Mathematical Sciences: Analysis of Mathematical Models of Ecological Problems

恒化器是一种实验室仪器， 在生态学中用来研究剥削性竞争。 (It是一 一个简单的湖）。 这是一个地方， 数学和生物实验都很容易掌握。所有 所需的系数在实验室中是可测量的。 的 基本模型不考虑养分吸收之间的延迟 细胞分裂（cell division） 一个模型，它允许 延迟以一种试图捕捉复杂的 生理学中的一个常数。 该项目将 发展新的数学方法， (or细胞生长），并将分析其行为。 模型 将是泛函微分方程（细胞周期）或 偏微分方程（细胞生长） 另一 感兴趣的修改是删除充分搅拌的假设 让营养物质和细胞移动 扩散或对流。导致反应扩散 或者说，是一种“一个人”，一种“一个人”。 项目 包括对这些模型的分析。 坚持的概念抓住了所有人都认为 一个生态系统模型的组成部分“生存”。 最近的结果扩展 这一思想的渐近光滑半动力系统， 包括时滞方程和一些偏微分方程。 Waltman教授将研究这项工作的应用 这类有意义的生物学问题， 那些描述竞争的。

The chemostat is a piece of laboratory apparatus that is used in ecology to study exploitative competition. (It is a model of a simple lake.) It is one place where both the mathematics and the biological experiments are tractable .. all of the needed coefficients are measurable in the laboratory. The basic model does not allow for the delay between nutrient uptake and reproduction (cell division). A model which does allow for a delay does so in a way which tries to capture the complicated physiology involved in a single constant. This project will develop new mathematical methods wich incorporate the cell cycle (or cell growth) and will analyze their behavior. The models will be either functional differential equations (cell cycle) or partial differential equations (cell growth). Another modification of interest is to remove the well stirred hypothesis of the chemostat and let the nutrient and the cells move by diffusion or convection .. resulting a reaction diffusion equation or (in the discretization) a gradostat. The project includes an analysis of these models as well. The notion of persistence captures the ideas that all components of a model ecosystem "survive". Recent results extend this idea to asymptotically smooth semi.dynamical systems which includes delay equations and some partial differential equations. Professor Waltman will investigate the applications of this work to meaningful biological problems of this type, particularly those describing competition.