American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317
[2]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[3]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457
[4]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464
[5]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[6]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[7]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384
[8]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316
[9]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219
[10]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344
[11]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349
[12]

Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459
[13]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339
[14]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[15]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[16]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[17]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454
[18]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448
[19]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[20]

Yolanda Guerrero–Sánchez, Muhammad Umar, Zulqurnain Sabir, Juan L. G. Guirao, Muhammad Asif Zahoor Raja. Solving a class of biological HIV infection model of latently infected cells using heuristic approach. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020431

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (83)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences