# 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  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
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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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