Dynamical Systems in Biology

Arizona State University, Scottsdale AZ

Investigators

Abstract

Smith 0107160 The investigator develops mathematical methods that exploit the special properties inherent in dynamical systems arising in the biological sciences and applies them to specific problems. The main focus is on the long-term behavior of the dynamics after transient effects have disappeared. The work on biofilms focuses on systems of advection-diffusion equations coupled to ordinary differential equations with coupled nonlinear boundary conditions in piecewise-smooth domains. Aside from well-posedness and the existence of multiple steady states, which are nontrivial questions, the important biological information comes from key eigenvalues for associated nonstandard eigenvalue problems and their dependence on parameters. Both mathematical analysis and numerical simulations are required for an understanding. The work on the paradox of the plankton focuses on the classical model of n-species exploitative competition for k essential (non-substitutable) nutrients. If n is larger than k there can be no equilibrium coexistence of all n species so one must seek oscillatory coexistence. The investigator exploits certain heteroclinic structures (cycles of equilibria) that arise naturally in such systems in order to seek periodic coexistence solutions. The theory of competitive and cooperative systems can play a significant role. Abstract dynamical systems theory comes into play in the work on robust persistence. Specifically, understanding chain recurrent sets and Morse decompositions of the boundary dynamics (where one or more species are absent) are key to showing that persistence for a dynamical system is robust to perturbation of that system or of its parameters. Dynamical systems that arise naturally in many areas in the biological sciences often have special features not shared by systems arising in the physical sciences. Broadly speaking, the investigator develops mathematical methods that exploit these special properties and applies them to specific problems. The main focus is on the long-term behavior of the dynamics after transient effects have disappeared. Progress in this area can help to answer important biological questions that are subject to mathematical modeling, such as whether an infectious disease becomes endemic in a population or becomes extinct, or which species in an ecosystem can survive and which will become extinct. A substantial effort is devoted to understanding the dynamics of mathematical models in three areas of the biosciences. The first is microbial growth and competition in environmental settings where biofilms may form on surfaces. Biofilms are of great importance in the health sciences and food 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 investigator studies under what conditions biofilms may form and what bacterial densities can be expected. The second is a fundamental issue in population biology, the so-called "paradox of the plankton": why can so many plankton species (and, more generally, species of other taxa) be supported by so few limiting resources? The investigator provides mathematically rigorous results for the existence of oscillatory coexistence states of the relevant mathematical models when the number of species exceeds the number of resources, as is typical in natural environments. The third is a more robust theory of persistence (also called permanence) for dynamical systems arising in population dynamics. In its simplest form, this theory seeks not to determine the global behavior of solutions of a dynamical system representing interacting populations but rather to answer the more basic question: What species are present at the end of the day? While past research in this area has primarily ignored the fact that population models are only approximately correct, the investigator seeks to provide answers to the basic question that hold true for all small perturbations of the model equations.