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

Roberto Natalini (1) , Thierry Paul (2)

1 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

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.

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]; Organisation et fonctions cellulaires [q-bio.SC]; Physique mathématique [math-ph]

Roberto Natalini, Thierry Paul. The Mean-Field limit for hybrid models of collective motions with chemotaxis. SIAM Journal on Mathematical Analysis, In press. ⟨hal-03310547v4⟩