Singular limits of elliptic and parabolic systems

椭圆和抛物线系统的奇异极限

• 批准号: 2227486
• 金额: --
• 依托单位: Swansea University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2227486
• 关键词: Singular; limits; elliptic; parabolic; systems

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

不断变化的环境中的抛物线系统分析 该项目涉及在域（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。这些问题可能对气候变化影响建模方面的应用感兴趣，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。该研究的总体目标是研究对此类反应扩散系统的解的行为的影响：边界的指定运动（例如，对于有界域，域的增长或收缩，或者对于圆柱域，墙壁的移动）；整个领域内有利栖息地的渐进移动；在不断变化的环境中物种的入侵前沿（即在至少一个维度上无界的领域中的行波）。解决方案的属性将可以通过单一反应扩散方程研究单物种动力学，也可以通过反应扩散系统的耦合系统研究多物种相互作用的系统。将用数学方法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？一个物种可以入侵未占领的领土吗？一个物种可以入侵另一个物种占领的区域吗？入侵速度如何根据环境以及环境变化的速率而变化？正稳态的存在性和稳定性属性是什么？预计对于这些问题中的每一个，都可能有不同的制度，具体取决于域相对于物种迁移的典型速率的变化速率。将通过应用和调整非线性偏微分方程分析工具来量化不同的可能性，以建立有关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来应对域变化带来的挑战当标准比较原则和子/超级解决方案论证（在固定域上）不立即适用时，EPSRC 研究领域：数学分析（初级）、数学生物学（次级）