Microscopic, kinetic and hydrodynamic hybrid models of collective motions withchemotaxis: a numerical study

TL;DR

This paper conducts a numerical analysis of hybrid microscopic, kinetic, and hydrodynamic models for collective cell movement influenced by chemotaxis, exploring their behavior beyond idealized initial conditions.

Contribution

It provides a numerical investigation of the Vlasov and Euler systems in hybrid chemotaxis models, extending understanding beyond monokinetic initial data.

Findings

Numerical solutions show diverse behaviors for different initial conditions.

The models effectively capture collective cell migration patterns.

Results validate the theoretical mean-field and pressureless Euler limits.

Abstract

A general class of hybrid models has been introduced recently, gathering the advantages multiscale descriptions. Concerning biological applications, the particular coupled structure fits to collective cell migrations and pattern formation scenarios. In this context, cells are modelled as discrete entities and their dynamics is given by ODEs, while the chemical signal influencing the motion is considered as a continuous signal which solves a diffusive equation. From the analytical point of view, this class of model has been proved to have a mean-field limit in the Wasserstein distance towards a system given by the coupling of a Vlasov-type equation with the chemoattractant equation. Moreover, a pressureless nonlocal Euler-type system has been derived for these models, rigorously equivalent to the Vlasov one for monokinetic initial data. In the present paper, we present a numerical study of the solutions to the Vlasov and Euler systems, exploring general settings for inital data, far from the monokinetic ones.