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

Cascade flocking with free-will

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

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.
Citation: Le Li, Lihong Huang, Jianhong Wu. Cascade flocking with free-will. Discrete & 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. doi: 10.1016/j.anbehav.2011.12.018. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/S0036139903437424. Google Scholar

[4]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bull. Math. Bio., 68 (2006), 1601. doi: 10.1007/s11538-006-9088-6. Google Scholar

[5]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model,, In: ACM SIGGRAPH Computer Graphics, 21 (1987), 25. doi: 10.1145/37401.37406. Google Scholar

[6]

D. Gu and Z. Wang, Leader-Follower flocking: Algorithms and experiments,, PNAS, 17 (2008), 1211. Google Scholar

[7]

D. Gu and Z. Wang, Leader-follower flocking: Algorithms and experiments,, IEEE Transactions on control systems technology, 17 (2009), 1211. Google Scholar

[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. doi: 10.1016/j.anbehav.2003.04.004. Google Scholar

[9]

F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852. doi: 10.1109/TAC.2007.895842. Google Scholar

[10]

F. Cucker and S. Smale, Lectures on emergence,, Japan J. Math., 2 (2007), 197. doi: 10.1007/s11537-007-0647-x. Google Scholar

[11]

F. Cucker, S. Smale and D. Zhou, Modeling language evolution,, Found. Comput. Math., 4 (2004), 315. doi: 10.1007/s10208-003-0101-2. Google Scholar

[12]

F. Cucker and C. Huepe, Flocking with informed agents,, Mathematics In Action, 1 (2008), 1. doi: 10.5802/msia.1. Google Scholar

[13]

F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierachical leadership and random interactions,, SIAM J. Aappl. Math., 71 (2011), 1307. doi: 10.1137/100785910. Google Scholar

[14]

G. Grgoire, H. Chat and Y. Tu, Moving and staying together without a leader,, Physica D, 181 (2003), 157. doi: 10.1016/S0167-2789(03)00102-7. Google Scholar

[15]

H. Su, X. Wang and Z. Lin, Flocking of multi-agents with a virtual leader,, IEEE Transactions on automatic control, 54 (2009), 293. doi: 10.1109/TAC.2008.2010897. Google Scholar

[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. doi: 10.1038/nature03236. Google Scholar

[17]

I. Couzin, J. Krause, R. James and G. Ruxton, Collective memory and spatial sorting in animal groups,, J. theor. Biol., 218 (2002), 1. doi: 10.1006/jtbi.2002.3065. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/090757290. Google Scholar

[20]

J. Dong, Flocking under hierarchical leadership with a free-will leader,, International journal of robust and nonlinear control, 23 (2013), 1891. Google Scholar

[21]

J. Shen, Cucker-Smale flocking under hierarchical leadership,, SIAM J. Appl. Math., 68 (2008), 694. doi: 10.1137/060673254. Google Scholar

[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. doi: 10.3934/krm.2011.4.1. Google Scholar

[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. doi: 10.1073/pnas.0711437105. Google Scholar

[24]

M. Nagy, Z. Akos, D. Biro and T. Vicsek, Hierarchical group dynamics in pigeon flocks,, Nature, 464 (2010), 890. doi: 10.1038/nature08891. Google Scholar

[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. doi: 10.4310/CMS.2009.v7.n2.a2. Google Scholar

[26]

S. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking,, Kinet. Relat.Models, 1 (2008), 415. doi: 10.3934/krm.2008.1.415. Google Scholar

[27]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior,, J. Stat. Phys., 144 (2011), 923. doi: 10.1007/s10955-011-0285-9. Google Scholar

[28]

S. Motsch and E. Tadmor, Asymptotic flocking dynamics for the kinetic Cucker-Smale model,, SIAM REVIEW, 56 (2014), 577. doi: 10.1137/120901866. Google Scholar

[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. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

[30]

Y. Liu and K. Passino, Stable social foraging swarms in a noisy environment,, IEEE Trans. Automat. Control, 49 (2004), 30. doi: 10.1109/TAC.2003.821416. Google Scholar

[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. Google Scholar

[32]

Z. Li and X. Xue, Cucher-Smale flocking under rooted leadership with fixed switching topologies,, SIAM J. Appl. Math., 70 (2010), 3156. doi: 10.1137/100791774. Google Scholar

[33]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with free-will agents,, Physica A, 410 (2014), 205. doi: 10.1016/j.physa.2014.05.008. Google Scholar

[34]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference,, Discrete and Continue Dynamical Systems, 34 (2014), 3683. doi: 10.3934/dcds.2014.34.3683. Google Scholar

[35]

Z. Wang and D. Gu, A local sensor based leader-follower flocking system,, in Proc. 2008 IEEE Int. Conf. Robot. Autom., (2008), 19. Google Scholar

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. doi: 10.1016/j.anbehav.2011.12.018. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/S0036139903437424. Google Scholar

[4]

C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bull. Math. Bio., 68 (2006), 1601. doi: 10.1007/s11538-006-9088-6. Google Scholar

[5]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model,, In: ACM SIGGRAPH Computer Graphics, 21 (1987), 25. doi: 10.1145/37401.37406. Google Scholar

[6]

D. Gu and Z. Wang, Leader-Follower flocking: Algorithms and experiments,, PNAS, 17 (2008), 1211. Google Scholar

[7]

D. Gu and Z. Wang, Leader-follower flocking: Algorithms and experiments,, IEEE Transactions on control systems technology, 17 (2009), 1211. Google Scholar

[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. doi: 10.1016/j.anbehav.2003.04.004. Google Scholar

[9]

F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852. doi: 10.1109/TAC.2007.895842. Google Scholar

[10]

F. Cucker and S. Smale, Lectures on emergence,, Japan J. Math., 2 (2007), 197. doi: 10.1007/s11537-007-0647-x. Google Scholar

[11]

F. Cucker, S. Smale and D. Zhou, Modeling language evolution,, Found. Comput. Math., 4 (2004), 315. doi: 10.1007/s10208-003-0101-2. Google Scholar

[12]

F. Cucker and C. Huepe, Flocking with informed agents,, Mathematics In Action, 1 (2008), 1. doi: 10.5802/msia.1. Google Scholar

[13]

F. Dalmao and E. Mordecki, Cucker-Smale flocking under hierachical leadership and random interactions,, SIAM J. Aappl. Math., 71 (2011), 1307. doi: 10.1137/100785910. Google Scholar

[14]

G. Grgoire, H. Chat and Y. Tu, Moving and staying together without a leader,, Physica D, 181 (2003), 157. doi: 10.1016/S0167-2789(03)00102-7. Google Scholar

[15]

H. Su, X. Wang and Z. Lin, Flocking of multi-agents with a virtual leader,, IEEE Transactions on automatic control, 54 (2009), 293. doi: 10.1109/TAC.2008.2010897. Google Scholar

[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. doi: 10.1038/nature03236. Google Scholar

[17]

I. Couzin, J. Krause, R. James and G. Ruxton, Collective memory and spatial sorting in animal groups,, J. theor. Biol., 218 (2002), 1. doi: 10.1006/jtbi.2002.3065. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/090757290. Google Scholar

[20]

J. Dong, Flocking under hierarchical leadership with a free-will leader,, International journal of robust and nonlinear control, 23 (2013), 1891. Google Scholar

[21]

J. Shen, Cucker-Smale flocking under hierarchical leadership,, SIAM J. Appl. Math., 68 (2008), 694. doi: 10.1137/060673254. Google Scholar

[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. doi: 10.3934/krm.2011.4.1. Google Scholar

[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. doi: 10.1073/pnas.0711437105. Google Scholar

[24]

M. Nagy, Z. Akos, D. Biro and T. Vicsek, Hierarchical group dynamics in pigeon flocks,, Nature, 464 (2010), 890. doi: 10.1038/nature08891. Google Scholar

[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. doi: 10.4310/CMS.2009.v7.n2.a2. Google Scholar

[26]

S. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking,, Kinet. Relat.Models, 1 (2008), 415. doi: 10.3934/krm.2008.1.415. Google Scholar

[27]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior,, J. Stat. Phys., 144 (2011), 923. doi: 10.1007/s10955-011-0285-9. Google Scholar

[28]

S. Motsch and E. Tadmor, Asymptotic flocking dynamics for the kinetic Cucker-Smale model,, SIAM REVIEW, 56 (2014), 577. doi: 10.1137/120901866. Google Scholar

[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. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

[30]

Y. Liu and K. Passino, Stable social foraging swarms in a noisy environment,, IEEE Trans. Automat. Control, 49 (2004), 30. doi: 10.1109/TAC.2003.821416. Google Scholar

[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. Google Scholar

[32]

Z. Li and X. Xue, Cucher-Smale flocking under rooted leadership with fixed switching topologies,, SIAM J. Appl. Math., 70 (2010), 3156. doi: 10.1137/100791774. Google Scholar

[33]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with free-will agents,, Physica A, 410 (2014), 205. doi: 10.1016/j.physa.2014.05.008. Google Scholar

[34]

Z. Li, Effectual leadership in flocks with hierarchy and individual preference,, Discrete and Continue Dynamical Systems, 34 (2014), 3683. doi: 10.3934/dcds.2014.34.3683. Google Scholar

[35]

Z. Wang and D. Gu, A local sensor based leader-follower flocking system,, in Proc. 2008 IEEE Int. Conf. Robot. Autom., (2008), 19. Google Scholar

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