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]

Invent. Math., 158 (2004), 227-260. doi: 10.1007/s00222-004-0367-2.  Google Scholar

[2]

Bulletin of the Japanese Society of Scientific Fisheries, 48 (1982), 1081-1088. 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]

Nonlinear Anal., 32 (1998), 891-933 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]

Kinetic and Related Models, 2 (2009), 363-378. doi: 10.3934/krm.2009.2.363.  Google Scholar

[7]

SIAM J. Math. Anal., 42 (2010), 218-219. doi: 10.1137/090757290.  Google Scholar

[8]

in G. Naldi, L. Pareshi and G. Toscani (eds). Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Series Modelling and Simulation in Science and Technology, Birkha\"user (2010), 297-336. doi: 10.1007/978-0-8176-4946-3_12.  Google Scholar

[9]

Math. Mod. Meth. Appl. Sci., 20 (2010), 1533-1552. doi: 10.1142/S0218202510004684.  Google Scholar

[10]

J. Theor. Biol., 218 (2002), 1-11. doi: 10.1006/jtbi.2002.3065.  Google Scholar

[11]

Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.  Google Scholar

[12]

IEEE Trans. Automat. Control., 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.  Google Scholar

[13]

Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.  Google Scholar

[14]

Funct. Anal. Appl., 13 (1979), 115-123. doi: 10.1007/BF01077243.  Google Scholar

[15]

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]

Ethology, 114 (2008), 245-254. doi: 10.1111/j.1439-0310.2007.01459.x.  Google Scholar

[18]

Behavioral Ecology, 16 (2005), 178-187. doi: 10.1093/beheco/arh149.  Google Scholar

[19]

J. Theor. Biol., 156 (1992), 365-385. doi: 10.1016/S0022-5193(05)80681-2.  Google Scholar

[20]

Artificial Life, 9 (2003), 237253. doi: 10.1162/106454603322392451.  Google Scholar

[21]

Phys. rev. Lett., 96 (2006), 104302-1/4. Google Scholar

[22]

Commun. Math. Sci., 7 (2009), 297-325.  Google Scholar

[23]

Kinetic and Related models, 1 (2008), 415-435.  Google Scholar

[24]

Phys. Rev. Lett. E., 63 (2000), 017101-1/4. Google Scholar

[25]

Bull Math Biol., 71 (2008), 352-382. doi: 10.1007/s11538-008-9365-7.  Google Scholar

[26]

Physica D., 237 (2008), 699-720. doi: 10.1016/j.physd.2007.10.009.  Google Scholar

[27]

Graduate Studies in Math, 75 (2006), AMS.  Google Scholar

[28]

Trans. Fluid Dynamics, 18 (1997), 663-678. Google Scholar

[29]

In: "Kinetic Theories and the Boltzmann Equation" (Montecatini 1981), Lecture Notes in Math., Springer Verlag Berlin, (1984), 60-110.  Google Scholar

[30]

Science, 294 (1999), 99-101. doi: 10.1126/science.284.5411.99.  Google Scholar

[31]

Physical Review E., 58 (1998), 4828-2858. doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[32]

Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[33]

Graduate Studies in Math., 58 (2003), AMS.  Google Scholar

show all references

References:
[1]

Invent. Math., 158 (2004), 227-260. doi: 10.1007/s00222-004-0367-2.  Google Scholar

[2]

Bulletin of the Japanese Society of Scientific Fisheries, 48 (1982), 1081-1088. 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]

Nonlinear Anal., 32 (1998), 891-933 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]

Kinetic and Related Models, 2 (2009), 363-378. doi: 10.3934/krm.2009.2.363.  Google Scholar

[7]

SIAM J. Math. Anal., 42 (2010), 218-219. doi: 10.1137/090757290.  Google Scholar

[8]

in G. Naldi, L. Pareshi and G. Toscani (eds). Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Series Modelling and Simulation in Science and Technology, Birkha\"user (2010), 297-336. doi: 10.1007/978-0-8176-4946-3_12.  Google Scholar

[9]

Math. Mod. Meth. Appl. Sci., 20 (2010), 1533-1552. doi: 10.1142/S0218202510004684.  Google Scholar

[10]

J. Theor. Biol., 218 (2002), 1-11. doi: 10.1006/jtbi.2002.3065.  Google Scholar

[11]

Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.  Google Scholar

[12]

IEEE Trans. Automat. Control., 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.  Google Scholar

[13]

Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.  Google Scholar

[14]

Funct. Anal. Appl., 13 (1979), 115-123. doi: 10.1007/BF01077243.  Google Scholar

[15]

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]

Ethology, 114 (2008), 245-254. doi: 10.1111/j.1439-0310.2007.01459.x.  Google Scholar

[18]

Behavioral Ecology, 16 (2005), 178-187. doi: 10.1093/beheco/arh149.  Google Scholar

[19]

J. Theor. Biol., 156 (1992), 365-385. doi: 10.1016/S0022-5193(05)80681-2.  Google Scholar

[20]

Artificial Life, 9 (2003), 237253. doi: 10.1162/106454603322392451.  Google Scholar

[21]

Phys. rev. Lett., 96 (2006), 104302-1/4. Google Scholar

[22]

Commun. Math. Sci., 7 (2009), 297-325.  Google Scholar

[23]

Kinetic and Related models, 1 (2008), 415-435.  Google Scholar

[24]

Phys. Rev. Lett. E., 63 (2000), 017101-1/4. Google Scholar

[25]

Bull Math Biol., 71 (2008), 352-382. doi: 10.1007/s11538-008-9365-7.  Google Scholar

[26]

Physica D., 237 (2008), 699-720. doi: 10.1016/j.physd.2007.10.009.  Google Scholar

[27]

Graduate Studies in Math, 75 (2006), AMS.  Google Scholar

[28]

Trans. Fluid Dynamics, 18 (1997), 663-678. Google Scholar

[29]

In: "Kinetic Theories and the Boltzmann Equation" (Montecatini 1981), Lecture Notes in Math., Springer Verlag Berlin, (1984), 60-110.  Google Scholar

[30]

Science, 294 (1999), 99-101. doi: 10.1126/science.284.5411.99.  Google Scholar

[31]

Physical Review E., 58 (1998), 4828-2858. doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[32]

Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[33]

Graduate Studies in Math., 58 (2003), AMS.  Google Scholar

[1]

Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021015
[2]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011
[3]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008
[4]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019
[5]

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
[6]

Andrea Tosin, Mattia Zanella. Uncertainty damping in kinetic traffic models by driver-assist controls. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021018
[7]

Zhisu Liu, Yicheng Liu, Xiang Li. Flocking and line-shaped spatial configuration to delayed Cucker-Smale models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3693-3716. doi: 10.3934/dcdsb.2020253
[8]

Lifen Jia, Wei Dai. Uncertain spring vibration equation. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021073
[9]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225
[10]

Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161
[11]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[12]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[13]

Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029
[14]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002
[15]

Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101
[16]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427
[17]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[18]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[19]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[20]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences