# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [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. 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##### 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. [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. [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. [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. [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. [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. [49] M. Moussaïd, D. 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. [50] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications. Third edition, Springer, 2003. [51] G. Naldi, L. Pareschi and G. 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$R_A:$ 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 $\tilde y = \tilde x$
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)
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
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