Singular limits of elliptic and parabolic systems

在变化的环境中对抛物型系统的分析这个项目涉及描述种群动态的反应扩散系统的严格的数学分析，在域(环境)随时间变化的情况下。在对气候变化的影响进行建模方面，这些问题可能会对应用程序感兴趣，气候变化可能会随着时间的推移影响栖息地的大小或形状，以及它们支持物种的能力。研究的总体目标是调查下列因素对这类反应扩散系统解的行为的影响：规定的边界移动(例如，对于有界域，域的增长或缩小，或对于圆柱域，墙的运动)；整个域内有利生境的渐进移动；在变化的环境中，一个物种的入侵前沿(即在一个至少一维无界的区域中的行波)。我们将通过单一的反应扩散方程来研究单物种动力学的解的性质，以及通过反应扩散系统的耦合系统来研究多物种相互作用系统的解的性质。将在数学上解决的具体类型的问题包括：在一个时变的领域中，种群的长期行为是什么？一个物种可以入侵无人占领的领土吗？一个物种可以入侵另一个物种占据的地区吗？入侵速度如何根据环境和环境变化的速度而变化？正稳态的存在和稳定性是什么？预计对于这些问题中的每一个，不同的机制将可能取决于区域相对于典型物种迁徙速度的变化速度。不同的可能性将通过应用和采用非线性偏微分方程分析的工具来量化，以建立关于不同环境下的反应扩散系统的解的严格结果。当标准的比较原理和(固定区域上的)下/上解论点不立即适用时，将设计新的分析方法来应对区域和边界改变所引起的挑战。EPSRC研究领域：数学分析(初级)，数学生物学(次级

Analysis of parabolic systems on a changing environment This project is concerned with the rigorous mathematical analysis of reaction-diffusion systems describing population dynamics, in circumstances where the domain (environment) is changing over time. Such questions may be of interest to applications in terms of modelling the effects of climate change, which may affect the size or shape of habitats over time, and also their ability to support a species. The overall objective of the research is to investigate the effect on the behaviour of solutions of such reaction-diffusion systems of:prescribed movement of the boundary (e.g., for bounded domains, the growth or shrinking of the domain, or for a cylindrical domain, the movement of the walls);the progressive movement of a favourable habitat within an overall domain;invasion fronts of a species (i.e. travelling waves in a domain which is unbounded in at least one dimension) in a changing environment.Properties of solutions will be studied both for single species dynamics, via single reaction-diffusion equations, and for interacting systems of multiple species, via coupled systems of reaction-diffusion systems. The specific types of questions that will be addressed mathematically are: What is the long-time behaviour of the populations in a time-varying domain?Can a species invade unoccupied territory?Can one species invade a region occupied by another? How do the invasion speeds vary according to the environment and according to the rate at which the environment is changing?What are the existence and stability properties of positive steady states?It is expected that for each of these questions, different regimes will be possible depending on the rate at which the domain changes relative to the typical rate of species migration. The different possiblilites will be quantified by applying and adapting tools from the analysis of nonlinear partial differential equations to establish rigorous results about solutions of reaction-diffusion systems in various settings.Novel methods of analysis will be devised to deal with the challenges caused by changing domain and boundaries, when the standard comparison principles and sub-/super-solution arguments (on a fixed domain) do not immediately apply.EPSRC Research areas: Mathematical Analysis (primary), Mathematical Biology (secondary