Nonlocal interaction equations and applications to collective motion of individuals

非局部相互作用方程是大型粒子系统的连续体模型，其中每个粒子不仅可以与其近邻相互作用，还可以与远处的粒子相互作用。在生物学中，它们作为大群个体（昆虫、细菌、鸟类、鱼类等）运动的模型而出现。这项研究将涉及非线性偏微分方程的严格分析、数值模拟和建模。更具体地说，将研究以下问题：1)研究非局部相互作用方程奇点的形成和模式的形成。2)研究具有边界的区域中的非局部相互作用方程，并使用最优输运技术来研究边界上的质量集中现象（例如，在蝗虫群的连续统模型中观察到这种现象）。3)蚁群“觅食路径”形成的建模。本项目研究描述大量个体集体行为的方程，如一群昆虫、一群鸟、一群鱼或一群细菌。了解这些蜂群行为的基本规则，有可能为无人驾驶车辆设计灵活而稳健的算法，并可能应用于危险区域的监视。本研究中研究的方程也出现在各种其他背景下：超导体中涡流密度的演变，粘附动力学，颗粒流动的简化模型和细胞中细丝的定向分布。

Nonlocal interaction equations are continuum models for large systems of particles where every particle can interact not only with its immediate neighbors but also with particles far away. In biology, they arise as model for the movement of large swarms of individuals (insects, bacteria, birds, fish, etc...). The research will involve rigorous analysis of nonlinear partial differential equations, numerical simulations, and modeling. More specifically the following problems will be investigated: 1) Study of the formation of singularities and the formation of patterns in nonlocal interaction equations. 2) Study of nonlocal interaction equations in a domain with boundaries and use of optimal transport techniques in order to study phenomena of mass concentration on the boundaries (such phenomena are observed, for example, in continuum models for swarms of locusts). 3) Modeling of the formation of "foraging trails'' in an ant colony.This project investigates equations describing the collective behavior of a large number of individuals, such as a swarm of insects, a flock of birds, a school of fish, or a colony of bacteria. Understanding basic rules of these swarming behaviors has potential for the design of flexible and robust algorithms for unmanned vehicles, with possible applications, for example, to the surveillance of hazardous areas. The equations investigated in this research also arise in various other contexts: evolution of vortex densities in superconductors, adhesion dynamics, simplified model for granular flow and orientational distribution of filaments in cells.