Mathematical Sciences: Applied Geometric Singular Perturbation Theory and Biofilm Modeling

[494160]研究者研究几个奇异摄动系统中的波动现象。这些模型来自生物学和化学。讨论了几个重要的主题。第一个主题是为这些模型提供行波和驻波模式的数学存在性证明。用几何奇异摄动的方法来确定这些存在性结果。与这个主题相关的是对这些波和模式的稳定性和分岔的研究。另一个项目致力于开发生物膜的预测模型。一些生物膜可以起到有益的作用，而另一些则会对自然或工业环境造成广泛的破坏。了解这些系统中发生的过程和相互作用，并制定概念和数学模型，使人们能够控制生物膜过程，从而减少它们的负面影响或增强它们的积极影响。目前，对生物和化学背景下的波传播的理解相当有限，特别是与其他物理系统（水波、声波等）中的波现象相比。其中一个原因是这些方程很难分析。这些方程通常表现出许多不同的时间尺度，即系统是奇异摄动的。尽管存在这些困难，但在这些系统中，波的传播是一个非常重要的现象。例如，心肌上的重入波被认为是引起室性心动过速的兴奋模式。研究者研究波动现象，动机是生物学和化学中的几个问题。特别有趣的是生物膜建模。通俗地说，生物膜可能被称为黏液或污泥。生物膜在自然界中没有人类的相互作用，或者可以人为地引入工业或自然系统。由于它们对水质、发电、能源效率和产品质量的普遍影响，它们的负面影响在大多数美国工业中都能感受到。生物膜工艺还可以提供独特的工艺优势，包括更高的工艺吞吐量，更有效的分离/净化和危险废物场地的生物修复能力。生物膜过程控制的潜力尚未完全实现，部分原因是缺乏对这些过程的基本知识。

9494160 Dockery The investigator studies wave phenomena in several singular perturbed systems. The models come from biology and chemistry. Several important topics are addressed. The first topic is to provide mathematical existence proofs for traveling waves and standing patterns for these models. The methods of geometric singular perturbations are used to ascertain these existence results. Related to this topic is the study of the stability and bifurcations of these waves and patterns. Another project is devoted to developing a predictive model of biofilms. Some biofilms can serve beneficial purposes, while others can cause extensive damage in the natural or industrial environment. Understanding the processes and interactions occurring in these systems, and formulating conceptual and mathematical models enable one to control biofilm processes, and thus reduce their negative effects or enhance their positive effects. At present an understanding of wave propagation in biological and chemical contexts is quite limited, especially in comparison to wave phenomena in other physical systems (water waves, acoustic waves, etc.). One reason for this is that the equations are difficult to analyze. These equations typically exhibit a number of different time scales, i.e., the systems are singularly perturbed. Despite these difficulties, wave propagation in these systems is a very important phenomena. By way of example, reentrant waves on myocardium are thought to be the pattern of excitation responsible for ventricular tachycardia. The investigator studies wave phenomena, motivated by several problems in biology and chemistry. Of particular interest is biofilm modeling. In the vernacular, a biofilm might be called slime or sludge. Biofilms occur in nature without human interaction, or can artificially be introduced to industrial or natural systems. Because of their pervasive effects on water quality, power generation, energy eff iciency, and product quality, their negative effect is felt in most U.S. industries. Biofilm processes can also offer distinct process advantages, including capacity for higher process throughput, more efficient separation/purification, and bioremediation of hazardous waste sites. The potential of biofilm process control has yet been fully realized due in part to a lack of fundamental knowledge of these processes.