Mathematical models for aggregation and self-collective behaviour

拟议的研究计划的重点是聚集/群集现象的数学模型的理论和数值研究。虽然这种现象出现在各个领域，我们特别感兴趣的应用程序的聚集模型的人口生物学（细胞的趋化性，群集或群集的动物），机器人和意见形成。 文献中存在各种数学模型，从基于粒子的（ODE）到动力学和连续介质（PDE）描述。大多数这些公式导致非线性和非局部微分方程，这是具有挑战性的分析和模拟。拟议中的研究将涉及扩展和概括以前的自我集体行为模型，以及开发和分析真正的新模型。一个特别的重点将放在长时间的行为的解决方案和由此产生的平衡配置。 一类重要的聚集模型是由宏观密度演化的积分-微分方程描述的。该方程表示由速度场的密度的主动传输，该速度场具有与其的函数依赖性，由与相互作用势的卷积给出。该模型也有相应的离散/常微分方程制定。相互作用势通常包括短程排斥和长程吸引，其性质在这类方程的分析和数值计算中是必不可少的。除了这样的相互作用之外，模型还可以包括外部电势和/或扩散项。离散和连续模型都可以表述为某些能量泛函的梯度流;因此，变分方法可以用来研究它们的平衡。 拟议的研究的一部分将解决与这类非常一般的模型有关的各个方面，这些模型到目前为止大多被忽视：i）边界和边界条件的作用，ii）各向异性（例如，一个有限的感知领域）及其影响，三）多个物种的模型，四）表面和流形上的聚集。这些方面在应用中都非常重要，但迄今为止在文献中很少受到关注。我们还计划研究聚合模型的几个具体应用（例如，意见形成和蛋白质吸附），以及开发和研究数学新模型，以解决最近对鱼群的实验观察。 拟议的项目为学生提供培训机会，从数值和形式方法（渐近）到严格的分析。此外，这些项目可以适应各种技能水平，从没有经验的本科生到高级博士生和PDF。该计划旨在推进基础数学研究和应用。

The focus of the proposed research program is a theoretical and numerical investigation of mathematical models for aggregation/swarming phenomena. While such phenomena arise in a variety of areas, we are particularly interested in applications of aggregation models to population biology (chemotaxis of cells, swarming or flocking of animals), robotics and opinion formation. Various mathematical models exist in the literature, ranging from particle-based (ODE) to kinetic and continuum (PDE) descriptions. Most of these formulations lead to nonlinear and nonlocal differential equations which are challenging to analyze and simulate. The proposed research will concern extensions and generalizations of previous models for self-collective behaviour, as well as development and analysis of genuinely new models. A particular emphasis will be placed on the long time behaviour of the solutions and the resulting equilibrium configurations. An important class of aggregation models is described by an integro-differential equation for the evolution of the macroscopic density. The equation represents the active transport of the density by a velocity field that has a functional dependence on it, given by a convolution with an interaction potential. The model also has a corresponding discrete/ODE formulation. The interaction potential typically incorporates short-range repulsion and long-range attraction, and its properties are essential in the analysis and numerics of this class of equations. In addition to such interactions, the model may also include an external potential and/or diffusion terms. Both the discrete and the continuum models can be formulated as gradient flows of certain energy functionals; consequently, variational methods can be used to investigate their equilibria. Part of the proposed research will address various aspects related to this very general class of models that have been mostly overlooked so far: i) the role of boundaries and boundary conditions, ii) anisotropy (e.g., a limited perception field) and its effects, iii) models for multiple species, iv) aggregation on surfaces and manifolds. All aspects are very important in applications, but received little attention in the literature so far. We also plan to investigate several specific applications of aggregations models (e.g., to opinion formation and protein adsorption), as well as develop and study mathematically new models to address recent experimental observations on schooling fish. The proposed projects offer training opportunities for students with a variety of interests ranging from numerics and formal methods (asymptotics) to rigorous analysis. In addition, the projects can accommodate a wide range of skill levels, from unexperienced undergraduates to advanced PhD students and PDFs. The proposed program aims to advance basic mathematical research as well as applications.