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    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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[26]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[26]

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

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