January  2020, 25(1): 443-472. doi: 10.3934/dcdsb.2019189

A hybrid model of collective motion of discrete particles under alignment and continuum chemotaxis

1. 

Istituto per le Applicazioni del Calcolo "M. Picone", – Consiglio Nazionale delle Ricerche, Via dei Taurini 19 00185 Rome, Italy

2. 

Università Campus Bio-Medico di Roma, Via Àlvaro del Portillo 00128 Rome, Italy

3. 

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico Ⅱ", Via Cintia 80126 Naples, Italy

4. 

Istituto per le Applicazioni del Calcolo "M. Picone", – Consiglio Nazionale delle Ricerche, Via Pietro Castellino 111 80131 Naples, Italy

Received  August 2018 Revised  March 2019 Published  January 2020 Early access  September 2019

In this paper we propose and study a hybrid discrete–continuous mathematical model of collective motion under alignment and chemotaxis effect. Starting from paper [23], in which the Cucker-Smale model [22] was coupled with other cell mechanisms, to describe the cell migration and self-organization in the zebrafish lateral line primordium, we introduce a simplified model in which the coupling between an alignment and chemotaxis mechanism acts on a system of interacting particles. In particular we rely on a hybrid description in which the agents are discrete entities, while the chemoattractant is considered as a continuous signal. The proposed model is then studied both from an analytical and a numerical point of view. From the analytic point of view we prove, globally in time, existence and uniqueness of the solution. Then, the asymptotic behaviour of a linearised version of the system is investigated. Through a suitable Lyapunov functional we show that for t → +∞, the migrating aggregate exponentially converges to a state in which all the particles have a same position with zero velocity. Finally, we present a comparison between the analytical findings and some numerical results, concerning the behaviour of the full nonlinear system.

Citation: Ezio Di Costanzo, Marta Menci, Eleonora Messina, Roberto Natalini, Antonia Vecchio. A hybrid model of collective motion of discrete particles under alignment and continuum chemotaxis. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 443-472. doi: 10.3934/dcdsb.2019189
References:
[1]

G. Albi and L. Pareschi, Modeling self-organized systems interacting with few individuals: From microscopic to macroscopic dynamics, Appl Math Lett, 26 (2013), 397-401.  doi: 10.1016/j.aml.2012.10.011.  Google Scholar

[2]

I. Aoki, A simulation study on the schooling mechanism in fish, Bullettin Of The Japanese Society Scientific Fischeries, 48 (1982), 1081-1088.  doi: 10.2331/suisan.48.1081.  Google Scholar

[3]

Y. Arboleda-EstudilloM. KriegJ. StühmerN. A. LicataD. J. Muller and C.-P. Heisenberg, Movement Directionality in Collective Migration of Germ Layer Progenitors, Curr Biology, 20 (2010), 161-169.  doi: 10.1016/j.cub.2009.11.036.  Google Scholar

[4]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, P Natl Acad Sci USA, 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105.  Google Scholar

[5]

J. M. Belmonte, G. L. Thomas, L. G. Brunnet, R. M. de Almeida and H. Chaté, Self-propelled particle model for cell-sorting phenomena, Phys Rev Lett, 100 (2008), 248702. doi: 10.1103/PhysRevLett.100.248702.  Google Scholar

[6] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, UK, 2004.  doi: 10.1017/CBO9780511543234.  Google Scholar
[7]

L. BrunoA. TosinP. Tricerri and F. Venuti, Non-local first-order modelling of crowd dynamics: A multidimensional framework with applications, Applied Mathematical Modelling, 35 (2011), 426-445.  doi: 10.1016/j.apm.2010.07.007.  Google Scholar

[8]

T. A. Burton, Volterra Integral and Differential Equations. Second Edition, Springer, 2005.  Google Scholar

[9]

J. A. CarrilloM. 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

[10]

T. ColinM.-C. DurrieuJ. JoieY. LeiY. MammeriC. Poignard and O. Saut, Modeling of the migration of endothelial cells on bioactive micropatterned polymers, Math. BioSci. and Eng., 10 (2013), 997-1015.  doi: 10.3934/mbe.2013.10.997.  Google Scholar

[11]

A. ColombiM. Scianna and A. Tosin, Differentiated cell behavior: A multiscale approach using measure theory, J Math Biol, 71 (2015), 1049-1079.  doi: 10.1007/s00285-014-0846-z.  Google Scholar

[12]

I. D. CouzinJ. KrauseN. R. Franks and S. A. Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516.  doi: 10.1038/nature03236.  Google Scholar

[13]

I. D. CouzinJ. KrauseR. JamesG. D. Ruxton and N. R. Franks, Collective memory and spatial sorting in animal groups, J Theor Biol, 218 (2002), 1-11.  doi: 10.1006/jtbi.2002.3065.  Google Scholar

[14]

E. CristianiP. Frasca and B. Piccoli, Effects of anisotropic interactions on the structure of animal groups, J. Math. Biol., 62 (2011), 569-588.  doi: 10.1007/s00285-010-0347-7.  Google Scholar

[15]

E. Cristiani, B. Piccoli and A. Tosin, Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints, in Mathematical Modeling of Collective Behavior in Socio-economic and Life-sciences (eds. G. Naldi, L. Pareschi and G. Toscani), Modeling and Simulation in Science, Engineering, and Technology, Birkhäuser Boston, 2010,337–364. doi: 10.1007/978-0-8176-4946-3_13.  Google Scholar

[16]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS & A: Modeling, Simulation and Applications, Springer International Publishing, 2014. doi: 10.1007/978-3-319-06620-2.  Google Scholar

[17]

E. CristianiF. S. Priuli and A. Tosin, Modeling rationality to control self-organization of crowds: an environmental approach, SIAM J Appl Math, 75 (2015), 605-629.  doi: 10.1137/140962413.  Google Scholar

[18]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Transactions on Automatic Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.  Google Scholar

[19]

F. Cucker and J.-G. Dong, A General Collision-Avoiding Flocking Framework, IEEE Transactions on Automatic Control, 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.  Google Scholar

[20]

F. Cucker and C. Huepe, Flocking with informed agents, MathematicS In Action, 1 (2008), 1-25.  doi: 10.5802/msia.1.  Google Scholar

[21]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[22]

F. Cucker and S. Smale, Emergent Behavior in Flocks, Ieee T Automat Contr, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[23]

E. Di CostanzoR. Natalini and L. Preziosi, A hybrid mathematical model for self-organizing cell migration in the zebrafish lateral line, J of Math Biol, 71 (2015), 171-214.  doi: 10.1007/s00285-014-0812-9.  Google Scholar

[24]

E. Di CostanzoA. GiacomelloE. MessinaR. NataliniG. PontrelliF. RossiR. Smits and M. Twarogowska, A discrete in continuous mathematical model of cardiac progenitor cells formation and growth as spheroid clusters (cardiospheres), Mathematical Medicine and Biology: A Journal of the IMA, 35 (2018), 121-144.  doi: 10.1093/imammb/dqw022.  Google Scholar

[25]

E. Di Costanzo, R. Natalini and L. Preziosi, A hybrid model of cell migration in zebrafish embryogenesis, in ITM Web of Conferences, EDP Sciences, 5 (2015), 00013. doi: 10.1051/itmconf/20150500013.  Google Scholar

[26]

M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys Rev Lett, 96 (2016), 104302. doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[27] M. Eisenbach and J. W. Lengeler, Chemotaxis, Imperial College Press, 2004.   Google Scholar
[28]

J. J. FariaJ. R. G. DyerC. R. Tosh and J. Krause, Leadership and social information use in human crowds, Animal Behaviour, 79 (2010), 895-901.  doi: 10.1016/j.anbehav.2009.12.039.  Google Scholar

[29]

F. E. Fish, Kinematics of ducklings swimming in formation: consequences of position, Journal of Experimental Zoology, 273 (1995), 1-11.  doi: 10.1002/jez.1402730102.  Google Scholar

[30]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964.  Google Scholar

[31]

G. Grégoire and H. Chaté, Onset of collective and cohesive motion, Phys. Rev. Lett., 92. Google Scholar

[32]

G. GrégoireH. Chaté and Y. Tu, Moving and staying together without a leader, Physica D, 181 (2013), 157-170.  doi: 10.1016/S0167-2789(03)00102-7.  Google Scholar

[33]

S. Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Comm. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[34]

S. Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete and Continuous Dynamical Systems - Series B, 12 (2009), 77-108.  doi: 10.3934/dcdsb.2009.12.77.  Google Scholar

[35]

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

[36]

H. Hatzikirou and A. Deutsch, Collective guidance of collective cell migration, Curr. Top. Dev. Biol., 81 (2007), 401-434.   Google Scholar

[37]

D. HelbingF. SchweitzerJ. Keltsch and P. Molnár, Active walker model for the formation of human and animal trail systems, Physical Review, 56 (1997), 2527-2539.  doi: 10.1103/PhysRevE.56.2527.  Google Scholar

[38]

C. K. Hemelrijk and H. Hildenbrandt, Self-organized shape and frontal density of fish schools, Ethology, 114 (2008), 245-254.  doi: 10.1111/j.1439-0310.2007.01459.x.  Google Scholar

[39]

W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Computational Mathematics, Springer, 2003. doi: 10.1007/978-3-662-09017-6.  Google Scholar

[40]

A. Huth and C. Wissel, The simulation of the movement of fish schools, J Theor Biol, 156 (1992), 365-385.  doi: 10.1016/S0022-5193(05)80681-2.  Google Scholar

[41]

C. C. IoannouC. R. ToshL. Neville and J. Krause, The confusion effect. from neural networks to reduced predation risk, Behavioral Ecology, 19 (2008), 126-130.  doi: 10.1093/beheco/arm109.  Google Scholar

[42]

A. JadbabaieJ. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans Autom Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781.  Google Scholar

[43]

J. JoieY. LeiM.-C. DurrieuT. ColinC. Poignard and O. Saut, Migration and orientation of endothelial cells on micropatterned polymers: A simple model based on classical mechanics, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1059-1076.  doi: 10.3934/dcdsb.2015.20.1059.  Google Scholar

[44]

H. K. Khalil, Nonlinear Systems. Third Edition, Prentice Hall, 2002. Google Scholar

[45]

V. Lakshmikantham and M. R. M. Rama, Theory of Integro-Differential Equations, vol. 1 of Stability and Control: Theory, Methods and Applications, Gordon and Breach Science Publishers, 1995.  Google Scholar

[46]

C. Lubich, On the stability of linear multistep methods for Volterra convolution equations, IMA J. Numer. Anal., 3 (1983), 439-465.  doi: 10.1093/imanum/3.4.439.  Google Scholar

[47]

E. Méhes and T. Vicsek, Collective motion of cells: from experiments to models, Integr Biol, 6 (2014), 831-854.   Google Scholar

[48]

M. Menci and M. Papi, Global solutions for a path-dependent hybrid system of differential equations under parabolic signal, Nonlinear Analysis, 184 (2019), 172-192.  doi: 10.1016/j.na.2019.01.034.  Google Scholar

[49]

M. MoussaïdD. Helbing and G. Theraulaz, How simple rules determine pedestrian behavior and crowd disasters, Proceeding of the National Academy of Sciences of the United States of America, 108 (2011), 6884-6888.   Google Scholar

[50]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications. Third edition, Springer, 2003.  Google Scholar

[51]

G. Naldi, L. Pareschi and G. Toscani (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life-Sciences, Modeling and Simulation in Science, Engineering, and Technology, Birkhäuser Boston, 2010. doi: 10.1007/978-0-8176-4946-3.  Google Scholar

[52]

M. Onitsuka, Uniform asymptotic stability for damped linear oscillators with variable parameters, Applied Mathematics and Computation, 218 (2011), 1436-1442.  doi: 10.1016/j.amc.2011.06.025.  Google Scholar

[53] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2014.   Google Scholar
[54]

B. Perthame, Transport Equations in Biology, Birkhäuser, 2007.  Google Scholar

[55]

B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107.  doi: 10.1007/s00161-009-0100-x.  Google Scholar

[56]

T. PitcherA. Magurran and I. Winfield, Fish in larger shoals find food faster, Behav Ecol and Sociobiology, 10 (1982), 149-151.  doi: 10.1007/BF00300175.  Google Scholar

[57]

N. Sepúlveda, L. Petitjean, O. Cochet, E. Grasland-Mongrain, P. Silberzan and V. Hakim, Collective cell motion in an epithelial sheet can be quantitatively described by a stochastic interacting particle model, PLOS Computational Biology, 9 (2013), e1002944, 12 pp. doi: 10.1371/journal.pcbi.1002944.  Google Scholar

[58]

D. Strömbom, Collective motion from local attraction, J Theor Biol, 283 (2011), 145-151.  doi: 10.1016/j.jtbi.2011.05.019.  Google Scholar

[59]

B. Szabò, G. J. Szöllösi, B. Gönci, Z. Jurànyi, D. Selmeczi and T. Vicsek, Phase transition in the collective migration of tissue cells: Experiment and model, Phys Rev E, 74. Google Scholar

[60]

J. Tsitsiklis, Problems in Decentralized Decision Making and Computation. Ph.D. Dissertation, Dept. of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, 1984. Google Scholar

[61]

T. VicsekA. CziròkE. Ben-JacobI. Cohen and O. Shochet, Novel Type of Phase Transition in a System of Self-Driven Particles, Phys Rev Lett, 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[62]

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

[63]

A.-M. Wazwaz, Linear and Nonlinear Integral Equations. Methods and Applications, Springer, 2011. doi: 10.1007/978-3-642-21449-3.  Google Scholar

show all references

References:
[1]

G. Albi and L. Pareschi, Modeling self-organized systems interacting with few individuals: From microscopic to macroscopic dynamics, Appl Math Lett, 26 (2013), 397-401.  doi: 10.1016/j.aml.2012.10.011.  Google Scholar

[2]

I. Aoki, A simulation study on the schooling mechanism in fish, Bullettin Of The Japanese Society Scientific Fischeries, 48 (1982), 1081-1088.  doi: 10.2331/suisan.48.1081.  Google Scholar

[3]

Y. Arboleda-EstudilloM. KriegJ. StühmerN. A. LicataD. J. Muller and C.-P. Heisenberg, Movement Directionality in Collective Migration of Germ Layer Progenitors, Curr Biology, 20 (2010), 161-169.  doi: 10.1016/j.cub.2009.11.036.  Google Scholar

[4]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, P Natl Acad Sci USA, 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105.  Google Scholar

[5]

J. M. Belmonte, G. L. Thomas, L. G. Brunnet, R. M. de Almeida and H. Chaté, Self-propelled particle model for cell-sorting phenomena, Phys Rev Lett, 100 (2008), 248702. doi: 10.1103/PhysRevLett.100.248702.  Google Scholar

[6] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, UK, 2004.  doi: 10.1017/CBO9780511543234.  Google Scholar
[7]

L. BrunoA. TosinP. Tricerri and F. Venuti, Non-local first-order modelling of crowd dynamics: A multidimensional framework with applications, Applied Mathematical Modelling, 35 (2011), 426-445.  doi: 10.1016/j.apm.2010.07.007.  Google Scholar

[8]

T. A. Burton, Volterra Integral and Differential Equations. Second Edition, Springer, 2005.  Google Scholar

[9]

J. A. CarrilloM. 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

[10]

T. ColinM.-C. DurrieuJ. JoieY. LeiY. MammeriC. Poignard and O. Saut, Modeling of the migration of endothelial cells on bioactive micropatterned polymers, Math. BioSci. and Eng., 10 (2013), 997-1015.  doi: 10.3934/mbe.2013.10.997.  Google Scholar

[11]

A. ColombiM. Scianna and A. Tosin, Differentiated cell behavior: A multiscale approach using measure theory, J Math Biol, 71 (2015), 1049-1079.  doi: 10.1007/s00285-014-0846-z.  Google Scholar

[12]

I. D. CouzinJ. KrauseN. R. Franks and S. A. Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516.  doi: 10.1038/nature03236.  Google Scholar

[13]

I. D. CouzinJ. KrauseR. JamesG. D. Ruxton and N. R. Franks, Collective memory and spatial sorting in animal groups, J Theor Biol, 218 (2002), 1-11.  doi: 10.1006/jtbi.2002.3065.  Google Scholar

[14]

E. CristianiP. Frasca and B. Piccoli, Effects of anisotropic interactions on the structure of animal groups, J. Math. Biol., 62 (2011), 569-588.  doi: 10.1007/s00285-010-0347-7.  Google Scholar

[15]

E. Cristiani, B. Piccoli and A. Tosin, Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints, in Mathematical Modeling of Collective Behavior in Socio-economic and Life-sciences (eds. G. Naldi, L. Pareschi and G. Toscani), Modeling and Simulation in Science, Engineering, and Technology, Birkhäuser Boston, 2010,337–364. doi: 10.1007/978-0-8176-4946-3_13.  Google Scholar

[16]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS & A: Modeling, Simulation and Applications, Springer International Publishing, 2014. doi: 10.1007/978-3-319-06620-2.  Google Scholar

[17]

E. CristianiF. S. Priuli and A. Tosin, Modeling rationality to control self-organization of crowds: an environmental approach, SIAM J Appl Math, 75 (2015), 605-629.  doi: 10.1137/140962413.  Google Scholar

[18]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Transactions on Automatic Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.  Google Scholar

[19]

F. Cucker and J.-G. Dong, A General Collision-Avoiding Flocking Framework, IEEE Transactions on Automatic Control, 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.  Google Scholar

[20]

F. Cucker and C. Huepe, Flocking with informed agents, MathematicS In Action, 1 (2008), 1-25.  doi: 10.5802/msia.1.  Google Scholar

[21]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[22]

F. Cucker and S. Smale, Emergent Behavior in Flocks, Ieee T Automat Contr, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[23]

E. Di CostanzoR. Natalini and L. Preziosi, A hybrid mathematical model for self-organizing cell migration in the zebrafish lateral line, J of Math Biol, 71 (2015), 171-214.  doi: 10.1007/s00285-014-0812-9.  Google Scholar

[24]

E. Di CostanzoA. GiacomelloE. MessinaR. NataliniG. PontrelliF. RossiR. Smits and M. Twarogowska, A discrete in continuous mathematical model of cardiac progenitor cells formation and growth as spheroid clusters (cardiospheres), Mathematical Medicine and Biology: A Journal of the IMA, 35 (2018), 121-144.  doi: 10.1093/imammb/dqw022.  Google Scholar

[25]

E. Di Costanzo, R. Natalini and L. Preziosi, A hybrid model of cell migration in zebrafish embryogenesis, in ITM Web of Conferences, EDP Sciences, 5 (2015), 00013. doi: 10.1051/itmconf/20150500013.  Google Scholar

[26]

M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys Rev Lett, 96 (2016), 104302. doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[27] M. Eisenbach and J. W. Lengeler, Chemotaxis, Imperial College Press, 2004.   Google Scholar
[28]

J. J. FariaJ. R. G. DyerC. R. Tosh and J. Krause, Leadership and social information use in human crowds, Animal Behaviour, 79 (2010), 895-901.  doi: 10.1016/j.anbehav.2009.12.039.  Google Scholar

[29]

F. E. Fish, Kinematics of ducklings swimming in formation: consequences of position, Journal of Experimental Zoology, 273 (1995), 1-11.  doi: 10.1002/jez.1402730102.  Google Scholar

[30]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964.  Google Scholar

[31]

G. Grégoire and H. Chaté, Onset of collective and cohesive motion, Phys. Rev. Lett., 92. Google Scholar

[32]

G. GrégoireH. Chaté and Y. Tu, Moving and staying together without a leader, Physica D, 181 (2013), 157-170.  doi: 10.1016/S0167-2789(03)00102-7.  Google Scholar

[33]

S. Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Comm. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[34]

S. Y. Ha and D. Levy, Particle, kinetic and fluid models for phototaxis, Discrete and Continuous Dynamical Systems - Series B, 12 (2009), 77-108.  doi: 10.3934/dcdsb.2009.12.77.  Google Scholar

[35]

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

[36]

H. Hatzikirou and A. Deutsch, Collective guidance of collective cell migration, Curr. Top. Dev. Biol., 81 (2007), 401-434.   Google Scholar

[37]

D. HelbingF. SchweitzerJ. Keltsch and P. Molnár, Active walker model for the formation of human and animal trail systems, Physical Review, 56 (1997), 2527-2539.  doi: 10.1103/PhysRevE.56.2527.  Google Scholar

[38]

C. K. Hemelrijk and H. Hildenbrandt, Self-organized shape and frontal density of fish schools, Ethology, 114 (2008), 245-254.  doi: 10.1111/j.1439-0310.2007.01459.x.  Google Scholar

[39]

W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Computational Mathematics, Springer, 2003. doi: 10.1007/978-3-662-09017-6.  Google Scholar

[40]

A. Huth and C. Wissel, The simulation of the movement of fish schools, J Theor Biol, 156 (1992), 365-385.  doi: 10.1016/S0022-5193(05)80681-2.  Google Scholar

[41]

C. C. IoannouC. R. ToshL. Neville and J. Krause, The confusion effect. from neural networks to reduced predation risk, Behavioral Ecology, 19 (2008), 126-130.  doi: 10.1093/beheco/arm109.  Google Scholar

[42]

A. JadbabaieJ. Lin and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans Autom Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781.  Google Scholar

[43]

J. JoieY. LeiM.-C. DurrieuT. ColinC. Poignard and O. Saut, Migration and orientation of endothelial cells on micropatterned polymers: A simple model based on classical mechanics, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1059-1076.  doi: 10.3934/dcdsb.2015.20.1059.  Google Scholar

[44]

H. K. Khalil, Nonlinear Systems. Third Edition, Prentice Hall, 2002. Google Scholar

[45]

V. Lakshmikantham and M. R. M. Rama, Theory of Integro-Differential Equations, vol. 1 of Stability and Control: Theory, Methods and Applications, Gordon and Breach Science Publishers, 1995.  Google Scholar

[46]

C. Lubich, On the stability of linear multistep methods for Volterra convolution equations, IMA J. Numer. Anal., 3 (1983), 439-465.  doi: 10.1093/imanum/3.4.439.  Google Scholar

[47]

E. Méhes and T. Vicsek, Collective motion of cells: from experiments to models, Integr Biol, 6 (2014), 831-854.   Google Scholar

[48]

M. Menci and M. Papi, Global solutions for a path-dependent hybrid system of differential equations under parabolic signal, Nonlinear Analysis, 184 (2019), 172-192.  doi: 10.1016/j.na.2019.01.034.  Google Scholar

[49]

M. MoussaïdD. Helbing and G. Theraulaz, How simple rules determine pedestrian behavior and crowd disasters, Proceeding of the National Academy of Sciences of the United States of America, 108 (2011), 6884-6888.   Google Scholar

[50]

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications. Third edition, Springer, 2003.  Google Scholar

[51]

G. Naldi, L. Pareschi and G. Toscani (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life-Sciences, Modeling and Simulation in Science, Engineering, and Technology, Birkhäuser Boston, 2010. doi: 10.1007/978-0-8176-4946-3.  Google Scholar

[52]

M. Onitsuka, Uniform asymptotic stability for damped linear oscillators with variable parameters, Applied Mathematics and Computation, 218 (2011), 1436-1442.  doi: 10.1016/j.amc.2011.06.025.  Google Scholar

[53] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2014.   Google Scholar
[54]

B. Perthame, Transport Equations in Biology, Birkhäuser, 2007.  Google Scholar

[55]

B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107.  doi: 10.1007/s00161-009-0100-x.  Google Scholar

[56]

T. PitcherA. Magurran and I. Winfield, Fish in larger shoals find food faster, Behav Ecol and Sociobiology, 10 (1982), 149-151.  doi: 10.1007/BF00300175.  Google Scholar

[57]

N. Sepúlveda, L. Petitjean, O. Cochet, E. Grasland-Mongrain, P. Silberzan and V. Hakim, Collective cell motion in an epithelial sheet can be quantitatively described by a stochastic interacting particle model, PLOS Computational Biology, 9 (2013), e1002944, 12 pp. doi: 10.1371/journal.pcbi.1002944.  Google Scholar

[58]

D. Strömbom, Collective motion from local attraction, J Theor Biol, 283 (2011), 145-151.  doi: 10.1016/j.jtbi.2011.05.019.  Google Scholar

[59]

B. Szabò, G. J. Szöllösi, B. Gönci, Z. Jurànyi, D. Selmeczi and T. Vicsek, Phase transition in the collective migration of tissue cells: Experiment and model, Phys Rev E, 74. Google Scholar

[60]

J. Tsitsiklis, Problems in Decentralized Decision Making and Computation. Ph.D. Dissertation, Dept. of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, 1984. Google Scholar

[61]

T. VicsekA. CziròkE. Ben-JacobI. Cohen and O. Shochet, Novel Type of Phase Transition in a System of Self-Driven Particles, Phys Rev Lett, 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[62]

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

[63]

A.-M. Wazwaz, Linear and Nonlinear Integral Equations. Methods and Applications, Springer, 2011. doi: 10.1007/978-3-642-21449-3.  Google Scholar

Figure 1.  $ R_A: $ vanishing region for $ \tilde a $ and $ \tilde b $ defined in (80), bounded by the curves $ \tilde y = \sqrt{\tilde x^2+2\frac{\alpha}{p}}, $ $ \tilde y = \frac{\alpha}{2p\tilde x} $ and $ \tilde y = \tilde x $
Figure 2.  Numerical test. Simulation with parameters $ \sigma = 0.5 $, $ \beta = 5 $, $ \gamma = 2\times 10^2 $, $ D = 2\times 10^2 $, $ \xi = 0.5 $, $ V_{0, \max} = 3 $, and $ \mathbf{X}_{0} $ randomly taken in the red square shown in the top panel (Section 6.2)
Figure 3.  Numerical test. Functions $ Fl_{X}(t) $, $ Fl_{V}(t) $ and $ \left\|\mathbf{V}_{\text{CM}}(t)\right\| $ versus time (x-axis shows only a part of the time domain), as defined in Section 6.2
[1]

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

[2]

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

[3]

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

[4]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[5]

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

[6]

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

[7]

Razvan C. Fetecau, Beril Zhang. Self-organization on Riemannian manifolds. Journal of Geometric Mechanics, 2019, 11 (3) : 397-426. doi: 10.3934/jgm.2019020

[8]

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 & Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028

[9]

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

[10]

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 & Heterogeneous Media, 2018, 13 (3) : 379-407. doi: 10.3934/nhm.2018017

[11]

Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic & Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045

[12]

Hyunjin Ahn, Seung-Yeal Ha, Jeongho Kim. Uniform stability of the relativistic Cucker-Smale model and its application to a mean-field limit. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4209-4237. doi: 10.3934/cpaa.2021156

[13]

Laure Pédèches. Asymptotic properties of various stochastic Cucker-Smale dynamics. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2731-2762. doi: 10.3934/dcds.2018115

[14]

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

[15]

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

[16]

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

[17]

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

[18]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

[19]

Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control & Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447

[20]

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

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (467)
  • HTML views (359)
  • Cited by (1)

[Back to Top]