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。许多过程涉及时间延迟，在数学可追溯性模型中被忽略。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不是机械地证明的，而只是从数据中提出它们的基本形式。我一直在研究在模型预测中使用不同选择的数学形式（具有相同属性）的含义。一种形式比另一种形式更有可能预示物种灭绝吗？是否有一种形式更有可能预测种群规模的波动而不是接近恒定的种群规模？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相信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。另一个目标是制定可测量的标准，使科学家能够预测在水净化、生物废物分解和生物补救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。最近，我对建模产生了兴趣：与动物粪便产生绿色能源相关的厌氧消化；用于水净化的自循环发酵；探索光照对浮游植物华度的影响；探索各种捕食者-猎物和竞争模型的预测。更一般地说，我感兴趣的是试图理解振荡行为和混沌动力学的原因。***为了探索模型的动力学，我们使用微分方程的定性理论来确定模型的局部动力学，如果可能的话，还可以确定模型的全局动力学；持久性理论适用于预测某些物种何时免于灭绝；分岔理论，以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，以提高预测。使用专门的软件来获取分岔图，计算机模拟来阐明复杂的动力学，测试猜想，并揭示有助于发展分析证明的特性，以及符号计算来验证冗长的计算。这些分析常常导致动力系统、普通、脉冲、积分和泛函微分方程中有趣的抽象数学问题。********

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.********