Dynamical Systems in Biology

Arizona State University, Scottsdale AZ

Investigators

Abstract

Broadly, the investigator studies specific mathematical models arising from the life sciences and develops new mathematical tools with which to analyze such models. Models of biofilms, previously developed by this investigator and collaborators, are used to study important phenomena such as gene transfer among micro-organisms and quorum sensing. These models are based on the classical chemical engineering bio-reactors, namely, the continuously stirred tank reactor and the tubular plug flow reactor with dispersion, and take the form of ordinary differential equations for the former and coupled parabolic equations with nonlinear boundary conditions for the latter. The aim is to understand the factors that facilitate biofilm formation, the factors involved in the successful introduction of new genes via plasmid transfer in biofilm communities, and the role of bacterial quorum sensing in these processes. The study also addresses a fundamental issue in population biology -- the so-called "paradox of the plankton": why can so many plankton species be supported by so few limiting resources in lakes and seas? The same paradox arises with the microbial denizens of arid soils that receive infrequent rains and deposition of nutrients -- another focus of the project. The investigator's aim is to introduce mathematical models based on the classical model of a multi-species community in which members compete exploitatively for several essential (nonoverlapping) resources (light, nitrogen, carbon, etc.) in a well-mixed environment and to provide mathematically rigorous results of an explanatory nature rather than numerical simulations. On the mathematical side, the theory of monotone dynamical systems and persistence theory have greatly contributed to the understanding of biological models and the study aims to enhance and build on these theories. A substantial effort is devoted to study mathematical models of microbial processes in environmental settings where biofilms may form on surfaces. Biofilms are of great importance in the health sciences and industry, where their formation typically results in negative outcomes. They are responsible for food and water contamination, dental caries and periodontal disease, and the contamination of medical implants. The focus is on understanding conditions favorable for biofilm formation, factors involved in gene transfer (e.g., antibiotic resistance) among microorganisms in biofilms, and the ability of organisms to chemically communicate their existence and current state to others organisms on these processes. Broadly, the study is characterized by a desire to obtain mathematically rigorous results along with computer simulations. Modelling and computer simulations are widely acknowledged to be important tools in the study of biological systems. The modelling process forces a careful description of what are the known and important features of a system. Computer simulations are important in order to explore the implications of assumptions and sensitivity of model behavior to parameter changes. However, computer simulations alone seldom lead to a comprehensive understanding of a model, let alone the underlying biological process, due to the inherent limitations of the computer itself -- one can only test a relatively small sample of the full range of model parameters, follow the time course of the model over a relatively small time window, for a relatively small number of possible initial data. Rigorous mathematical analysis can lead to real understanding of a model and the underlying biological processes by identifying mechanisms responsible for model behavior. Unfortunately, the mathematical tools necessary to provide a careful mathematical analysis are often not currently available. The study seeks to develop such tools and to apply them to problems in the life sciences.