American Institute of Mathematical Sciences

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March  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

Martial Agueh 1, , Reinhard Illner 1, and  Ashlin Richardson 1,

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.
Keywords: swarming, particle model, kinetic equation., Flocking.
Mathematics Subject Classification: 92D50, 97M10, 82C22, 82C7.
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. Google Scholar

[2]

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

[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, (). Google Scholar

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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), (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[14]

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

[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). Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

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

[24]

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

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

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. Google Scholar

[2]

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

[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, (). Google Scholar

[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. Google Scholar

[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), (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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. Google Scholar

[14]

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

[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). Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

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

[24]

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

[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. Google Scholar

[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. Google Scholar

[27]

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

[28]

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

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

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