Singular limits of elliptic and parabolic systems

在变化的环境中抛物系统的分析本项目关注在域（环境）随时间变化的情况下，描述种群动态的反应扩散系统的严格数学分析。这些问题可能对模拟气候变化影响的应用程序有意义，气候变化可能随着时间的推移影响生境的大小或形状，以及它们支持一个物种的能力。本研究的总体目标是研究以下因素对此类反应扩散系统解的行为的影响：边界的规定运动（例如，对于有界域，域的生长或收缩，或对于圆柱形域，壁的运动）;有利的栖息地在整个域内的渐进运动;物种入侵锋（即在至少一维上无界的区域中的行波）在变化的环境中。将通过单个反应扩散方程研究单物种动力学的解的性质，对于多物种的相互作用系统，通过反应扩散系统的耦合系统。具体类型的问题，将解决数学是：什么是长期的行为，人口在一个随时间变化的域？一个物种可以入侵未被占领的领土吗？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率而变化？正平衡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，不同的制度将是可能的，这取决于该领域的变化相对于物种迁移的典型速度的速度。通过应用和调整非线性偏微分方程的分析工具，将不同的可能性量化，以建立各种设置下反应扩散系统解的严格结果。将设计新的分析方法，以应对变化的域和边界所带来的挑战，当标准的比较原则和下/上解决方案参数（在一个固定的域）不立即适用。EPSRC研究领域：数学分析（小学），数学生物学（中学）

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