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.
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