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

我的研究重点是开发和分析数学模型来研究人口动态。我感兴趣的是模型假设如何影响预测，特别是整个可能的动态范围。 许多过程涉及在数学易处理性模型中被忽略的时间延迟。我一直在研究忽略这些延迟的后果以及如何对延迟进行建模，即，使用不同的分布。模型中使用的许多数学函数并没有机械地证明是合理的，而是从数据中提出了它们的基本形式。我一直在研究对模型预测的数学形式（具有相同属性）使用不同选择的含义。一种形式比另一种形式更有可能预测物种灭绝吗？是否有一种形式更有可能预测人口规模的振荡，而不是接近恒定的人口规模？这种影响是一致的还是依赖于模型的？ 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，并提出新的或修改的原则。另一个目标是开发可测量的标准，使科学家能够预测哪种微生物组合在水净化，生物废物分解和生物修复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。最近，我对建模产生了兴趣：与从动物粪便中生产绿色能源有关的厌氧消化;用于水净化的自循环发酵;探索光对浮游植物水华的影响;以及探索各种捕食者-猎物和竞争模型的预测。更一般地说，我有兴趣尝试理解振荡行为和混沌动力学的原因。 为了探索模型的动态，微分方程的定性理论被用来确定模型的局部动态，如果可能的话，还可以使用全局动态;持久性理论在适当的时候预测某些物种何时避免灭绝;分叉理论确定模型行为的全谱，并帮助确定哪些关键参数需要测量以及如何准确，以改善预测。 使用专门的软件来获得分叉图，计算机模拟来阐明复杂的动力学，测试结构，并揭示有助于开发分析证明的属性，以及符号计算来验证冗长的计算。分析往往会导致有趣的抽象数学问题的动力系统，普通，脉冲，积分和泛函微分方程。

My research focuses on developing and analyzing mathematical models to study population dynamics. I am interested in how model assumptions influence the predictions, especially the entire range of possible dynamics. Many processes involve time delays that are ignored in models for mathematical tractability. I have been investigating the consequences of ignoring these delays as well as how the delays are modeled, i.e., using different distributions. Many mathematical functions used in models are not mechanistically justified, but rather only their basic form is suggested from data. I have been investigating the implications of using different choices for the mathematical forms (with the same properties) on model predictions. Is one form more likely to predict extinction of species than another? Is one form more likely to predict that a population size oscillates as opposed to approaching a constant population size? Is this influence consistent or model dependent? One goal is to reconcile commonly believed general principles with conflicting experimental observations, and to 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 waste decomposition, and biological remediation. Other potential applications include pest control on the one hand and the prevention of the extinction of endangered species on the other. Recently, I have become interested in modelling: anaerobic digestion related to the production of green energy from animal waste; self-cylcing fermentation used for water purification; exploring the effect of access to light on phytoplankton blooms; and exploring the predictions of various predator-prey and competition models. More generally, I am interested in trying to understand the causes of oscillatory behavior and chaotic dynamics. To explore the dynamics of the models, the qualitative theory of differential equations is used to determine local and if possible global dynamics of the models; persistence theory when appropriate to predict when certain species avoid extinction; bifurcation theory to determine the full spectrum of model behavior and to help identify which key parameters need to be measured and how accurately, in order to improve predictions. Use is made of specialized software to obtain bifurcation diagrams, computer simulations to elucidate complex dynamics, to test conjectures, and to reveal properties that help to develop analytic proofs, and symbolic computation to verify lengthy calculations. The analyses often lead to interesting abstract mathematical problems in dynamical systems, ordinary, impulsive, integro, and functional differential equations.