Finite element methods for nonlinear reaction-diffusion systems with applications in biology.

非线性反应扩散系统的有限元方法及其在生物学中的应用。

• 批准号: 340739-2013
• 负责人: Garvie, Marcus
• 金额: $ 0.8万
• 依托单位: University of Guelph
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=595337
• 关键词: Finite; element; methods; nonlinear; reaction

The world is nonlinear and in order to make sense of such a complicated system we use mathematical models to capture the key features of what we study. The most successful mathematical models employ nonlinear partial differential equations, which can be tailored to fit a wide variety of situations. An example of this type are reaction-diffusion equations, which attempt to capture the essential behaviour of a number of different interacting 'species' that also disperse in space. The 'species' in this case can be chemical, biological, or some physical quantity whose density varies in space and time. For example, we may be interested in tracking the fluctuating numbers of foxes and rabbits in a forest, or understand how certain chemicals react and diffuse to produce patterns in a beaker during a laboratory experiment. The problem with such models is that they are based on a continuous representation of time and space, and are usually highly nonlinear. Thus these models almost never have exact solutions that can be worked out on paper. Instead, we must approximate these continuum models using discrete algorithms that can be programmed on a computer. It is then the job of the numerical analyst to rigorously prove that the discrete algorithm approximates the original continuum model with sufficient accuracy, provided we make our computer programs work hard enough. The most widely used, successful and flexible family of discrete algorithms used for approximating nonlinear reaction-diffusion equations are called finite element methods. The research proposal addresses how to construct efficient and accurate finite element methods for specific applications, without sacrificing the key physical features of the model concerned. The physical situations that this work applies to are varied, and include metapopulation dynamics (populations living in fragmented landscapes), pattern formation models (e.g., models designed to understand how skin patterns develop), and stochastic differential equations (e.g., models that incorporate random environmental factors).

世界是非线性的，为了理解如此复杂的系统，我们使用数学模型来捕捉我们研究的关键特征。最成功的数学模型采用非线性偏微分方程，可以对其进行定制以适应各种情况。这种类型的一个例子是反应扩散方程，它试图捕捉同样分散在空间中的许多不同的相互作用“物种”的基本行为。在这种情况下，“物种”可以是化学的、生物的或密度随空间和时间变化的一些物理量。例如，我们可能有兴趣跟踪森林中狐狸和兔子的数量波动，或者了解某些化学物质在实验室实验期间如何在烧杯中发生反应和扩散以产生模式。此类模型的问题在于它们基于时间和空间的连续表示，并且通常是高度非线性的。因此，这些模型几乎从来没有可以在纸上算出的精确解决方案。相反，我们必须使用可以在计算机上编程的离散算法来近似这些连续模型。如果我们让计算机程序足够努力地工作，那么数值分析师的工作就是严格证明离散算法以足够的精度逼近原始连续统模型。用于逼近非线性反应扩散方程的最广泛使用、成功且灵活的离散算法系列称为有限元方法。该研究提案解决了如何在不牺牲相关模型的关键物理特征的情况下，为特定应用构建高效、准确的有限元方法。这项工作适用的物理情况多种多样，包括集合种群动态（生活在碎片化景观中的种群）、模式形成模型（例如，旨在了解皮肤模式如何发展的模型）和随机微分方程（例如，包含随机环境因素的模型）。