Singular limit analysis for aggregation generated by random motion

为了理解生物种群以自组织方式形成的主动聚集，在不同领域提出了两种不同类型的模型；一种是偏微分方程领域的宏观模型，另一种是物理学和数学生物学领域的微观模型。然而，为了揭示聚集中产生自组织的机制，我们需要了解这两个模型之间的关系。在这一建议中，我们关注的是生物种群的主动聚集，它们自己分泌聚集信息素。对于聚集的形成，既有趋化扩散模型（宏观模型），又有独立的基于个体的模型（微观模型）。对于这些模型的一个环节，我们发展了奇异极限和水动力极限两种极限程序，并通过它们的互补，成功地揭示了这两种模型之间的关系。

For understanding of active aggregation of biological populations which are formed in a self-organized way, two different types of models have been proposed in different fields; One is macroscopic models in the field of PDEs and the other is microscopic models in the fields of physics and mathematical biology. However, In order to reveal the mechanism of self-organization arising in aggregation, we need to understand the relation between these two models.In this proposal, we focus on active aggregation of biological populations which secret aggregating pheromone by themselves. For the formation of aggregation, there are already a chemotaxis-diffusion model (macroscopic model) and independently an individual-based model (microscopic model). For a link of these models, we develop two limiting procedures, that is, a singular limit and a hydrodynamic limit and by complementarily using them, we succeed in revealing the relation between these two models.