# American Institute of Mathematical Sciences

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

 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 and 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), 227-260. 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), 1081-1088. [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, arXiv:0709.1916v1. [4] F. Bouchut and F. James, One-dimensional transport equation with discontinuous coefficients, Nonlinear Anal., 32 (1998), 891-933 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), arXiv:0907.3901v2. [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), 363-378. 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), 218-219. 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, 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. [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), 1533-1552. 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), 1-11. doi: 10.1006/jtbi.2002.3065. [11] F. Cucker and S. Smale, On the mathematics of emergence, Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x. [12] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control., 52 (2007), 852-862. 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), 1193-1215. doi: 10.1142/S0218202508003005. [14] R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123. 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), 245-254. 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), 178-187. doi: 10.1093/beheco/arh149. [19] H. Huth and C. Wissel, The simulation of the movement of fish schools, J. Theor. Biol., 156 (1992), 365-385. 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), 237253. 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), 104302-1/4. [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), 297-325. [23] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamics description of flocking, Kinetic and Related models, 1 (2008), 415-435. [24] H. Levine, W.-J. Rappel and I. Cohen, Self-organization in systems of self-propelled particles, Phys. Rev. Lett. E., 63 (2000), 017101-1/4. [25] R. Lukeman, Y. X. Li and L. Edelstein-Keshet, A conceptual model for milling formations in bilological aggregates, Bull Math Biol., 71 (2008), 352-382. 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), 699-720. doi: 10.1016/j.physd.2007.10.009. [27] P. D. Miller, "Applied Asymptotic Analysis," Graduate Studies in Math, 75 (2006), AMS. [28] H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles, Trans. Fluid Dynamics, 18 (1997), 663-678. [29] H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, In: "Kinetic Theories and the Boltzmann Equation" (Montecatini 1981), Lecture Notes in Math., Springer Verlag Berlin, (1984), 60-110. [30] J. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation, Science, 294 (1999), 99-101. 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), 4828-2858. 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), 1226-1229. doi: 10.1103/PhysRevLett.75.1226. [33] C. Villani, "Topics in Optimal Transportation," Graduate Studies in Math., 58 (2003), AMS.

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##### References:
 [1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), 227-260. 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), 1081-1088. [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, arXiv:0709.1916v1. [4] F. Bouchut and F. James, One-dimensional transport equation with discontinuous coefficients, Nonlinear Anal., 32 (1998), 891-933 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), arXiv:0907.3901v2. [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), 363-378. 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), 218-219. 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, 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. [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), 1533-1552. 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), 1-11. doi: 10.1006/jtbi.2002.3065. [11] F. Cucker and S. Smale, On the mathematics of emergence, Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x. [12] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control., 52 (2007), 852-862. 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), 1193-1215. doi: 10.1142/S0218202508003005. [14] R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123. 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), 245-254. 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), 178-187. doi: 10.1093/beheco/arh149. [19] H. Huth and C. Wissel, The simulation of the movement of fish schools, J. Theor. Biol., 156 (1992), 365-385. 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), 237253. 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), 104302-1/4. [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), 297-325. [23] S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamics description of flocking, Kinetic and Related models, 1 (2008), 415-435. [24] H. Levine, W.-J. Rappel and I. Cohen, Self-organization in systems of self-propelled particles, Phys. Rev. Lett. E., 63 (2000), 017101-1/4. [25] R. Lukeman, Y. X. Li and L. Edelstein-Keshet, A conceptual model for milling formations in bilological aggregates, Bull Math Biol., 71 (2008), 352-382. 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), 699-720. doi: 10.1016/j.physd.2007.10.009. [27] P. D. Miller, "Applied Asymptotic Analysis," Graduate Studies in Math, 75 (2006), AMS. [28] H. Neunzert, The Vlasov equation as a limit of Hamiltonian classical mechanical systems of interacting particles, Trans. Fluid Dynamics, 18 (1997), 663-678. [29] H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, In: "Kinetic Theories and the Boltzmann Equation" (Montecatini 1981), Lecture Notes in Math., Springer Verlag Berlin, (1984), 60-110. [30] J. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation, Science, 294 (1999), 99-101. 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), 4828-2858. 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), 1226-1229. doi: 10.1103/PhysRevLett.75.1226. [33] C. Villani, "Topics in Optimal Transportation," Graduate Studies in Math., 58 (2003), AMS.
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