, 11 (4) : 735-756. doi: 10.3934/krm.2018030

[7]

Hélène Hivert. Numerical schemes for kinetic equation with diffusion limit and anomalous time scale. Kinetic & Related Models, 2018, 11 (2) : 409-439. doi: 10.3934/krm.2018019

[8]

Jingwei Hu, Shi Jin. On kinetic flux vector splitting schemes for quantum Euler equations. Kinetic & Related Models, 2011, 4 (2) : 517-530. doi: 10.3934/krm.2011.4.517

[9]

Ming Mei, Bruno Rubino, Rosella Sampalmieri. Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain. Kinetic & Related Models, 2012, 5 (3) : 537-550. doi: 10.3934/krm.2012.5.537

[10]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[11]

Nicolas Crouseilles, Mohammed Lemou, SV Raghurama Rao, Ankit Ruhi, Muddu Sekhar. Asymptotic preserving scheme for a kinetic model describing incompressible fluids. Kinetic & Related Models, 2016, 9 (1) : 51-74. doi: 10.3934/krm.2016.9.51

[12]

Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146

[13]

Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles. Kinetic & Related Models, 2012, 5 (4) : 787-816. doi: 10.3934/krm.2012.5.787

[14]

Roman Czapla, Vladimir V. Mityushev. A criterion of collective behavior of bacteria. Mathematical Biosciences & Engineering, 2017, 14 (1) : 277-287. doi: 10.3934/mbe.2017018

[15]

Stefka Bouyuklieva, Zlatko Varbanov. Some connections between self-dual codes, combinatorial designs and secret sharing schemes. Advances in Mathematics of Communications, 2011, 5 (2) : 191-198. doi: 10.3934/amc.2011.5.191

[16]

Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093

[17]

Paolo Aluffi. Segre classes of monomial schemes. Electronic Research Announcements, 2013, 20: 55-70. doi: 10.3934/era.2013.20.55

[18]

Marx Chhay, Aziz Hamdouni. On the accuracy of invariant numerical schemes. Communications on Pure & Applied Analysis, 2011, 10 (2) : 761-783. doi: 10.3934/cpaa.2011.10.761

[19]

Benjamin Seibold, Rodolfo R. Rosales, Jean-Christophe Nave. Jet schemes for advection problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1229-1259. doi: 10.3934/dcdsb.2012.17.1229

[20]

Florent Berthelin, Thierry Goudon, Bastien Polizzi, Magali Ribot. Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams. Networks & Heterogeneous Media, 2017, 12 (4) : 591-617. doi: 10.3934/nhm.2017024

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]