Mathematical models for aggregation and self-collective behaviour

聚合和自我集体行为的数学模型

• 批准号: RGPIN-2018-04180
• 负责人: Fetecau, Razvan
• 金额: $ 1.31万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712769
• 关键词: Mathematical; models; aggregation; self; collective

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.

拟议的研究计划的重点是对聚集/蜂群现象的数学模型的理论和数值研究。尽管这种现象出现在各个领域，但我们对聚集模型在种群生物学（细胞的趋化性，动物的趋化或植入），机器人技术和舆论形成的应用特别感兴趣。 文献中存在各种数学模型，从基于粒子的（ODE）到动力学和连续性（PDE）描述。这些配方中的大多数导致非线性和非局部微分方程，这些方程式在分析和模拟方面具有挑战性。拟议的研究将涉及以前模型的扩展和概括，以及对真正新模型的开发和分析。特别重点将放在解决方案的长时间行为和所得的平衡配置上。 重要的聚集模型由宏观密度演化的全差分方程描述。该方程表示密度通过具有相互作用电位的卷积给出的速度依赖性的速度场的活动转运。该模型还具有相应的离散/ODE公式。相互作用潜力通常包含短程排斥和远程吸引力，其属性在此类方程的分析和数字中至关重要。除了这种相互作用外，该模型还可能包括外部电位和/或扩散项。离散模型和连续模型都可以作为某些能量功能的梯度流进行配合。因此，变异方法可用于研究其平衡。 拟议的研究的一部分将解决与迄今为止大多被忽视的非常通用模型类别相关的各个方面：i）边界和边界条件的作用，ii）各向异性（例如，感知领域有限）及其影响，其影响及iii）多种物种的模型，iv）表面和歧管上的聚集。在应用程序中，所有方面都非常重要，但是到目前为止，文献中很少关注。我们还计划调查聚集模型的几种特定应用（例如意见形成和蛋白质吸附），并开发和研究数学上的新模型，以解决有关教育鱼类的最新实验观察。 拟议的项目为具有各种兴趣的学生提供了培训机会，从数字和形式方法（渐近学）到严格的分析。此外，这些项目可以容纳各种技能水平，从不经验的大学生到高级博士生和PDF。拟议的计划旨在推进基本的数学研究和应用。