Mathematical Sciences: Analysis of Mathematical Models in Ecology

研究人员沃尔特曼开发和分析了一类生物模型，通常被描述为类恒化器模型。虽然常微分方程组技术对基本模型是有效的，但扩散和输运等效应需要反应扩散方程组。基本技术是无穷维空间上的单调动力系统的技术。延迟的影响(例如细胞周期)也可以在这样的上下文中描述。在以前的工作中发展的关于有序Banach空间乘积上的竞争系统的一般技巧似乎是适用的。在转基因生物的产品生产过程中，经常使用的是质粒体。因为质粒在繁殖过程中可能丢失，而且没有质粒的有机体通常是更好的竞争者，所以编码抗生素耐药性通常包括在质粒上和饲料瓶中使用的抗生素上。描述含有抑制剂和质粒的药物的微分方程式模型不再是单调的。该项目负责对这些模型进行数学分析。恒化器是模拟生物反应器的基本实验室设备。这种模型的数学描述采用常微分方程组的形式。如果从模型中去掉良好混合的基本假设(有证据表明它在总体上并不满足)，就会产生更复杂的方程。该项目利用现代动力系统技术对这些模型方程进行分析。一般的观点是用数学为生物(或化学工程)实验指明方向，希望建造更好的生物反应堆。

Waltman The investigator develops and analyzes a class of biological models generally described as chemostat-like models. While ordinary differential equations techniques have been effective for the basic models, effects such as diffusion and transport require system of reaction diffusion equations. The basic techniques are those of monotone dynamical systems on infinite dimensional spaces. The effect of delays (for example the cell cycle) can also be described in such a context. General techniques for competitive systems on products of ordered Banach spaces, developed in previous work, seem to be applicable. Plasmids are often used in the manufacture of products by genetically altered organisms. Because the plasmid can be lost in reproduction and because the plasmid-free organism is often a better competitor, coding for antibiotic resistance is often included on the plasmid and an antibiotic used in the feed bottle. The differential equation models describing chemostats with inhibitors and plasmids are no longer monotone. The project undertakes the mathematical analysis of such models. The chemostat is a basic piece of laboratory apparatus that models a bio-reactor. The mathematical description of such models takes the form of systems of ordinary differential equations. If the basic assumption of well mixing is removed from the model (there is evidence that it is not satisfied in general) more complicated equations result. The project undertakes the analysis of such model equations using the modern techniques of dynamical systems. The general view is to use the mathematics to point the way to biological (or chemical engineering) experiments with the hope of building better bio-reactors.