Applications of Moving Mesh Finite Elements to Population Dynamics

该项目涉及定量种群生态学中在时间和空间尺度上的生态系统的数学模型。该项目的目的是开发动态领域中具有聚集和聚集效应的竞争-扩散系统的某些二维数学模型。我们将研究描述两相反应扩散系统的一类偏微分Lotka-Volterra竞争方程。特别令人感兴趣的是多相系统，特别是那些用相之间的动态界面分隔的系统，例如在不同的物理状态之间或在交战物种之间。一种有效的数值方法是Baines，Hubbard和Jimack的移动网格有限元方法，它使用几何守恒方法进行网格自适应。Watkins最近在她的博士论文(阅读2017)中利用了这一技术，该论文考虑了MMFE在具有聚集动力学和集群的种内和种间相互作用系统中的应用。试金石是具有动态界面行为的斯特凡问题，就像希尔霍斯特的工作一样。在熟悉了背景和数值技术之后，学生将把模型的行为与经验数据集进行比较。这些模型可以动态地适应区域的大小和形状，以及逻辑项的改变和参数的变化，而不需要进一步的计算。这种适应性意味着有广泛的现实生物和生态系统可以应用和验证模型。与显示竞争-扩散-聚集行为的物种的数据集进行比较将是这项研究的一个特殊目标。这些模型非常适合于研究资源空间的变化如何改变动态环境中的行为。目的是从数学和定量的角度来理解这些要求。后续的开发工作将朝着那些将从具有这种建模能力的研究中受益的生态系统的研究需求的方向进行。主要参考文献：Baines，Hubbard和Jimack，(I)自适应求解具有移动边界的含时偏微分方程组的移动网格有限元算法}，Appl.诺默。数学，54，第450-469页，2005，(Ii)移动网格有限元方法及其在相变问题数值解中的应用)，通用。《计算物理》，第6页，第595-624页(2009)。(与R.Mahmoud)Watkins，一种移动网格有限元方法及其在人口动力学中的应用，英国雷丁大学博士论文(2017)。Hilhorst，在生物学中产生的类Stefan问题中的潜热极限消失，(非线性分析：现实世界应用)，4，第261-285页，2003。

The project concerns mathematical modelling of ecological systems on temporal and spatial scales in quantitative population ecology. The purpose of the project is to exploit certain two-dimensional mathematical models of competition-diffusion systems, with aggregation and clustering effects, in dynamic domains. A version of the partilal differential Lotka-Volterra competition equations will be studied that describes a two-phase reaction-diffusion system. Of special interest is multi-phase systems, particularly those segregated with a dynamic interface between phases, for example between different physical states or between warring species. An effective numerical technique is the moving mesh finite element method (MMFEM) of Baines, Hubbard and Jimack, that uses a geometric conservation approach to mesh adaptation. This technique has recently been exploited by Watkins in her PhD thesis (Reading 2017) which considers the application of the MMFEM to systems of intraspecies and interspecies interactions with aggregating dynamics and clustering. A touchstone is the Stefan problem with dynamical interface behaviour, as in the work of Hilhorst. Following on from familiarisation with the background and the numerical technique, the student will compare the behaviour of the models against empirical data sets. The models lend themselves to dynamical adaptations in the sizes and shapes of the domains, as well as to alterations to the logistic terms and changes in parameters, without the need for further calculations. This adaptability means there is a wide range of realistic biological and ecological systems to which the models can be applied and validated. Comparison with data sets for species which show competition-diffusion-aggregation behaviour will be a particular objective of the research.The models are highly suitable for tackling how changes in the resource space might alter behaviour in a dynamical context. An aim will be to understand these requirements from both a mathematical and quantitative perspective. The subsequent development work will be in the direction of the research requirements of those ecological systems which would most benefit from a study which has access to this modelling capability.Key References:Baines, Hubbard and Jimack, (i) A Moving Mesh Finite Element Algorithm for the Adaptive Solution of Time-Dependent Partial Differential Equations with Moving Boundaries}, Appl. Numer. Math., 54, pp. 450-469, 2005, (ii) A moving-mesh finite element method and its application to the numerical solution of phase-change problems), Commun. in Comput.Phys., 6. pp. 595-624 (2009). (with R.Mahmoud)Watkins, A Moving Mesh Finite Element Method and its Application to Population Dynamics, PhD thesis, University of Reading, UK (2017).Hilhorst, Vanishing latent heat limit in a Stefan-like problem arising in biology, (Nonlinear analysis: real world applications), 4, pp.261-285, 2003.