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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 |
In this paper we propose and study a hybrid discrete–continuous mathematical model of collective motion under alignment and chemotaxis effect. Starting from paper [
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. |
[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. |
[3] |
Y. Arboleda-Estudillo, M. Krieg, J. Stühmer, N. A. Licata, D. 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. |
[4] |
M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. 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. |
[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. |
[6] |
H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, UK, 2004.
doi: 10.1017/CBO9780511543234.![]() ![]() |
[7] |
L. Bruno, A. Tosin, P. 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. |
[8] |
T. A. Burton, Volterra Integral and Differential Equations. Second Edition, Springer, 2005. |
[9] |
J. A. Carrillo, M. Fornasier, J. 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. |
[10] |
T. Colin, M.-C. Durrieu, J. Joie, Y. Lei, Y. Mammeri, C. 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. |
[11] |
A. Colombi, M. 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. |
[12] |
I. D. Couzin, J. Krause, N. 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. |
[13] |
I. D. Couzin, J. Krause, R. James, G. 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. |
[14] |
E. Cristiani, P. 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. |
[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. |
[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. |
[17] |
E. Cristiani, F. 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. |
[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. |
[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. |
[20] |
F. Cucker and C. Huepe,
Flocking with informed agents, MathematicS In Action, 1 (2008), 1-25.
doi: 10.5802/msia.1. |
[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. |
[22] |
F. Cucker and S. Smale,
Emergent Behavior in Flocks, Ieee T Automat Contr, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[23] |
E. Di Costanzo, R. 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. |
[24] |
E. Di Costanzo, A. Giacomello, E. Messina, R. Natalini, G. Pontrelli, F. Rossi, R. 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. |
[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. |
[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. |
[27] | M. Eisenbach and J. W. Lengeler, Chemotaxis, Imperial College Press, 2004. Google Scholar |
[28] |
J. J. Faria, J. R. G. Dyer, C. 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. |
[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. |
[30] |
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964. |
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G. Grégoire and H. Chaté, Onset of collective and cohesive motion, Phys. Rev. Lett., 92. Google Scholar |
[32] |
G. Grégoire, H. 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. |
[33] |
S. Y. Ha, K. 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. |
[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. |
[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. |
[36] |
H. Hatzikirou and A. Deutsch, Collective guidance of collective cell migration, Curr. Top. Dev. Biol., 81 (2007), 401-434. Google Scholar |
[37] |
D. Helbing, F. Schweitzer, J. 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. |
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C. K. Hemelrijk and H. Hildenbrandt,
Self-organized shape and frontal density of fish schools, Ethology, 114 (2008), 245-254.
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W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Computational Mathematics, Springer, 2003.
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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. |
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C. C. Ioannou, C. R. Tosh, L. 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. |
[42] |
A. Jadbabaie, J. 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. |
[43] |
J. Joie, Y. Lei, M.-C. Durrieu, T. Colin, C. 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. |
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H. K. Khalil, Nonlinear Systems. Third Edition, Prentice Hall, 2002. Google Scholar |
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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. |
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On the stability of linear multistep methods for Volterra convolution equations, IMA J. Numer. Anal., 3 (1983), 439-465.
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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. |
[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. |
[3] |
Y. Arboleda-Estudillo, M. Krieg, J. Stühmer, N. A. Licata, D. 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. |
[4] |
M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. 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. |
[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. |
[6] |
H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, UK, 2004.
doi: 10.1017/CBO9780511543234.![]() ![]() |
[7] |
L. Bruno, A. Tosin, P. 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. |
[8] |
T. A. Burton, Volterra Integral and Differential Equations. Second Edition, Springer, 2005. |
[9] |
J. A. Carrillo, M. Fornasier, J. 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. |
[10] |
T. Colin, M.-C. Durrieu, J. Joie, Y. Lei, Y. Mammeri, C. 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. |
[11] |
A. Colombi, M. 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. |
[12] |
I. D. Couzin, J. Krause, N. 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. |
[13] |
I. D. Couzin, J. Krause, R. James, G. 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. |
[14] |
E. Cristiani, P. 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. |
[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. |
[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. |
[17] |
E. Cristiani, F. 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. |
[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. |
[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. |
[20] |
F. Cucker and C. Huepe,
Flocking with informed agents, MathematicS In Action, 1 (2008), 1-25.
doi: 10.5802/msia.1. |
[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. |
[22] |
F. Cucker and S. Smale,
Emergent Behavior in Flocks, Ieee T Automat Contr, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[23] |
E. Di Costanzo, R. 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. |
[24] |
E. Di Costanzo, A. Giacomello, E. Messina, R. Natalini, G. Pontrelli, F. Rossi, R. 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. |
[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. |
[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. |
[27] | M. Eisenbach and J. W. Lengeler, Chemotaxis, Imperial College Press, 2004. Google Scholar |
[28] |
J. J. Faria, J. R. G. Dyer, C. 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. |
[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. |
[30] |
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964. |
[31] |
G. Grégoire and H. Chaté, Onset of collective and cohesive motion, Phys. Rev. Lett., 92. Google Scholar |
[32] |
G. Grégoire, H. 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. |
[33] |
S. Y. Ha, K. 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. |
[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. |
[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. |
[36] |
H. Hatzikirou and A. Deutsch, Collective guidance of collective cell migration, Curr. Top. Dev. Biol., 81 (2007), 401-434. Google Scholar |
[37] |
D. Helbing, F. Schweitzer, J. 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. |
[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. |
[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. |
[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. |
[41] |
C. C. Ioannou, C. R. Tosh, L. 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. |
[42] |
A. Jadbabaie, J. 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. |
[43] |
J. Joie, Y. Lei, M.-C. Durrieu, T. Colin, C. 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. |
[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. |
[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. |
[47] |
E. Méhes and T. Vicsek, Collective motion of cells: from experiments to models, Integr Biol, 6 (2014), 831-854. Google Scholar |
[48] |
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