American Institute of Mathematical Sciences

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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

Ezio Di Costanzo 1, , Marta Menci 2, , Eleonora Messina 3, , Roberto Natalini 1, and  Antonia Vecchio 4,

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.

Keywords: Differential equations, existence and uniqueness of solution, asymptotic stability, Lyapunov function, collective motion, Cucker-Smale model, flocking behaviour, chemotaxis, self-organization, finite differences.
Mathematics Subject Classification: 45J05, 92B05, 37N25, 92C17.
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:
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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

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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.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

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[10]

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[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.  Google Scholar

[12]

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[14]

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[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 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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

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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 $
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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)
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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
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