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.

翻译：近年来，一类融合多尺度描述优势的通用混合模型被提出。针对生物学应用场景，该耦合结构特别适用于描述集体细胞迁移与模式形成过程。在此框架下，细胞被建模为离散实体，其动力学由常微分方程组描述；而影响运动方向的化学信号则被视为连续信号，满足扩散方程。在理论分析层面，已证明此类模型在Wasserstein距离下存在平均场极限，收敛于由Vlasov型方程与趋化因子方程耦合构成的系统。进一步地，基于该模型推导出无压非局部欧拉型系统，该方程在单动能初始条件下严格等价于Vlasov方程。本文通过数值方法研究Vlasov与欧拉系统的解，探索远离单动能条件的广义初始参数设定。