Mathematical Sciences: Project in Mathematical Modeling in Population Biology

根据这一奖项，将开展一个REU场地项目，重点是人口生物学领域的数学建模问题。将向学生介绍动力系统理论、科学计算和统计方法应用于当前数学生物学中感兴趣的问题的技术。具体活动包括研究简单相互作用的种群的动态，包括捕食者-猎物系统和进化稳定的策略。将介绍几个系统，如在实验室恒化器中研究的系统。还将开展关于单一和相互作用的年龄结构种群的动态的工作，包括简单的寄主-寄生虫系统。其他学生将学习动物和人类疾病传播的流行病学模型。基于传染病病原体在同质人群中传播的基本假设，将建立SIR(易感-感染-清除)模型。所得到的非线性常微分方程组的全局行为的图像将利用相空间分析来建立。还将在资源管理和最佳收获战略方面开展工作。这一研究领域的主要目标是建立不同收获策略下单一物种的连续种群模型。在一开始，我们将对没有收获的物种进行研究，以展示人们如何试图找到一个在生物学上真实和数学上容易处理的模型。然后将各种收获策略引入到模型中并进行测试。这些项目将包括让学生制定一个原始的数学模型或修改一个现有的数学模型，尽可能对参数进行定性分析和统计估计，设计一个数值方案并进行数值模拟。随后将在模型框架内理解数值结果，并在必要时修改模型。

An REU site project will be conducted under this award focusing on problems of mathematical modeling in the area of population biology. Students will be introduced to techniques of dynamical systems theory, scientific computing and statistical methods applied to problems of current interest in mathematical biology. Specific activities include studies of the dynamics of simple interacting populations, including predator-prey systems and evolutionarily stable strategies. Several systems such as those studied in laboratory chemostats will be introduced. Work on the dynamics of single and interacting age-structured populations, including simple host-parasite systems, will also be carried out. Other students will be involved with epidemiological models for animal and human disease transmission. Based on basic assumptions about the transmission of infectious disease agents in a homogeneous population, SIR (Susceptible-Infected-Removed) models will be developed. A picture of the global behavior of the resulting nonlinear ordinary differential equations systems will be built up using phase space analysis. Work will also be done on resource management and optimal harvesting strategies. The main object of this research area is to develop continuous population models for a single species under different harvesting strategies. At the beginning, a study of the species in the absence of harvesting will be made to show how one attempts to find a model that is biologically realistic and mathematically tractable. Then various harvesting strategies will be introduced into the model and tested. The projects will be selected to involve the students in formulating an original mathematical model or modifying an existing one, doing qualitative analysis and statistical estimation of the parameters where possible, devising a numerical scheme and performing the numerical simulations. Understanding the numerical results in the framework of the model and modifying the model where necessary will follow.