Mathematical Sciences: Project in Mathematical Modeling in Population Biology

一个REU网站项目将在此奖项下进行 重点研究数学建模领域的问题， 种群生物学 学生将被介绍到技术， 动力系统理论、科学计算和统计 数学中当前感兴趣的问题的方法 生物学 具体活动包括研究 简单相互作用种群，包括捕食者-被捕食者系统 和进化稳定策略。一些系统，如 将介绍在实验室恒化器中进行的研究。 工作 单一和相互作用的年龄结构的动态 种群，包括简单的宿主-寄生虫系统，也将被 贯彻 其他学生将参与流行病学模型 动物和人类疾病的传播。 基于基本 关于传染病病原体传播的假设 在同质人群中，SIR（易感-感染-移除） 将开发模型。 一张全球行为的图片， 由此产生的非线性常微分方程组 将使用相空间分析建立。 还将在资源管理和优化方面开展工作。 收获策略 本研究的主要对象是 建立单一物种的连续种群模型 在不同的收获策略下。 一开始，一项研究 在没有收获的情况下， 一个人如何试图找到一个生物学上真实的模型 数学上也很容易处理 然后各种收割 战略将被引入模型并进行测试。 这些项目将被选中，让学生参与 制定原始数学模型或修改 现有的一个，做定性分析和统计 在可能的情况下估计参数，设计一个数值 方案并进行数值模拟。 理解 模型框架下的数值结果和修正 在有需要时，将采用该模式。

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.