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来预测种群的平均丰度如何变化，以及丰度是否有望随着时间的推移而波动或保持相对恒定。特别是，我研究模型公式中的某些微小差异如何影响模型所展示的整个动力学范围的预测。许多过程都包含延迟，而这些延迟往往由于数学上的可追溯性而被忽略。我研究了如何制定模型来适当地包括延迟和忽略延迟的后果。通过比较无延迟模型与类似的离散模型或具有不同分布的固定或分布延迟的连续模型的动力学，研究了以不同方式包含延迟对动力学的影响。此外，模型中的许多函数不是机械地证明的，而是它们的基本形式是从数据中建议的，我一直在比较用具有相同基本属性的不同数学形式制定的模型可能的动态范围的差异。是一种形式更有可能预测物种灭绝，还是更有可能预测种群规模的平衡或无限振荡？更一般地说，我对试图理解振荡行为和混沌动力学的原因特别感兴趣。一个目标是调和普遍认为的一般原则与相互矛盾的实验观察，从而提出新的或修改的原则。另一个目标是制定可测量的标准，使科学家能够预测在水净化、生物修复、生物废物分解、利用微生物厌氧消化从动物废物中生产绿色能源以及防止有害浮游植物大量繁殖等过程中使用哪种微生物组合最有效和最安全。其他潜在的应用包括虫害防治和防止濒危物种灭绝。微分方程的定性理论用于确定模型的局部和可能时的全局动力学。学生将学习使用持续性理论来预测在什么情况下某些物种会避免灭绝。分岔理论有助于确定所有适当参数范围和初始数据的全谱行为，并有助于确定需要测量的关键参数，以改进预测。如果在任何相互作用中有明显的时间延迟，则使用积分和泛函微分方程。学生发展计算技能，以测试猜想和揭示模型的性质，有助于发展分析证明。符号计算用于复杂的计算。这种分析常常导致动力系统中有趣的抽象数学问题，包括差分方程、常微分方程、脉冲微分方程、积分微分方程和泛函微分方程。