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

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振荡或保持相对恒定。特别是，我研究了模型配方中的某些微小差异如何影响模型所表现出的整个动态范围的预测。许多过程涉及延迟，为了数学上的易处理性，这些延迟常常被忽略。我研究如何制定模型，包括延误和忽视延误的后果。我研究的动态，包括延迟以不同的方式通过比较动态的模型没有延迟与类似的离散模型或连续模型具有固定或分布的延迟与不同的分布。 此外，模型中的许多功能并不是机械地证明的，而是它们的基本形式是从数据中提出的，我一直在比较具有相同基本属性的不同数学形式的模型的动态范围的差异。是一种更有可能预测物种灭绝的形式，还是更有可能预测种群规模的平衡或无限期振荡。更一般地说，我特别感兴趣的是试图了解振荡行为和混沌动力学的原因。一个目标是协调通常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衡量的标准，使科学家能够预测哪种微生物组合在水净化、生物补救、生物废物分解、利用微生物厌氧消化从动物废物生产绿色能源以及防止有害浮游植物大量繁殖等过程中最有效和最安全。其他潜在的应用包括一方面控制害虫，另一方面防止濒危物种灭绝。 微分方程的定性理论被用来确定本地和可能的情况下，全球动态的模型。学生学习使用持久性理论来预测某些物种在什么情况下避免灭绝。分叉理论有助于确定所有适当参数范围和初始数据的完整行为谱，并有助于确定需要测量以改进预测的关键参数。如果在任何相互作用中涉及显著的时间延迟，则使用积分和泛函微分方程。学生发展计算技能，以测试结构，并揭示在开发分析证明有用的模型的属性。符号计算用于复杂的计算。分析往往导致有趣的抽象数学问题的动力系统，包括差分方程，普通，脉冲，积分和功能微分方程。

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.