• Previous Article
    Well-posedness for boundary value problems for coagulation-fragmentation equations
  • KRM Home
  • This Issue
  • Next Article
    Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field
August  2020, 13(4): 795-813. doi: 10.3934/krm.2020027

Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime

1. 

Computer, Electrical and Mathematical Sciences & Engineering, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

2. 

Institute of Applied and Computational Mathematics (IACM-FORTH), N. Plastira 100, Vassilika Vouton, GR - 700 13 Heraklion, Crete, Greece

* Corresponding author: Jan Haskovec

Received  September 2019 Revised  February 2020 Published  May 2020

Fund Project: The first author is supported by KAUST baseline funds

We study a variant of the Cucker-Smale system with reaction-type delay. Using novel backward-forward and stability estimates on appropriate quantities we derive sufficient conditions for asymptotic flocking of the solutions. These conditions, although not explicit, relate the velocity fluctuation of the initial datum and the length of the delay. If satisfied, they guarantee monotone decay (i.e., non-oscillatory regime) of the velocity fluctuations towards zero for large times. For the simplified setting with only two agents and constant communication rate the Cucker-Smale system reduces to the delay negative feedback equation. We demonstrate that in this case our method provides the sharp condition for the size of the delay such that the solution be non-oscillatory. Moreover, we comment on the mathematical issues appearing in the formal macroscopic description of the reaction-type delay system.

Citation: Jan Haskovec, Ioannis Markou. Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime. Kinetic and Related Models, 2020, 13 (4) : 795-813. doi: 10.3934/krm.2020027
References:
[1]

J. CañizoJ. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.

[2]

J. A. CarriloM. FornasierJ. 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.

[3]

J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, Vol. 533, Springer, Vienna, 2014, 1–46. doi: 10.1007/978-3-7091-1785-9_1.

[4]

Y.-P. Choi, S.-Y. Ha and Z. Li, Emergent dynamics of the Cucker–Smale flocking model and its variants, in Active Particles, Vol. 1, Birkhäuser/Springer, Cham, 2017,299–331.

[5]

Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033.  doi: 10.3934/krm.2017040.

[6]

Y.-P. Choi and J. Haskovec, Hydrodynamic Cucker-Smale model with normalized communication weights and time delay, SIAM J. Math. Anal., 51 (2019), 2660-2685.  doi: 10.1137/17M1139151.

[7]

Y.-P. Choi and Z. Li, Emergent behavior of Cucker-Smale flocking particles with heterogeneous time delays, Appl. Math. Lett., 86 (2018), 49-56.  doi: 10.1016/j.aml.2018.06.018.

[8]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[9]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[10]

J.-G. DongS.-Y. Ha and D. Kim, Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5569-5596.  doi: 10.3934/dcdsb.2019072.

[11]

R. ErbanJ. Haskovec and Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557.  doi: 10.1137/15M1030467.

[12] I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. 
[13]

S.-Y. Ha.J. KimJ. Park and X. Zhang, Complete cluster predictability of the cucker-smale flocking model on the real line, Arch. Rational Mech. Anal., 231 (2019), 319-365.  doi: 10.1007/s00205-018-1281-x.

[14]

S.-Y. HaC. LattanzioB. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.  doi: 10.1090/S0033-569X-2010-01200-7.

[15]

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.

[16]

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.

[17]

D. KaliseJ. PeszekA. Peters and Y.-P. Choi, A collisionless singular Cucker-Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., 18 (2019), 1954-1981.  doi: 10.1137/19M1241799.

[18]

Y. Liu and J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Anal. Appl., 415 (2014), 53-61.  doi: 10.1016/j.jmaa.2014.01.036.

[19]

I. Markou, Collision avoiding in the singular Cucker-Smale model with nonlinear velocity couplings, Discrete Contin. Dyn. Syst., 38 (2018), 5245-5260.  doi: 10.3934/dcds.2018232.

[20]

G. Naldi, L. Pareschi and G. Toscani (eds.), Mathematical Modeling of Collective Behaviour in Socio-Economic and Life Sciences, Birkhäuser Boston, Ltd., Boston, MA, 2010. doi: 10.1007/978-0-8176-4946-3.

[21] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods, Oxford University Press, 2014. 
[22]

C. Pignotti and E. Trelat, Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays, Commun. Math. Sci., 16 (2018), 2053-2076.  doi: 10.4310/CMS.2018.v16.n8.a1.

[23]

C. Pignotti and I. Reche Vallejo, Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, J. Math. Anal. Appl., 464 (2018), 1313-1332.  doi: 10.1016/j.jmaa.2018.04.070.

[24]

C. Pignotti and I. Reche Vallejo, Asymptotic Analysis of a Cucker-Smale System with Leadership and Distributed Delay, in Trends in Control Theory and Partial Differential Equations

[25]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2011. doi: 10.1007/978-1-4419-7646-8.

[26]

T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.

show all references

References:
[1]

J. CañizoJ. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.

[2]

J. A. CarriloM. FornasierJ. 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.

[3]

J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, Vol. 533, Springer, Vienna, 2014, 1–46. doi: 10.1007/978-3-7091-1785-9_1.

[4]

Y.-P. Choi, S.-Y. Ha and Z. Li, Emergent dynamics of the Cucker–Smale flocking model and its variants, in Active Particles, Vol. 1, Birkhäuser/Springer, Cham, 2017,299–331.

[5]

Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033.  doi: 10.3934/krm.2017040.

[6]

Y.-P. Choi and J. Haskovec, Hydrodynamic Cucker-Smale model with normalized communication weights and time delay, SIAM J. Math. Anal., 51 (2019), 2660-2685.  doi: 10.1137/17M1139151.

[7]

Y.-P. Choi and Z. Li, Emergent behavior of Cucker-Smale flocking particles with heterogeneous time delays, Appl. Math. Lett., 86 (2018), 49-56.  doi: 10.1016/j.aml.2018.06.018.

[8]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[9]

F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.

[10]

J.-G. DongS.-Y. Ha and D. Kim, Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5569-5596.  doi: 10.3934/dcdsb.2019072.

[11]

R. ErbanJ. Haskovec and Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557.  doi: 10.1137/15M1030467.

[12] I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. 
[13]

S.-Y. Ha.J. KimJ. Park and X. Zhang, Complete cluster predictability of the cucker-smale flocking model on the real line, Arch. Rational Mech. Anal., 231 (2019), 319-365.  doi: 10.1007/s00205-018-1281-x.

[14]

S.-Y. HaC. LattanzioB. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.  doi: 10.1090/S0033-569X-2010-01200-7.

[15]

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.

[16]

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.

[17]

D. KaliseJ. PeszekA. Peters and Y.-P. Choi, A collisionless singular Cucker-Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., 18 (2019), 1954-1981.  doi: 10.1137/19M1241799.

[18]

Y. Liu and J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Anal. Appl., 415 (2014), 53-61.  doi: 10.1016/j.jmaa.2014.01.036.

[19]

I. Markou, Collision avoiding in the singular Cucker-Smale model with nonlinear velocity couplings, Discrete Contin. Dyn. Syst., 38 (2018), 5245-5260.  doi: 10.3934/dcds.2018232.

[20]

G. Naldi, L. Pareschi and G. Toscani (eds.), Mathematical Modeling of Collective Behaviour in Socio-Economic and Life Sciences, Birkhäuser Boston, Ltd., Boston, MA, 2010. doi: 10.1007/978-0-8176-4946-3.

[21] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods, Oxford University Press, 2014. 
[22]

C. Pignotti and E. Trelat, Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays, Commun. Math. Sci., 16 (2018), 2053-2076.  doi: 10.4310/CMS.2018.v16.n8.a1.

[23]

C. Pignotti and I. Reche Vallejo, Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, J. Math. Anal. Appl., 464 (2018), 1313-1332.  doi: 10.1016/j.jmaa.2018.04.070.

[24]

C. Pignotti and I. Reche Vallejo, Asymptotic Analysis of a Cucker-Smale System with Leadership and Distributed Delay, in Trends in Control Theory and Partial Differential Equations

[25]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2011. doi: 10.1007/978-1-4419-7646-8.

[26]

T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.

[1]

Mauro Rodriguez Cartabia. Cucker-Smale model with time delay. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2409-2432. doi: 10.3934/dcds.2021195

[2]

Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168

[3]

Young-Pil Choi, Jan Haskovec. Cucker-Smale model with normalized communication weights and time delay. Kinetic and Related Models, 2017, 10 (4) : 1011-1033. doi: 10.3934/krm.2017040

[4]

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

[5]

Martin Friesen, Oleksandr Kutoviy. Stochastic Cucker-Smale flocking dynamics of jump-type. Kinetic and Related Models, 2020, 13 (2) : 211-247. doi: 10.3934/krm.2020008

[6]

Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419

[7]

Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223

[8]

Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062

[9]

Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5569-5596. doi: 10.3934/dcdsb.2019072

[10]

Yu-Jhe Huang, Zhong-Fu Huang, Jonq Juang, Yu-Hao Liang. Flocking of non-identical Cucker-Smale models on general coupling network. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1111-1127. doi: 10.3934/dcdsb.2020155

[11]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic and Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023

[12]

Seung-Yeal Ha, Doheon Kim, Weiyuan Zou. Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field. Kinetic and Related Models, 2020, 13 (4) : 759-793. doi: 10.3934/krm.2020026

[13]

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

[14]

Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure and Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028

[15]

Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang, Xiongtao Zhang. Emergent dynamics in the interactions of Cucker-Smale ensembles. Kinetic and Related Models, 2017, 10 (3) : 689-723. doi: 10.3934/krm.2017028

[16]

Roberto Natalini, Thierry Paul. On the mean field limit for Cucker-Smale models. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2873-2889. doi: 10.3934/dcdsb.2021164

[17]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023

[18]

Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. On the Cucker-Smale ensemble with $ q $-closest neighbors under time-delayed communications. Kinetic and Related Models, 2020, 13 (4) : 653-676. doi: 10.3934/krm.2020022

[19]

Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks and Heterogeneous Media, 2018, 13 (3) : 379-407. doi: 10.3934/nhm.2018017

[20]

Young-Pil Choi, Cristina Pignotti. Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays. Networks and Heterogeneous Media, 2019, 14 (4) : 789-804. doi: 10.3934/nhm.2019032

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (240)
  • HTML views (106)
  • Cited by (2)

Other articles
by authors

[Back to Top]