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2016, 13(2): 369-380. doi: 10.3934/mbe.2015007

## Flocking and invariance of velocity angles

 1 College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082, China 2 College of Mathematics and Econometrics, Hunan University & Hunan Women's University, Changsha, Hunan, 410004 3 Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3

Received  November 2014 Revised  May 2015 Published  December 2015

Motsch and Tadmor considered an extended Cucker-Smale model to investigate the flocking behavior of self-organized systems of interacting species. In this extended model, a cone of the vision was introduced so that outside the cone the influence of one agent on the other is lost and hence the corresponding influence function takes the value zero. This creates a problem to apply the Motsch-Tadmor and Cucker-Smale method to prove the flocking property of the system. Here, we examine the variation of the velocity angles between two arbitrary agents, and obtain a monotonicity property for the maximum cone of velocity angles. This monotonicity permits us to utilize existing arguments to show the flocking property of the system under consideration, when the initial velocity angles satisfy some minor technical constraints.
Citation: Le Li, Lihong Huang, Jianhong Wu. Flocking and invariance of velocity angles. Mathematical Biosciences & Engineering, 2016, 13 (2) : 369-380. doi: 10.3934/mbe.2015007
##### References:
 [1] 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), 218-236. doi: 10.1137/090757290. [2] I. D. Couzin, J. Krause, N. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516. doi: 10.1038/nature03236. [3] F. Cucker and S. Smale, Lectures on emergence, Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x. [4] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842. [5] F. Cucker, S. Smale and D. Zhou, Modeling language evolution, Found. Comput. Math., 4 (2004), 315-343. doi: 10.1007/s10208-003-0101-2. [6] 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. doi: 10.4310/CMS.2009.v7.n2.a2. [7] S. Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat.Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415. [8] Y. Liu and K. Passino, Stable social foraging swarms in a noisy environment, IEEE Trans. Automat. Control, 49 (2004), 30-44. doi: 10.1109/TAC.2003.821416. [9] S. Motsch and E. Tadmor, A new model for Self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9. [10] C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, In: ACM SIGGRAPH Computer Graphics, 21 (1987), 25-34. doi: 10.1145/37401.37406. [11] J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719. doi: 10.1137/060673254. [12] C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174. doi: 10.1137/S0036139903437424. [13] C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Bio., 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6. [14] T. Vicsek, A. Czirók, 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), 1226-1225. doi: 10.1103/PhysRevLett.75.1226.

show all references

##### References:
 [1] 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), 218-236. doi: 10.1137/090757290. [2] I. D. Couzin, J. Krause, N. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516. doi: 10.1038/nature03236. [3] F. Cucker and S. Smale, Lectures on emergence, Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x. [4] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842. [5] F. Cucker, S. Smale and D. Zhou, Modeling language evolution, Found. Comput. Math., 4 (2004), 315-343. doi: 10.1007/s10208-003-0101-2. [6] 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. doi: 10.4310/CMS.2009.v7.n2.a2. [7] S. Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat.Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415. [8] Y. Liu and K. Passino, Stable social foraging swarms in a noisy environment, IEEE Trans. Automat. Control, 49 (2004), 30-44. doi: 10.1109/TAC.2003.821416. [9] S. Motsch and E. Tadmor, A new model for Self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9. [10] C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, In: ACM SIGGRAPH Computer Graphics, 21 (1987), 25-34. doi: 10.1145/37401.37406. [11] J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719. doi: 10.1137/060673254. [12] C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174. doi: 10.1137/S0036139903437424. [13] C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Bio., 68 (2006), 1601-1623. doi: 10.1007/s11538-006-9088-6. [14] T. Vicsek, A. Czirók, 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), 1226-1225. doi: 10.1103/PhysRevLett.75.1226.
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