The record of my research reflects my dual interests in mathematical research in applications of dynamical systems and in the development of projects and software to be used in undergraduate instruction of dynamical systems and differential equations. I view both of these facets of my professional agenda as equally important and there is a clear synergy between them.
A quick reading of my research publications might suggest that my research agenda is not very focused. Applications described in these papers include population genetics, neuroscience, and computer science. However, each of these projects uses the mathematical tools of dynamical systems to model and understand the underlying phenomenon. Each model is a system that varies over time (a dynamical system) and the objective in each of these papers was to determine the long-term behavior (or convergence) of the system. For example, the paper “Authority rankings from HITS, PageRank, and SALSA: existence, uniqueness, and effect of initialization” considers various methods for ordering the web pages found when doing a web search. We determine conditions under which each of the given algorithms will converge to a ranking that is independent of an initial seeding and is meaningful in the context of a web search. A second common question that is addressed in these papers is “how does the convergence change if parameters in the model are changed.” This is known as bifurcation theory. In the paper “Population Models of Genomic Imprinting II. Maternal and Fertility Selection” we use these techniques to show that in one particular model of genomic imprinting, a bifurcation occurs that causes the behavior of the system to bifurcate from a stable equilibrium solution (in this case a fixed allele frequency vector) to an oscillatory solution (where the allele frequencies will vary over time.)
The “Genomic Imprinting” paper illustrates another important facet of my research as well. It was done in collaboration with a former undergraduate student as part of his Senior Honors Thesis. Mathematical modeling provides a wonderful platform to engage students in research. The applied nature of a modeling problem is naturally appealing to many students. Moreover, it is often the case that developing a mathematical model and conducting a fairly thorough analysis of it is possible with the tools that a well-prepared mathematics major possesses. The publication of this work in a high quality research journal (Genetics) demonstrates the benefits of engaging undergraduate students in mathematical modeling research.
Two of my presentations focus on the evolution of cooperation. This work differs somewhat from my earlier research in that the system being modeled is a dynamical system with some intrinsic randomness (or stochasticity). The system models the interactions of two populations: the first will never cooperate while the second will cooperate with another individual until that individual takes advantage of them. The model equations combine game theory (in the form of the Prisoner’s Dilemma), population genetics (game payoffs are interpreted as fitnesses), and probability theory. I have proven that in the absence of randomness, the dynamics of a population that consists of mostly non-cooperators will, over time, tend to a solution that contains no cooperators. In other words, without randomness, cooperation cannot evolve. Simulations and basic analysis of the stochastic model suggest that there are certain conditions on the game payoffs that make the evolution of cooperation more likely.
Throughout my career I have been extremely active in developing, implementing, and assessing projects and software for use in differential equations and dynamical systems courses. This began at Washington State University with the award of two National Science Foundation grants for the project “IDEA: Internet Differential Equations Activities” (http://www.sci.wsu.edu/idea/). Although the NSF funding for this project expired years ago, my collaborator Kevin Cooper and I continue to monitor and update the site as technology advances and new ideas and projects arise. More recently, Kevin and I were asked to author nine projects that are included in the differential equations text book Differential Equations with Boundary-Value Problems 6th ed. by D.G. Zill and M.R. Cullin and published by Brooks/Cole.
I am currently involved in the NSF funded CODEE project. CODEE is a consortium of colleges and universities that are collaborating on making projects and tools such as the ones I have developed easily available to a wide audience of teachers. The primary investigator of this project is Darryl Yong of Harvey Mudd College. I have been asked to write a brief paper describing strategies for assigning, motivating, and grading modeling projects in differential equations courses. I will also be involved in training others in using the CODEE resources at upcoming national and regional math conferences.
My work in the pedagogy of differential equations education has been extensive and continues to be an integral facet of my career. I helped develop one of the first web sites that provided instructors with both differential equations projects and the software to help understand the dynamics of these models. I participated in the development of an award-winning piece of software (ODE Architect) that was one of the first to bring multimedia tools to an educational audience.