2011, 4(1): 1-16. doi: 10.3934/krm.2011.4.1

Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type

1. 

Department of Mathematics and Statistics, University of Victoria, PO BOX 3060 STN CSC, Victoria, BC V8W 3R4, Canada, Canada, Canada

Received  August 2010 Revised  November 2010 Published  January 2011

The Cucker-Smale model for flocking or swarming of birds or insects is generalized to scenarios where a typical bird will be subject to a) a friction force term driving it to fly at optimal speed, b) a repulsive short range force to avoid collisions, c) an attractive "flocking" force computed from the birds seen by each bird inside its vision cone, and d) a "boundary" force which will entice birds to search for and return to the flock if they find themselves at some distance from the flock. We introduce these forces in detail, discuss the required cutoffs and their implications and show that there are natural bounds in velocity space. Well-posedness of the initial value problem is discussed in spaces of measure-valued functions. We conclude with a series of numerical simulations.
Citation: Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1
References:
[1]

L. Ambrosio, Transport equation and Cauchy problem for BV vector fields,, Invent. Math., 158 (2004), 227. doi: 10.1007/s00222-004-0367-2.

[2]

I. Aoki, A simulation study on the schooling mechanism in fish,, Bulletin of the Japanese Society of Scientific Fisheries, 48 (1982), 1081.

[3]

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behaviour depends on topological rather than metric distance: Evidence from a field study,, Preprint, ().

[4]

F. Bouchut and F. James, One-dimensional transport equation with discontinuous coefficients,, Nonlinear Anal., 32 (1998), 891. doi: 10.1016/S0362-546X(97)00536-1.

[5]

J. A. Canizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion,, Math. Mod. Meth. Appl. Sci. (to appear), ().

[6]

J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory,, Kinetic and Related Models, 2 (2009), 363. doi: 10.3934/krm.2009.2.363.

[7]

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.

[8]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming,, in G. Naldi, (2010), 297. doi: 10.1007/978-0-8176-4946-3_12.

[9]

J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force,, Math. Mod. Meth. Appl. Sci., 20 (2010), 1533. doi: 10.1142/S0218202510004684.

[10]

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

[11]

F. Cucker and S. Smale, On the mathematics of emergence,, Japan J. Math., 2 (2007), 197. doi: 10.1007/s11537-007-0647-x.

[12]

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

[13]

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

[14]

R. Dobrushin, Vlasov equations,, Funct. Anal. Appl., 13 (1979), 115. doi: 10.1007/BF01077243.

[15]

V. V. Filippov, On the theory of the Cauchy problem for an ordinary differential equation with discontinuous right-hand side,, Russ. Acad. Sci. Sb. Math., 383 (1995).

[16]

H. Hildenbrandt, C. Carere and C. K. Hemelrijk, Self-organized aerial displays of thousands of starlings: A model,, Behavioral Ecology (to appear)., ().

[17]

C. K. Hemelrijk and H. Hildenbrandt, Self-organized shape pand frontal density of fish schools,, Ethology, 114 (2008), 245. doi: 10.1111/j.1439-0310.2007.01459.x.

[18]

C. K. Hemelrijk and H. Kunz, Density distribution and size sorting in fish schools: An individual-based model,, Behavioral Ecology, 16 (2005), 178. doi: 10.1093/beheco/arh149.

[19]

H. Huth and C. Wissel, The simulation of the movement of fish schools,, J. Theor. Biol., 156 (1992), 365. doi: 10.1016/S0022-5193(05)80681-2.

[20]

H. Kunz and C. K. Hemelrijk, Artificial fish schools: Collective effects of school size, body size, and body form,, Artificial Life, 9 (2003). doi: 10.1162/106454603322392451.

[21]

M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. rev. Lett., 96 (2006), 104302.

[22]

S-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit,, Commun. Math. Sci., 7 (2009), 297.

[23]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamics description of flocking,, Kinetic and Related models, 1 (2008), 415.

[24]

H. Levine, W.-J. Rappel and I. Cohen, Self-organization in systems of self-propelled particles,, Phys. Rev. Lett. E., 63 (2000), 017101.

[25]

R. Lukeman, Y. X. Li and L. Edelstein-Keshet, A conceptual model for milling formations in bilological aggregates,, Bull Math Biol., 71 (2008), 352. doi: 10.1007/s11538-008-9365-7.

[26]

R. Lukeman and L. Edelstein-Keshet, Minimal mechanisms for school formation in self-propelled particles,, Physica D., 237 (2008), 699. doi: 10.1016/j.physd.2007.10.009.

[27]

P. D. Miller, "Applied Asymptotic Analysis,", Graduate Studies in Math, 75 (2006).

[28]

H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles,, Trans. Fluid Dynamics, 18 (1997), 663.

[29]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation,, In:, (1984), 60.

[30]

J. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation,, Science, 294 (1999), 99. doi: 10.1126/science.284.5411.99.

[31]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking,, Physical Review E., 58 (1998), 4828. doi: 10.1103/PhysRevE.58.4828.

[32]

T. Vicsek, A. Czirok, 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. doi: 10.1103/PhysRevLett.75.1226.

[33]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Math., 58 (2003).

show all references

References:
[1]

L. Ambrosio, Transport equation and Cauchy problem for BV vector fields,, Invent. Math., 158 (2004), 227. doi: 10.1007/s00222-004-0367-2.

[2]

I. Aoki, A simulation study on the schooling mechanism in fish,, Bulletin of the Japanese Society of Scientific Fisheries, 48 (1982), 1081.

[3]

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behaviour depends on topological rather than metric distance: Evidence from a field study,, Preprint, ().

[4]

F. Bouchut and F. James, One-dimensional transport equation with discontinuous coefficients,, Nonlinear Anal., 32 (1998), 891. doi: 10.1016/S0362-546X(97)00536-1.

[5]

J. A. Canizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion,, Math. Mod. Meth. Appl. Sci. (to appear), ().

[6]

J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory,, Kinetic and Related Models, 2 (2009), 363. doi: 10.3934/krm.2009.2.363.

[7]

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.

[8]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming,, in G. Naldi, (2010), 297. doi: 10.1007/978-0-8176-4946-3_12.

[9]

J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force,, Math. Mod. Meth. Appl. Sci., 20 (2010), 1533. doi: 10.1142/S0218202510004684.

[10]

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

[11]

F. Cucker and S. Smale, On the mathematics of emergence,, Japan J. Math., 2 (2007), 197. doi: 10.1007/s11537-007-0647-x.

[12]

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

[13]

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

[14]

R. Dobrushin, Vlasov equations,, Funct. Anal. Appl., 13 (1979), 115. doi: 10.1007/BF01077243.

[15]

V. V. Filippov, On the theory of the Cauchy problem for an ordinary differential equation with discontinuous right-hand side,, Russ. Acad. Sci. Sb. Math., 383 (1995).

[16]

H. Hildenbrandt, C. Carere and C. K. Hemelrijk, Self-organized aerial displays of thousands of starlings: A model,, Behavioral Ecology (to appear)., ().

[17]

C. K. Hemelrijk and H. Hildenbrandt, Self-organized shape pand frontal density of fish schools,, Ethology, 114 (2008), 245. doi: 10.1111/j.1439-0310.2007.01459.x.

[18]

C. K. Hemelrijk and H. Kunz, Density distribution and size sorting in fish schools: An individual-based model,, Behavioral Ecology, 16 (2005), 178. doi: 10.1093/beheco/arh149.

[19]

H. Huth and C. Wissel, The simulation of the movement of fish schools,, J. Theor. Biol., 156 (1992), 365. doi: 10.1016/S0022-5193(05)80681-2.

[20]

H. Kunz and C. K. Hemelrijk, Artificial fish schools: Collective effects of school size, body size, and body form,, Artificial Life, 9 (2003). doi: 10.1162/106454603322392451.

[21]

M. R. D'Orsogna, Y.-L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse,, Phys. rev. Lett., 96 (2006), 104302.

[22]

S-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit,, Commun. Math. Sci., 7 (2009), 297.

[23]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamics description of flocking,, Kinetic and Related models, 1 (2008), 415.

[24]

H. Levine, W.-J. Rappel and I. Cohen, Self-organization in systems of self-propelled particles,, Phys. Rev. Lett. E., 63 (2000), 017101.

[25]

R. Lukeman, Y. X. Li and L. Edelstein-Keshet, A conceptual model for milling formations in bilological aggregates,, Bull Math Biol., 71 (2008), 352. doi: 10.1007/s11538-008-9365-7.

[26]

R. Lukeman and L. Edelstein-Keshet, Minimal mechanisms for school formation in self-propelled particles,, Physica D., 237 (2008), 699. doi: 10.1016/j.physd.2007.10.009.

[27]

P. D. Miller, "Applied Asymptotic Analysis,", Graduate Studies in Math, 75 (2006).

[28]

H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles,, Trans. Fluid Dynamics, 18 (1997), 663.

[29]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation,, In:, (1984), 60.

[30]

J. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation,, Science, 294 (1999), 99. doi: 10.1126/science.284.5411.99.

[31]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking,, Physical Review E., 58 (1998), 4828. doi: 10.1103/PhysRevE.58.4828.

[32]

T. Vicsek, A. Czirok, 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. doi: 10.1103/PhysRevLett.75.1226.

[33]

C. Villani, "Topics in Optimal Transportation,", Graduate Studies in Math., 58 (2003).

[1]

Seung-Yeal Ha, Eitan Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic & Related Models, 2008, 1 (3) : 415-435. doi: 10.3934/krm.2008.1.415

[2]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381

[3]

Pierre Degond, Simone Goettlich, Axel Klar, Mohammed Seaid, Andreas Unterreiter. Derivation of a kinetic model from a stochastic particle system. Kinetic & Related Models, 2008, 1 (4) : 557-572. doi: 10.3934/krm.2008.1.557

[4]

Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure & Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028

[5]

Felipe Cucker, Jiu-Gang Dong. A conditional, collision-avoiding, model for swarming. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1009-1020. doi: 10.3934/dcds.2014.34.1009

[6]

Seung-Yeal Ha, Doron Levy. Particle, kinetic and fluid models for phototaxis. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 77-108. doi: 10.3934/dcdsb.2009.12.77

[7]

Karsten Matthies, George Stone, Florian Theil. The derivation of the linear Boltzmann equation from a Rayleigh gas particle model. Kinetic & Related Models, 2018, 11 (1) : 137-177. doi: 10.3934/krm.2018008

[8]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[9]

Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062

[10]

Reiner Henseler, Michael Herrmann, Barbara Niethammer, Juan J. L. Velázquez. A kinetic model for grain growth. Kinetic & Related Models, 2008, 1 (4) : 591-617. doi: 10.3934/krm.2008.1.591

[11]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223

[12]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[13]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

[14]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[15]

Nicolas Vauchelet. Numerical simulation of a kinetic model for chemotaxis. Kinetic & Related Models, 2010, 3 (3) : 501-528. doi: 10.3934/krm.2010.3.501

[16]

Leif Arkeryd. A kinetic equation for spin polarized Fermi systems. Kinetic & Related Models, 2014, 7 (1) : 1-8. doi: 10.3934/krm.2014.7.1

[17]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

[18]

Wolfgang Wagner. Some properties of the kinetic equation for electron transport in semiconductors. Kinetic & Related Models, 2013, 6 (4) : 955-967. doi: 10.3934/krm.2013.6.955

[19]

Bertram Düring, Ansgar Jüngel, Lara Trussardi. A kinetic equation for economic value estimation with irrationality and herding. Kinetic & Related Models, 2017, 10 (1) : 239-261. doi: 10.3934/krm.2017010

[20]

Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503

2016 Impact Factor: 1.261

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (17)

[Back to Top]