Formulation and Analysis of Nonlinear Mathematical Models with Applications to Population Dynamics

非线性数学模型的制定和分析及其在人口动态中的应用

• 批准号: RGPIN-2022-05067
• 负责人: Wolkowicz, Gail
• 金额: $ 2.7万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759170
• 关键词: Formulation; Analysis; Nonlinear; Mathematical; Models

I develop and analyze mathematical models to predict how the average abundance of populations changes and if the abundance is expected to oscillate or remain relatively constant over time. In particular, I study how certain minor differences in model formulations influence the predictions of the entire range of dynamics exhibited by models. Many processes involve delays that are often ignored for the sake of mathematical tractability. I investigate how to formulate models to include delays properly and the consequences of ignoring delays. I study the impact on the dynamics of including delays in different ways by comparing the dynamics of models without delays with the analogous discrete models or continuous models with fixed or distributed delays with different distributions.  Also, many functions in models are not mechanistically justified, but rather their basic form is suggested from data, I have been comparing the differences in the range of dynamics possible for models formulated with different choices of  mathematical forms that have the same basic properties. Is one form more likely to predict extinction of species or  more likely to predict that a population size equilibrates or oscillates indefinitely. More generally, I am especially interested in trying to understand the causes of oscillatory behavior and chaotic dynamics. One goal is to reconcile what are commonly believed general principles with conflicting experimental observations, and hence suggest new or modified principles. Another goal is to develop 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 remediation, biological waste decomposition, green energy production from animal waste using anaerobic digestion by microbes, and prevention of harmful phytoplankton blooms. Other potential applications include pest control on the one hand and prevention of extinction of endangered species on the other. The qualitative theory of differential equations is used to determine local and when possible global dynamics of the models. Students learn to use persistence theory to predict under what circumstances certain species avoid extinction. Bifurcation theory helps to determine the full spectrum of behavior for all appropriate parameter ranges and initial data and helps to identify key parameters that need to be measured to improve predictions. If there are significant time delays involved in any interactions, integro- and functional differential equations are used. Students develop computational skills to test conjectures and reveal properties of the models useful in developing analytic proofs. Symbolic computation is used for complicated calculations. The analysis often leads to interesting abstract mathematical problems in dynamical systems, including difference equations, ordinary, impulsive, integro- and functional differential equations.

我开发和分析数学模型，以预测人口的平均丰度如何变化，以及随着时间的推移的振荡或仍然相对恒定的丰度。特别是，我研究模型公式的某些小差异如何影响模型暴露的整个动力学范围的预测。许多过程都涉及延迟，这些延迟通常是为了数学障碍而被忽略的。我研究了如何制定模型以正确地包括延迟和忽略延迟的后果。我通过比较没有延迟的模型的动力学与类似的离散模型或具有不同分布的固定或分布式延迟的连续模型，研究了以不同方式延迟的动态的影响。同样，模型中的许多功能并不是机械合理的，而是从数据中提出了它们的基本形式，我一直在比较具有具有相同基本属性的数学形式的不同选择的模型的动态范围差异。是一种更有可能预测物种扩展的形式，或者更有可能预测人口大小当量或无限期振荡。更普遍地，我特别有兴趣尝试了解振荡行为和混乱动态的原因。一个目标是协调通常认为的一般原则与相互冲突的实验观察，因此提出了新的或修改的原则。另一个目标是制定可衡量的标准，使科学家能够预测微生物的哪种组合最有效，最安全，用于在诸如水纯化，生物修复，生物废物分解，生物学废物分解，使用微生物厌​​氧消化的动物废物产生绿色能源以及有害的phytoplank血液的预防。其他潜在的应用包括一方面的害虫控制以及防止另一方面的濒危物种扩展。微分方程的定性理论用于确定局部，并在可能的模型的全局动力学时。学生学会使用持久理论来预测某些物种避免扩展的情况。分叉理论有助于确定所有适当的参数范围和初始数据的全部行为范围，并有助于确定需要测量的关键参数以改善预测。如果任何相互作用涉及大量时间延迟，则使用整数和功能差分方程。学生开发计算技能来测试猜想并揭示有助于开发分析证明的模型的特性。符号计算用于复杂的计算。该分析通常会导致动态系统中有趣的抽象数学问题，包括差异方程，普通，冲动，整数和功能差分方程。