A hybrid mathematical model of collective motion under alignment and chemotaxis

TL;DR

This paper introduces a hybrid discrete-continuous mathematical model for collective cell motion influenced by alignment and chemotaxis, analyzing its behavior both analytically and numerically.

Contribution

It presents a simplified hybrid model coupling particle alignment with chemotaxis, proving existence, uniqueness, and asymptotic convergence of solutions.

Findings

Solutions exist and are unique globally in time.

The system's aggregate converges exponentially to a single position with zero velocity.

Numerical results support the analytical findings.

Abstract

In this paper we propose and study a hybrid discrete in continuous mathematical model of collective motion under alignment and chemotaxis effect. Starting from the paper by Di Costanzo et al (2015a), in which the Cucker-Smale model (Cucker and Smale, 2007) 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 $t\rightarrow +\infty$, 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.