Mathematical Sciences: Analysis of Mathematical Models of Ecological Problems

恒化器是生态学中用来研究剥削竞争的一种实验室仪器。(这是一个简单湖泊的模型。)这是一个数学和生物实验都很容易处理的地方。所有所需的系数都可以在实验室中测量。基本模型不考虑营养吸收和繁殖(细胞分裂)之间的延迟。一个确实允许延迟的模型以一种试图捕捉单个常量所涉及的复杂生理的方式做到了这一点。这个项目将开发新的数学方法，其中包括细胞周期(或细胞生长)，并将分析它们的行为。这些模型要么是泛函微分方程(细胞周期)，要么是偏微分方程(细胞生长)。另一个令人感兴趣的修改是取消恒化器的充分搅拌假说，让营养物质和细胞通过扩散或对流移动。得到反应扩散方程或(在离散化中)梯度仪。该项目还包括对这些模型的分析。持久化的概念抓住了模型生态系统的所有组件都“存活”的想法。最近的结果将这一思想推广到渐近光滑的半动力系统，其中包括时滞方程和一些偏微分方程组。沃尔特曼教授将研究这项工作在这类有意义的生物学问题上的应用，特别是那些描述竞争的问题。

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.