Analysis of Mathematical Models in Ecology

沃尔特曼9801622恒化器(也称为连续培养或生物反应器)是开放系统的基本模型。它是一个简单湖泊的模型，是商业生物反应堆的实验室版本，也是废水处理模型和哺乳动物肠道模型的起点。微生物以剥削的方式竞争输入的养分。控制该模型的方程采用非线性常微分方程组或偏微分方程组的形式。在这个项目中，调查者在基本模型上添加了新的功能，以使其更加逼真。这使方程变得复杂，从而使数学分析变得复杂。具体地说，需要研究壁生长、扩散、运输、抑制物(自然和外部)以及质粒动力学对竞争结果的影响。数学分析的目的是根据系统参数(可测量的或可操作的参数)来确定这些方程解的全局渐近行为。抑制剂是自然产生的(例如，抗竞争毒素)或以污染物的形式出现。墙壁(或其他表面)上的微生物生长可能会阻碍生化反应器的效率或污染任何流动过程。对这类问题的渐近分析可以更好地理解自然现象或反应堆技术。对这一行为的了解使人们能够预测固定(即自然)环境的结果，或者在人工(商业)环境的情况下控制结果。上述所有影响汇集在一起的是哺乳动物肠道的模型，其中壁的生长对这一过程至关重要，抑制剂影响竞争，运输和扩散不能被忽视。这项研究的目的之一是提供一个更好的人类大肠模型，并更好地了解所涉及的每一种现象的相对影响。

Waltman 9801622 The chemostat (also called a continuous culture or a bioreactor) is a basic model of an open system. It is a model of a simple lake, a laboratory version of the commercial bio-reactor, and the starting point for models of waste water treatment and models of the mammalian intestine. Microorganisms compete for an input nutrient in an exploitative manner. The equations governing the model take the form of systems of nonlinear ordinary or partial differential equations. In this project, the investigator adds new features to the basic model in order to make it more realistic. This complicates the equations and hence the mathematical analysis. Specifically, the effects of wall growth, diffusion, transport, inhibitors (natural and external), and plasmid dynamics on the outcome of the competition are to be studied. The goal of the mathematical analysis is to identify the global asymptotic behavior of the solutions to these equations in terms of the system parameters (measurable or operational parameters). Inhibitors occur naturally (for example, anti-competitor toxins) or as pollutants. Growth of microorganisms on the wall (or other surface) can impede the efficiency of a biochemical reactor or contaminate any flow process. The asymptotic analysis of such problems can lead to a better understanding of natural phenomena or of reactor technology. Knowledge of this behavior allows one to predict the outcome of a fixed (i.e., natural) environment or to control the outcome in the case of an artificial (commercial) environment. All of the effects mentioned above come together is models of the mammalian intestine where wall growth is essential to the process, inhibitors affect the competition and where transport and diffusion cannot be ignored. One of the objectives of the study is to provide a better model of the human large intestine and a better understanding of the relative effect of each of the phenomena involved.