Mathematical modelling in population dynamics

人口动态的数学模型

• 批准号: 9358-2011
• 负责人: Wolkowicz, Gail
• 金额: $ 1.09万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568665
• 关键词: Mathematical; modelling; population; dynamics

The focus of my research programme is on the development and analysis of mathematical models to study population dynamics. Instead of analyzing models of specific organisms, I try to determine robust classes of models that exhibit similar dynamics and to investigate what modifications result in significant changes to these dynamics. In an attempt to better understand the structure of communities, and in particular, what factors promote or limit the diversity of natural ecosystems, I use a resource-based approach that results in mathematical models of species interactions that often make predictions that can be tested in the laboratory in such devices as chemostats or self cycling fermentors. I consider both this mechanistic, resource-based approach, as well as classical models that are often more general, and require fewer equations, but involve parameters that may not be as easily measured. Comparing the range of dynamics predicted by different models is important, to understand how sensitive model predictions are to the various modelling approaches. One goal is to reconcile commonly believed general principles with conflicting experimental observations, and hence suggest new or modified principles. Another goal is the development of measurable criteria that would enable scientists to predict which combination of microorganisms would be most effective and safest for use in such processes as water purification, biological waste decomposition, biological remediation, and the production of bio-fuels. Other potential applications include pest control on the one hand and the prevention of the extinction of endangered species on the other. The analyses of the different models often lead to interesting mathematical problems in dynamical systems, ordinary, impulsive, integro- and functional differential equations, and the qualitative theory of differential equations including bifurcation, stability, and persistence theory.

我的研究计划的重点是开发和分析研究人口动态的数学模型。我不是分析特定生物体的模型，而是试图确定表现出类似动力学的稳健的模型类别，并调查哪些修改会导致这些动力学的重大变化。为了更好地了解群落的结构，特别是什么因素促进或限制了自然生态系统的多样性，我使用了一种基于资源的方法，这种方法产生了物种相互作用的数学模型，这些模型经常做出预测，可以在实验室里用化学调节器或自我循环发酵器等设备进行测试。我既考虑了这种机械的、基于资源的方法，也考虑了传统的模型，这些模型通常更通用，需要的方程式更少，但涉及的参数可能不那么容易测量。比较不同模型预测的动态范围，了解模型预测对各种建模方法的敏感度是很重要的。一个目标是调和普遍认为的一般原则与相互冲突的实验观察，从而提出新的或修改后的原则。另一个目标是制定可测量的标准，使科学家能够预测哪种微生物组合在水净化、生物废物分解、生物修复和生产生物燃料等过程中使用最有效和最安全。其他潜在的应用包括一方面控制虫害，另一方面防止濒危物种灭绝。 对不同模型的分析常常会产生一些有趣的数学问题，如动力系统中的常微分方程组、脉冲微分方程组、积分微分方程组和泛函微分方程组，以及微分方程组的定性理论，包括分支理论、稳定性理论和持久性理论。