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

世界是非线性的，为了理解这样一个复杂的系统，我们使用数学模型来捕捉我们研究的关键特征。最成功的数学模型使用了非线性偏微分方程式，可以根据不同的情况量身定做。这种类型的一个例子是反应扩散方程，它试图捕捉一些也分散在空间中的不同相互作用的‘物种’的基本行为。这种情况下的“物种”可以是化学的、生物的或某种密度在空间和时间上不同的物理量。例如，我们可能对跟踪森林中狐狸和兔子数量的波动感兴趣，或者在实验室实验中了解某些化学物质如何在烧杯中反应和扩散产生模式。这类模型的问题在于，它们基于对时间和空间的连续表示，而且通常是高度非线性的。因此，这些模型几乎从来没有纸上可以计算出的精确解。取而代之的是，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统模型。如果我们让我们的计算机程序足够努力，那么数值分析员的工作就是严格地证明离散算法以足够的精度逼近原始的连续介质模型。用于逼近非线性反应扩散方程的最广泛、最成功、最灵活的离散算法家族称为有限元方法。该研究方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和准确的有限元方法。这项工作适用于不同的物理情况，包括集合种群动态(生活在零散景观中的种群)、图案形成模型(例如，为了解皮肤图案如何发展而设计的模型)和随机微分方程式(例如，包含随机环境因素的模型)。

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).