Cascade flocking with free-will

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March 2016, 21(2): 497-522. doi: 10.3934/dcdsb.2016.21.497

Cascade flocking with free-will

Le Li ^1, , Lihong Huang ^2, and Jianhong Wu ^3,

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, China
3.	Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3

Received February 2015 Revised August 2015 Published November 2015

Abstract
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We consider a self-organized system with a hierarchy structure to allow multiple leaders in the highest rank, and with free-will. In the model, we use both Cucker-Smale and Motsch-Tadmor functions for the pair influence of agents, and we derive suffcient conditions for such a system to converge to a flock, where agents ultimately move in the same velocity. We provide examples to show our suffcient conditions are sharp, and we numerically observe that such a self-organized system may have agents moving in different (final) velocities but maintain finite distance from each other due to the free-will.

Keywords: biomathematics., hierarchy, free-will, Flocking.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Le Li, Lihong Huang, Jianhong Wu. Cascade flocking with free-will. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 497-522. doi: 10.3934/dcdsb.2016.21.497

References:

[1]	A. Flack, B. Pettit and R. Freeman, What are leaders made of? The role of individual experience in determining leader-follower relations in homing pigeons, Animal Behaviour, 83 (2012), 703-709. doi: 10.1016/j.anbehav.2011.12.018.
[2]	A. Flack, B. Pettit and R. Freeman, Fault-tolerant flocking for a group of autonomous mobile robots, The Journal of Systems and Software, 84 (2011), 29-36.
[3]	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.
[4]	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.
[5]	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.
[6]	D. Gu and Z. Wang, Leader-Follower flocking: Algorithms and experiments, PNAS, 17 (2008), 1211-1219.
[7]	D. Gu and Z. Wang, Leader-follower flocking: Algorithms and experiments, IEEE Transactions on control systems technology, 17 (2009), 1211-1219.
[8]	D. J. Hoare, I. D. Couzin, J.-G. J. Godin and J. Krause, Context-dependent group size choice in fish, Animal Behaviour, 67 (2007), 155-164. doi: 10.1016/j.anbehav.2003.04.004.
[9]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.
[10]	F. Cucker and S. Smale, Lectures on emergence, Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.
[11]	F. Cucker, S. Smale and D. Zhou, Modeling language evolution, Found. Comput. Math., 4 (2004), 315-343. doi: 10.1007/s10208-003-0101-2.
[12]	F. Cucker and C. Huepe, Flocking with informed agents, Mathematics In Action, 1 (2008), 1-25. doi: 10.5802/msia.1.
[13]	F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierachical leadership and random interactions, SIAM J. Aappl. Math., 71 (2011), 1307-1316. doi: 10.1137/100785910.
[14]	G. Grgoire, H. Chat and Y. Tu, Moving and staying together without a leader, Physica D, 181 (2003), 157-170. doi: 10.1016/S0167-2789(03)00102-7.
[15]	H. Su, X. Wang and Z. Lin, Flocking of multi-agents with a virtual leader, IEEE Transactions on automatic control, 54 (2009), 293-307. doi: 10.1109/TAC.2008.2010897.
[16]	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.
[17]	I. Couzin, J. Krause, R. James and G. Ruxton, Collective memory and spatial sorting in animal groups, J. theor. Biol., 218 (2002), 1-11. doi: 10.1006/jtbi.2002.3065.
[18]	I. D. Couzin, J. Krause, R. James and G. D. Ruxton, Complex spatial group patterns result from different animal communication mechanisms, PNAS, 104 (2007), 6974-6979.
[19]	J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cuker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236. doi: 10.1137/090757290.
[20]	J. Dong, Flocking under hierarchical leadership with a free-will leader, International journal of robust and nonlinear control, 23 (2013), 1891-1898.
[21]	J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719. doi: 10.1137/060673254.
[22]	M. Agueh, R. Illner and A. Richardson, Analysis and simulations of a refined flocking and swarming model of Cuker-Smale type, Kinetic and Related Models, 4 (2011), 1-16. doi: 10.3934/krm.2011.4.1.
[23]	M. Ballerini, N. Cabibbo, R. Candelier and et al., Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, PNAS, 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.
[24]	M. Nagy, Z. Akos, D. Biro and T. Vicsek, Hierarchical group dynamics in pigeon flocks, Nature, 464 (2010), 890-894. doi: 10.1038/nature08891.
[25]	S. Ha and J. 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.
[26]	S. 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.
[27]	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.
[28]	S. Motsch and E. Tadmor, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM REVIEW, 56 (2014), 577-621. doi: 10.1137/120901866.
[29]	T. Vicsek, A. Czirk, 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), 1226-1235. doi: 10.1103/PhysRevLett.75.1226.
[30]	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.
[31]	Z. Akos, T. Vicsek and E. Kubinyi, Leadership and path characteristics during walks are linked to dominance order and individual traits in Dogs, Computational biology, 10 (2014), 1-9.
[32]	Z. Li and X. Xue, Cucher-Smale flocking under rooted leadership with fixed switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174. doi: 10.1137/100791774.
[33]	Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with free-will agents, Physica A, 410 (2014), 205-217. doi: 10.1016/j.physa.2014.05.008.
[34]	Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete and Continue Dynamical Systems, 34 (2014), 3683-3702. doi: 10.3934/dcds.2014.34.3683.
[35]	Z. Wang and D. Gu, A local sensor based leader-follower flocking system, in Proc. 2008 IEEE Int. Conf. Robot. Autom., (2008), 19-23.

show all references

References:

[1]	A. Flack, B. Pettit and R. Freeman, What are leaders made of? The role of individual experience in determining leader-follower relations in homing pigeons, Animal Behaviour, 83 (2012), 703-709. doi: 10.1016/j.anbehav.2011.12.018.
[2]	A. Flack, B. Pettit and R. Freeman, Fault-tolerant flocking for a group of autonomous mobile robots, The Journal of Systems and Software, 84 (2011), 29-36.
[3]	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.
[4]	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.
[5]	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.
[6]	D. Gu and Z. Wang, Leader-Follower flocking: Algorithms and experiments, PNAS, 17 (2008), 1211-1219.
[7]	D. Gu and Z. Wang, Leader-follower flocking: Algorithms and experiments, IEEE Transactions on control systems technology, 17 (2009), 1211-1219.
[8]	D. J. Hoare, I. D. Couzin, J.-G. J. Godin and J. Krause, Context-dependent group size choice in fish, Animal Behaviour, 67 (2007), 155-164. doi: 10.1016/j.anbehav.2003.04.004.
[9]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.
[10]	F. Cucker and S. Smale, Lectures on emergence, Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.
[11]	F. Cucker, S. Smale and D. Zhou, Modeling language evolution, Found. Comput. Math., 4 (2004), 315-343. doi: 10.1007/s10208-003-0101-2.
[12]	F. Cucker and C. Huepe, Flocking with informed agents, Mathematics In Action, 1 (2008), 1-25. doi: 10.5802/msia.1.
[13]	F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierachical leadership and random interactions, SIAM J. Aappl. Math., 71 (2011), 1307-1316. doi: 10.1137/100785910.
[14]	G. Grgoire, H. Chat and Y. Tu, Moving and staying together without a leader, Physica D, 181 (2003), 157-170. doi: 10.1016/S0167-2789(03)00102-7.
[15]	H. Su, X. Wang and Z. Lin, Flocking of multi-agents with a virtual leader, IEEE Transactions on automatic control, 54 (2009), 293-307. doi: 10.1109/TAC.2008.2010897.
[16]	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.
[17]	I. Couzin, J. Krause, R. James and G. Ruxton, Collective memory and spatial sorting in animal groups, J. theor. Biol., 218 (2002), 1-11. doi: 10.1006/jtbi.2002.3065.
[18]	I. D. Couzin, J. Krause, R. James and G. D. Ruxton, Complex spatial group patterns result from different animal communication mechanisms, PNAS, 104 (2007), 6974-6979.
[19]	J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cuker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236. doi: 10.1137/090757290.
[20]	J. Dong, Flocking under hierarchical leadership with a free-will leader, International journal of robust and nonlinear control, 23 (2013), 1891-1898.
[21]	J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719. doi: 10.1137/060673254.
[22]	M. Agueh, R. Illner and A. Richardson, Analysis and simulations of a refined flocking and swarming model of Cuker-Smale type, Kinetic and Related Models, 4 (2011), 1-16. doi: 10.3934/krm.2011.4.1.
[23]	M. Ballerini, N. Cabibbo, R. Candelier and et al., Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, PNAS, 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.
[24]	M. Nagy, Z. Akos, D. Biro and T. Vicsek, Hierarchical group dynamics in pigeon flocks, Nature, 464 (2010), 890-894. doi: 10.1038/nature08891.
[25]	S. Ha and J. 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.
[26]	S. 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.
[27]	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.
[28]	S. Motsch and E. Tadmor, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM REVIEW, 56 (2014), 577-621. doi: 10.1137/120901866.
[29]	T. Vicsek, A. Czirk, 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), 1226-1235. doi: 10.1103/PhysRevLett.75.1226.
[30]	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.
[31]	Z. Akos, T. Vicsek and E. Kubinyi, Leadership and path characteristics during walks are linked to dominance order and individual traits in Dogs, Computational biology, 10 (2014), 1-9.
[32]	Z. Li and X. Xue, Cucher-Smale flocking under rooted leadership with fixed switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174. doi: 10.1137/100791774.
[33]	Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with free-will agents, Physica A, 410 (2014), 205-217. doi: 10.1016/j.physa.2014.05.008.
[34]	Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete and Continue Dynamical Systems, 34 (2014), 3683-3702. doi: 10.3934/dcds.2014.34.3683.
[35]	Z. Wang and D. Gu, A local sensor based leader-follower flocking system, in Proc. 2008 IEEE Int. Conf. Robot. Autom., (2008), 19-23.

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