Singular limits of elliptic and parabolic systems

该项目关注的是在领域（环境）随时间变化的情况下，对描述种群动态的反应扩散系统进行严格的数学分析。这些问题可能对模拟气候变化影响方面的应用有兴趣，气候变化可能随着时间的推移影响栖息地的大小或形状，以及它们支持一个物种的能力。该研究的总体目标是研究对这类反应-扩散系统的解的行为的影响：规定的边界运动（例如，对于有界域，区域的增长或缩小，或对于圆柱形域，墙壁的运动）；迁移：在整个领域内有利栖息地的逐渐移动；物种在不断变化的环境中的入侵前沿（即在至少一个维度上无界的领域中的行波）。我们将通过单一反应-扩散方程研究单物种动力学，以及通过反应-扩散系统的耦合系统研究多物种相互作用系统的解的性质。将以数学方式解决的具体问题类型是：在时变域中种群的长期行为是什么？一个物种可以入侵无人居住的领土吗？一个物种能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率而变化？正稳态的存在性和稳定性是什么？预计对于这些问题中的每一个，不同的制度将可能取决于相对于物种迁移的典型速度的领域变化的速度。不同的可能性将通过应用和调整非线性偏微分方程分析的工具来量化，以建立各种设置下反应扩散系统解的严格结果。当标准的比较原则和亚/超解参数（在固定的域上）不能立即适用时，将设计新的分析方法来处理由变化的域和边界引起的挑战。研究领域：数学分析（初级），数学生物学（二级）

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