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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676897
• 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/聚集现象的数学模型进行理论和数值研究。虽然这种现象出现在不同的领域，但我们特别感兴趣的是聚集模型在种群生物学(细胞的趋化性、动物的成群或成群)、机器人学和观点形成中的应用。*文献中存在各种数学模型，从基于粒子的(ODE)描述到动力学和连续统(PDE)描述。这些公式大多导致了非线性和非局部微分方程组，这给分析和模拟带来了挑战。拟议的研究将涉及对以前的自我集体行为模型的扩展和概括，以及真正新模型的开发和分析。我们将特别强调溶液的长时间行为和由此产生的平衡构型。*一类重要的聚集模型由宏观密度演化的积分-微分方程描述。该方程表示速度场对密度的主动输运，速度场与速度场有函数依赖关系，速度场由与相互作用势的卷积给出。该模型也有相应的离散/常数公式。相互作用势通常包括短程排斥和长程吸引，它的性质在这类方程的分析和数值计算中是必不可少的。除了这种相互作用之外，该模型还可以包括外部势和/或扩散项。离散的和连续的模型都可以表示为某些能量泛函的梯度流；因此，可以使用变分方法来研究它们的平衡。*拟议的研究的一部分将涉及与这类到目前为止大多被忽略的非常一般的模型相关的各个方面：i)边界和边界条件的作用，ii)各向异性(例如，有限的感知场)及其影响，iii)多物种的模型，iv)表面和流形上的聚集。各个方面在应用中都是非常重要的，但到目前为止在文献中很少受到关注。我们还计划调查聚集体模型的几个具体应用(例如，在意见形成和蛋白质吸附方面)，以及开发和研究数学上的新模型，以解决最近关于鱼群聚集的实验观察。*拟议的项目为具有从数值和形式方法(渐近性)到严格分析的各种兴趣的学生提供培训机会。此外，这些项目可以容纳广泛的技能水平，从没有经验的本科生到高级博士生和PDF。拟议的计划旨在促进基础数学研究和应用。