Mathematical Biology

The Mathematics Department has several faculty with strong interests in mathematical biology, especially ecology. They regularly conduct seminars and many of them collaborate with colleagues in the School of Biological Sciences and the School of Natural Resources on east campus. Their broad research activities involve modeling on organizational scales from molecules, to populations, to global processes. In particular, their interests are in: population ecology and conservation; the effects of global climate change, like temperature variation and carbon dioxide levels, on interactions in ecosystems; disease dynamics in consumer-resource interactions; food webs and chaos; neural networks; applications of communication theory to genetics; optimal molecular ratios and stoichiometry. The mathematical models involve continuous and discrete dynamical systems, partial differential equations, optimization, probability and stochastic processes, and statistics and parameter identification. Research and coursework leading to a PhD in mathematics with an emphasis in mathematical biology is a core part of the program.

Faculty

Chad Brassil, a colleague in the School of Biological Sciences, does research in the area of theoretical ecology by utilizing mathematics, principally nonlinear differential equations, as a tool for understanding ecology and evolution. A primary theoretical interest of his has been examining the implications of temporal variation for fundamental ecological theory. Current work is incorporating evolutionary dynamics into models of tropical diversity. In addition, he utilizes maximum likelihood techniques to bridge the gap between theoretical and empirical ecology.

Bo Deng has interests in Mathematical Biology which include: the origins and the evolution of DNA codes, electrical neurophysiology and neural communication, foodweb chaos and ecological stability, disease dynamics and epidemic modeling. The main tools which Professor Deng uses in his research activities include: information and communication theory, circuitry, differential equations, qualitative theory of dynamical systems, and applied nonlinear analysis. Through modeling, Professor Deng hopes to use mathematics to gain better understanding on biological processes.

Steve Dunbar has research interests in nonlinear differential equations, and applied dynamical systems, particularly those which arise in mathematical biology. In conjunction with his work with differential equation models and systems of mathematical biology, he is also interested in stochastic processes, the numerical and computer-aided solution of differential equations, and mathematical modeling. He also is interested in issues of mathematical education at the high school and collegiate level. He is the Director of the American Mathematics Competitions program of the Mathematical Association of America which sponsors middle school and high school mathematical competitions leading to the selection and training of the USA delegation to the annual International Mathematical Olympiad. In addition, he has interests in documenting trends in collegiate mathematics course enrollments and using mathematical software to teach and learn mathematics.

Wendy Hines does research in dynamical systems. She is interested in the general theory and also applications to delay equations and partial differential equations. Currently she is working on a a reaction-diffusion equation with nonlocal diffusion which models gene propogation through a population. This is a very interesting problem as very little has be one on it and it defies the application of standard reaction-diffusion methods.

Glenn Ledder works in mathematical modeling for life sciences and physical sciences. His current interests include population dynamics and dynamic energy budget models. He is also active in developing an undergraduate mathematics curriculum for biology students and in mentoring REU student groups. He is the director of the RUTE (Research for Undergraduates in Theoretical Ecology) program.

David Logan works in the areas of applied mathematics and ecological modeling. His interests include ordinary and partial differential equations, difference equations, and stochastic processes. His current research in mathematical ecology includes work on nutrient cycling, physiologically-structured population dynamics, the effects of global climate change on ecosystems and food webs, and insect eco-physiology.

Irakli Loladze combines the techniques of mathematical modeling with the principles of ecological stoichiometry to understand biological problems. His recent interests include the response of plants and crops to globally rising atmospheric carbon dioxide concentrations, the link between RNA:protein ratios and nitrogen to phosphorus ratio in oceans, the role of variable food quality in structuring food webs and its effect on the spread of infectious diseases.

Richard Rebarber does research in Mathematical Ecology and in Distributed Parameter Control Theory. His research in Ecology is in the analysis of discrete and continuous population models, including: the effect of data uncertainty on long-term and transient population growth; the application of robust control theory methods to population management; and the mathematics of biological invasions.

Tom Shores is interested in the numerical solutions of ordinary and partial differential equations, especially those singular and nonlinear equations which are amenable to sinc methods. He is also interested in numerical methods for problems in inverse theory, especially parameter identification problems in ordinary and partial differential equations. More generally, he has research interests in issues dealing with scientific computation. Finally, he is interested in the mathematical modeling of populations and porous medium problems.

Brigitte Tenhumberg uses stochastic, discrete time models tailored to specific biological systems to advance the understanding of ecological processes. The models she uses include stochastic dynamic programming, matrix models, and agent based simulation models. One area of research emphasis is optimal decision making of animals (foraging or life history decisions) or humans (management of wildlife populations). Recent work addresses topics in invasion ecology, in particular understanding ecological mechanisms promoting ecosystem resistance to invasions.

Graduate Students

Joan Lubben Richard Rebarber

Amy Parrott David Logan