Models in Mathematical Biology: A Feedback Perspective

University Of Louisville Research Foundation Inc, Louisville KY

Investigators

Abstract

De Leenheer Recent advances in the life sciences have been aimed at developing and analyzing mathematical models for complex biological systems. This has been triggered by the need for knowledge of their long term behavior. Moreover, a better understanding of these systems might lead to insights in the underlying mechanisms, which are usually much more complicated and less well understood than in the physical sciences. An in-depth analysis sometimes reveals unsatisfactory behavior, e.g. extinction of a certain species or -- at the opposite end of the spectrum -- persistence of an infectious disease, and require appropriate modifications. In this project the investigator uses control theory in both analysis and possible modification of the behavior of biological models. The main idea is to consider these models from a feedback perspective -- provided this is possible of course --, which is a central topic of interest in control theory. It is argued that certain biological (sub)systems are actually feedback systems. Two examples are studied from this point of view. The first is the chemostat model with crowding effects, while the second concerns certain classes of predator-prey systems. Stability issues are of primary concern in this study. A second part of the work is devoted to control (or modification) of the behavior of chemostat models with possibly multiple resources. The novelty of the approach lies in the way the control is implemented, by means of feedback control. The very principle of this type of control is that the control action is based on the state of the system. The usual strategies appearing in the literature consider constant controls leading to bifurcation analysis or control signals that depend only on time -- but not on the state of the system. Many biological systems are under close examination these days. The human genome project, infectious diseases such as AIDS, the flu, or even those caused by a bio-terror attack are but few of the compelling examples. Usually these systems are very complex and hard to understand, let alone control. The aim of this project is to use the perspective of control theory in dealing with these problems. Ideas from control theory have been very successful in applications in more traditional areas. Chemical and nuclear plants, airplanes, vehicles and satellites, robots and audio-visual equipment are nowadays equipped with computers controlling the smooth operation of the specific application. The question is whether control theory can contribute to rapid developments in the biological sciences. As an example, consider a biological system in which a number of organisms compete against each other. A basic question is whether or not the species survive. Depending on the application one would like the answer to this question to be yes (for example if some of the species are endangered) or no (for instance if one of the species is a malignant virus). Sometimes the answer turns out to be no where a yes is desired or vice-versa, and then the issue becomes whether this can be modified by suitably manipulating the system. This type of problem is central in control theory. The lessons learned in the areas mentioned above should prove valuable in a biological setting.