A hybrid model of collective motion of discrete particles under alignment and continuum chemotaxis

Ezio Di Costanzo; Marta Menci; Eleonora Messina; Roberto Natalini; Antonia Vecchio

doi:10.3934/dcdsb.2019189

Article Contents

2020, Volume 25, Issue 1. Doi: 10.3934/dcdsb.2019189

This issue Previous Article Next Article Singular perturbations and scaling

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: July 31, 2018

Revised: February 28, 2019

Early access: September 2019

Published: January 2020

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Abstract

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.

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Mathematics Subject Classification: 45J05, 92B05, 37N25, 92C17.

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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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