Applications of Moving Mesh Finite Elements to Population Dynamics

移动网格有限元在群体动力学中的应用

• 批准号: 2112774
• 金额: --
• 依托单位: University of Reading
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2112774
• 关键词: Applications; Moving; Mesh; Finite; Elements

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.

该项目涉及数量种群生态学在时间和空间尺度上的生态系统数学建模。该项目的目的是利用某些二维数学模型的竞争-扩散系统，聚集和集群效应，在动态领域。将研究描述两相反应扩散系统的Lotka-Volterra偏微分竞争方程的一个版本。特别令人感兴趣的是多相系统，特别是那些在不同物理状态之间或交战物种之间具有动态界面的系统。Baines， Hubbard和Jimack的运动网格有限元法（MMFEM）是一种有效的数值方法，该方法采用几何守恒方法进行网格自适应。沃特金斯最近在她的博士论文（阅读2017）中利用了这项技术，该论文考虑了MMFEM在具有聚集动力学和聚类的种内和种间相互作用系统中的应用。一个试金石是动态界面行为的Stefan问题，就像Hilhorst的工作一样。在熟悉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。这些模型使自己能够动态适应域的大小和形状，以及改变逻辑项和参数的变化，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，这些模型可以应用和验证。与显示竞争-扩散-聚集行为的物种的数据集进行比较将是研究的一个特定目标。这些模型非常适合处理资源空间的变化如何改变动态上下文中的行为。我们的目标是从数学和定量的角度来理解这些需求。随后的开发工作将朝着那些生态系统的研究要求的方向进行，这些生态系统将最受益于利用这种建模能力的研究。[关键文献]Baines， Hubbard and Jimack, (i)带移动边界的时变偏微分方程自适应解的移动网格有限元算法}，applied。号码。数学。(2)移动网格有限元法及其在相变问题数值求解中的应用[j] .岩石力学与工程学报，2005，第4期，第450-469页。在Comput.Phys。6。第595-624页（2009）。（with r.m hamud）Watkins，一种移动网格有限元法及其在人口动力学中的应用，博士论文，英国雷丁大学（2017）。李志强，张志强，李志强，等。基于非线性分析的生物系统研究进展，（自然科学版），第4期，第1-2页，2003。