Singular limit analysis for aggregation generated by random motion

为了更好地理解生物种群的主动聚集行为，在不同的领域提出了两类不同的模型：一类是偏微分方程领域的宏观模型，另一类是物理学和数学生物学领域的微观模型。然而，为了揭示聚集过程中的自组织机制，我们需要了解这两种模型之间的关系，在本论文中，我们主要研究生物种群的主动聚集行为，这些生物种群自身分泌聚集信息素。对于聚集体的形成，已经有趋化-扩散模型（宏观模型）和独立的基于个体的模型（微观模型）。对于这两种模型的联系，我们提出了两种极限方法，即奇异极限和流体力学极限，并通过互补使用，成功地揭示了这两种模型之间的联系。

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.