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

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.