The Mean-Field limit for hybrid models of collective motions with chemotaxis

TL;DR

This paper establishes the mean-field limit for hybrid models of collective cell motion influenced by chemotaxis, connecting discrete cell dynamics with continuous chemoattractant signals, and introduces a new pressureless Euler-type model.

Contribution

It proves the mean-field limit for hybrid cell-chemotaxis models using a novel approach based on marginals, and derives a new pressureless Euler-type model in the monokinetic case.

Findings

Proved mean-field limit with explicit bounds.

Established existence and uniqueness of the limit system.

Derived a new pressureless Euler-type model with chemotaxis.

Abstract

In this paper we study a general class of hybrid mathematical models of collective motions of cells under the influence of chemical stimuli. The models are hybrid in the sense that cells are discrete entities given by ODE, while the chemoattractant is considered as a continuous signal which solves a diffusive equation. For this model we prove the mean-field limit in the Wasserstein distance to a system given by the coupling of a Vlasov-type equation with the chemoattractant equation. Our approach and results are not based on empirical measures, but rather on marginals of large number of individuals densities, and we show the limit with explicit bounds, by proving also existence and uniqueness for the limit system. In the monokinetic case we derive new pressureless nonlocal Euler-type model with chemotaxis.