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  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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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.  Google Scholar

[10]

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[11]

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[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

[38]

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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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