Mathematical Sciences: Analysis of Mathematical Models in Ecology

研究者继续研究动态系统思想在生态学和种群生物学问题中的应用，重点是持久性的概念和趋化器的模型。在反应扩散方程和泛函微分方程的背景下研究了持续性定理。恒化器是生态学中用于研究剥削性竞争的实验室仪器。（这是一个简单湖泊的模型。）这是一个数学和生物实验都易于处理的地方；数学模型方程所需的所有参数在实验室中都是可测量的。基本模型没有考虑到在趋化器中有机体摄取营养和繁殖（细胞分裂）之间的时间差。模型中的一个标准假设是槽体搅拌良好；因此，在水箱的不同部分存在的营养物质的量是不同的。为了研究营养梯度的影响，生物学家提出了另一种被称为梯度计的装置。这个装置的数学原理现在已经相当清楚了。另一种诱导梯度的方法是允许营养物在某一点进入水箱，并随着生物体扩散。持久性的概念试图抓住一个生态概念，即一个模型生态系统的所有组成部分都能生存。这些定理在常微分方程模型中被证明是有用的；例如，在分析渐缓抑制剂和具有抑制剂的趋化抑制剂时。这个概念现在已经以一般形式出现在数学文献中，特别是对于渐近光滑半动力系统。这个项目的目的是利用持久性定理的全部潜力，将其应用于无限维的情况下，反应扩散方程和具有延迟的微分方程。如果改变关于恒化器的假设，去掉充分搅拌的假设，让营养物质和细胞通过扩散或对流移动，那么就会得到反应扩散方程或恒化器。延迟的化学调节模型已经被研究过，但是没有引入生理机制来解释延迟。明显的机制是细胞周期，即营养摄取和细胞分裂之间的时间。

The investigator continues research on applications of dynamical systems ideas to problems in ecology and population biology, focussing on the notion of persistence and models of the chemostat. The persistence theorem is studied in the context of reaction-diffusion equations and functional differential equations. The chemostat is a 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 the parameters needed for the equations of the mathematical model are measurable in the laboratory. The basic model does not allow for the delat between nutrient uptake by the organisms in the chemostat and reproduction (cell division). A standard assumption in the model is that the tank is well stirred; hence there are differences in the amounts of nutrient present in different parts of the tank. To study the effect of a nutrient gradient, another device called the gradostat has been proposed by biologists. The mathematics of that device is now fairly well understood. Another way to induce a gradient is to allow the nutrient to enter the tank at one point and to diffuse along with the organisms. The notion of persistence tries to capture the ecological idea that all components of a model ecosystem survive. Such theorems have proved useful in the context of ordinary differential equation models; e.g., in the analysis of the gradostat and of a chemostat with an inhibitor. The concept has now found its way into the mathematical literature in a general form, in particular for an asymptotically smooth semi-dynamical system. The aim of this project is to use the full potential of the persistence theorem by applying it in an infinite dimensional context, to reaction-diffusion equations and differential equations with delays. If the assumptions about the chemostat are changed to remove the well-stirred hypothesis and to let the nutrient and the cells move by diffusion or convection, then a reaction-diffusion equation or a gradostat results. Delayed chemostat models have been studied, but no physiological mechanism has been introduced to account for the delay. The obvious mechanism is the cell cycle, the time between nutrient uptake and cell division.