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，以及丰度是否会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整个范围的预测。许多过程都涉及延迟，但为了数学上的可处理性，这些延迟常常被忽略。我研究了如何正确地构建模型以包含延迟，以及忽略延迟对动态的影响。通过比较无延迟模型的动力学此外，模型中的许多函数在机械上并不是合理的，而是从数据中建议了它们的基本形式，我一直在比较可能的动态范围的差异。用具有相同基本属性的不同数学形式制定的模型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无限期平衡或振荡。更一般地说，我特别有兴趣尝试理解。这一个目标是调和普遍认为的一般原理与相互矛盾的实验观察结果，从而提出新的或修改的原理，使科学家能够预测哪种微生物组合会产生振荡行为和混沌动力学。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预防有害浮游植物等过程中最有效和最安全其他潜在的应用包括一方面防治害虫，另一方面防止濒危物种灭绝，微分方程的定性理论用于确定模型的局部动力学和全局动力学（如果可能）。预测在什么情况下某些物种可以避免灭绝。分叉理论有助于确定所有适当参数范围和初始数据的全部行为，并有助于确定需要测量的关键参数，以改善预测。任何相互作用、整合和功能差异学生发展计算技能来测试猜想并揭示可用于开发分析证明的模型的属性。分析通常会导致动态系统中有趣的抽象数学问题，包括差分方程、普通方程、脉冲、积分和函数微分方程。