Smith 9700910 Dynamical systems that arise in the biological sciences, and especially in population biology, often have special features that can be exploited in the study of their asymptotic behavior. They are typically dissipative and often the different components have a fixed feedback relation to other components. Systems of competitive or cooperative type and monotone cyclic feedback systems often arise. In this project the investigator exploits these special features, particularly monotonicity properties, in order to describe the asymptotic behavior of the system. He explores the possibility that some classes of systems with definite feedback relations between components have simpler long-term dynamics than their dimension implies. For example, the asymptotic behavior of cooperative systems of differential equations is convergence to equilibrium, while competitive systems behave like general systems of one less dimension and monotone cyclic feedback systems satisfy the Poincare-Bendixson trichotomy whatever their dimension. The application of these ideas to mathematical models of microbial cultures in bio-reactors is pursued. Of particular interest is a model of microbial competition for limiting resource in a distributed environment mimicking the large intestine of a mammal. The central questions of interest to physiologists are what factors determine the set of micro-organisms able to colonize the gut and what are the chief factors determining the stability of the colony to invasion by ingested organisms. Modeling issues arise in order to model both advection of the lumen and the effects of bacterial adherence to the intestinal walls. The model is intended to focus on these questions. In a different direction, the investigator continues earlier work aimed at simplifying complicated structured population models (those models that take account of biological structure such as age, size, physiological state, etc of a population) to simpler mode ls consisting of delay differential systems, possibly state dependent. The rationale is that there are fewer tools that apply to the hyperbolic partial differential equations that arise in structured population models than there are for delay differential equations. As a particular example, a model of microbial growth in a chemostat incorporating effects of cell-cycle phase is investigated. Mathematical models play a major role in the theoretical understanding of many issues in the biological sciences. Mathematical models, as simple caricatures of the real phenomena, allow an investigator to more easily and inexpensively answer the kinds of questions: "what if we change this to that?" Many such models that arise in physiology, population biology, bioengineering and epidemiology take the form of differential equations where it is important for understanding the applications to determine what happens to the solutions of these equations over the long run. The equations that arise in the biological sciences typically have special features not found in mathematical models in the physical sciences. This project aims at developing methods that exploit these special features to determine the long-time behavior of the solutions and to apply these methods to problems of current interest in biotechnology, population biology and physiology. Improved methods will lead to more accurate predictions from existing mathematical models and the ability to analyze more complicated, realistic models. As one example among several to be considered in the proposed work, the investigator and P. Waltman propose to study a mathematical model of microbial growth and competition for nutrients taking place in the large intestine of a mammal. As the intestines are often a source of infection from micro-organisms, it is important to understand the factors involved in determining the kinds of bacteria normally present in the intestine and the factors that determine whether an invadin g organism (e..g. a pathogen) can successfully establish itself as a member of the colony. They intend to construct a mathematical model based on a flow reactor that will exhibit the main features of the intestine (flow of lumen through the intestine and bacterial adherence to the walls of the intestine) in order to better understand the answers to the questions posed above.
