The Department of Mathematics has a prominent program in mathematical biology, with current strength in two major areas: Mathematical Ecology and Mathematical Neuroscience. Core faculty in ecology include Professors Bo Deng, Steve Dunbar, Glenn Ledder, David Logan, Richard Rebarber and Brigitte Tenhumberg (joint appointment with SBS); core faculty in neuroscience include Professors Carina Curto, Bo Deng and Vladimir Itskov. The group has significant collaborative relationships with colleagues in the life sciences across campus and at other institutions. Group members have mathematical backgrounds in several areas of pure and applied mathematics, including dynamical systems, partial differential equations, algebraic and differential geometry, topology, control theory, game theory, operations research and mathematical modeling. The group runs a very successful weekly research seminar that typically involves faculty and graduate students from mathematics, the School of Biological Sciences, and the School of Natural Resources.
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.
Carina Curto works in mathematics applied to and arising from theoretical and computational neuroscience. Research problems are in neural coding, neural networks, modeling and analysis of electrophysiological data. The mathematical tools come from a variety of areas in algebra, analysis and topology. Carina is one of the co-heads of the Mathematical Neuroscience Lab.
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.
Chad Giusti is interested in applications of recent advances in pure mathematics to problems in neuroscience. In particular, he is interested in how the topological properties and symmetries of a data set inform and restrict the structure of a network which encodes it. He also maintains an active interest in pure mathematics, with ongoing work in group cohomology and topology of embedding spaces.
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 reaction-diffusion equation with nonlocal diffusion which models gene propagation 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.
Vladimir Itskov works at the interface of mathematics and neuroscience (i.e., understanding of how the brain works). On the neuroscience side, his research is concerned with neural networks and representations of stimuli in the brain. The mathematical methods used include dynamical systems, as well as techniques coming from algebra and algebraic topology. Vladimir is one of the co-heads of the Mathematical Neuroscience Lab.
Yu Jin has research interest in applied mathematics with the main focus on dynamical systems and mathematical biology. Her research work is the conjoining of nonlinear dynamics and biology. This includes the establishment of appropriate mathematical models (mainly ordinary/partial/functional differential equations� and difference equations) for phenomena in spatial ecology, population dynamics, and epidemiology, as well as mathematical and computational analysis for models. Her current research is mainly focused on spatial population dynamics, especially on population spread and persistence in streams or rivers.
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.
Richard Rebarber has research interests in Mathematical Ecology and in Distributed Parameter Control Theory. His research in Ecology is in population dynamics, including: the effect of parameter uncertainty (such as modeling error) and perturbations (such as global warming) on long-term and transient population growth; the application of robust control theory methods to population analysis and management; stability properties of nonlinear models; and analysis of models with stochasticity.
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.
Drew Tyre is a colleague in the School of Natural Resources. His work focuses on using statistical and mechanistic models of single species population dynamics to help managers make better informed decisions about both game and non-game species. He is interested in applying robust control methods to structured population models, optimization methods to conservation and utilization decisions, and Bayesian Hierarchical models to survey and harvest data.
Alan Veliz-Cuba is interested in using and developing algebraic tools (algebraic geometry, Boolean algebra, combinatorics, graph theory, symbolic computation) to study problems arising in systems biology; he is also interested in the relationship between discrete and continuous dynamical systems. In particular, he is interested in modeling biological systems such as gene regulatory networks and studying how the network topology constrains the dynamics.
Graduate Students
Eric Eager (PhD 2012) Richard Rebarber
Brittney HindsBo Deng
Ben NoltingDavid Logan
Caitlyn ParmeleeCarina Curto
Sara ReynoldsChad Brassil and Glenn Ledder
Nora YoungsCarina Curto and Judy Walker