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

Marta Menci, Roberto Natalini and Thierry Paul

Vol. 12 (2024), No. 1, 47–83

A general class of hybrid models has been introduced recently, gathering the advantages of multiscale descriptions. Concerning biological applications, the particular coupled structure fits to collective cell migrations and pattern formation phenomena due to intercellular and chemotactic stimuli. In this context, cells are modeled as discrete entities and their dynamics are 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 models has been recently 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. For applications, the monokinetic assumption is quite strong and far from a real experimental setting. The aim of this paper is to introduce a numerical approach to the hybrid coupled structure at the different scales, investigating the case of general initial data. Several scenarios will be presented, aiming at exploring the role of the different terms on the overall dynamics. Finally, the pressureless nonlocal Euler-type system is generalized by means of an additional pressure term.

hybrid systems, mean-field limit, chemotaxis, hydrodynamic model, numerical simulations
Mathematical Subject Classification
Primary: 35Q83, 35Q92, 65M06, 82C22, 92C17
Received: 22 June 2023
Revised: 19 September 2023
Accepted: 2 November 2023
Published: 20 December 2023

Communicated by Francesco dell'Isola
Marta Menci
Università Campus Bio-medico di Roma
Roberto Natalini
Instituto per le Applicazioni del Calcolo “M. Picone”
Consiglio Nazionale delle Ricerche
Thierry Paul
Laboratoire Ypatia des Sciences Mathématiques
Laboratoire Jacques-Louis Lions
Sorbonne Université