Gregory J. Davis
Office: LS 416
Davis, Greogory, Professor, Natural and Applied Sciences (Mathematics).
M.A., Ph.D. Northwestern University
B.S. UW-Green Bay
Fields of interest: smooth, discrete, and chaotic dynamical systems; mathematical modeling of biological and physical systems.
- Ph.D. in Mathematics, 1987. Northwestern University, Evanston, Illinois.
- Discipline: Dynamical Systems
Advisor: Clark Robinson
Thesis Title: Homoclinic Tangencies and Infinitely Many Sinks
M.A. in Mathematics, 1985. Northwestern University, Evanston, Illinois.
B.S. in Science and Environmental Change (concentrations in Mathematics and Physics), 1981. University of Wisconsin-Green Bay.
University of Wisconsin-Green Bay, Dept. of Natural and Applied Sciences.
Chair, Natural and Applied Science (present)
Chair, Mathematics Program (1993-1998)
Associate Professor (1992)
Assistant Professor (1987-1992)
- STATEMENT OF TEACHING
- I teach mathematics at various levels and for different audiences; in all of my courses my teaching philosophy is the same - quality teaching based upon mutual respect between students and instructor. Because of this philosophy, I have very high standards, not just for the students, but for myself as well. To me, proficiency for students in a course means that the students not only have a good handle on the computational aspects of the material but also have the ability to apply their knowledge. Applications may be very concrete, such as doing a mixture problem in intermediate algebra or a related rates exercise in calculus; or the applications may be more abstract, such as deducing results particular to a certain ring or group (based on general theorems and definitions) in algebraic structures.
As an instructor, being proficient requires that I present material to my classes in an orderly and mathematically correct manner at a level which is comprehensive to my students. This does not mean that the course material should, in any way or form, be watered down. When students are properly motivated, mathematical concepts can be quite transparent. I try to illustrate all of the concepts that we study through examples from which I attempt to bring out the commonality and help my students conjecture what might be true in general. In this way, I help students increase their ability to think mathematically and instill in them a degree of mathematical insight. I believe in the use of technology in the classroom, such as graphing calculators and advanced mathematical software packages, in order to maximize my ability to convey information to my students. I do my best to inspire my students to continue their studies of mathematics and related subjects.
- COURSES TAUGHT
|Survey of Modern Mathematics
Foundations of Mathematics
Algebra and Trigonometry
Business and Economic Statistics
Calculus for Management/Social Sciences
Calculus and Analytic Geometry I, II
Ordinary Differential Equations
Environmental Modeling and Analysis
Linear Algebra I, II
Set Theory and Logic
Foundations of Geometry
Algebraic Structures (Abstract Algebra)
Systems of Ordinary Differential Equations
Mathematical Applications in Ecology
- GRADUATE PROGRAM
- Graduate Faculty member for Environmental Science Masters program in the areas of Metapopulation Dynamics and Mathematical Modeling.
- Theses supervised:
- Design and implementation of a decision support system for the Nicolet National Forest bird survey data. May 1998. Norine Dobiesz.
- Mathematical model to incorporate backyard composting into the city of Green Bay, Wisconsin waste management program. Dennis Rohr.
- Modeling a white-tailed deer population using an age-structured BIDE model. In progress. Bruce LaPlante.
- RESEARCH INTERESTS
- Presently, there are two mathematical areas in which I am engaged in active research: an investigation of the dynamics which are generated by homoclinic bifurcations in low dimensional maps; and an interdisciplinary investigation of metapopulation dynamics in population biology.
My work having to do with homoclinic bifurcations is a continuation of the research I began in my doctoral thesis. The problem of understanding homoclinic bifurcations is relatively new. Over the past twenty years, several major theories have been presented in conjunction with homoclinic bifurcations. Along with the possibility of strange attractors, some of the phenomena associated with homoclinic bifurcations include omega explosions, infinitely coexisting sinks, and antimonotonicity. I am interested in creating a theory which will integrate omega explosions, infinitely many coexisting sinks, and antimonotonicity; in doing so it is necessary to understand how Cantor sets intersect in general. Over the past few years, I have explored how middle third Cantor sets can intersect. My fellow mathematician, Tian-You Hu, and I are in the process of extending these results to a more general setting where we hope to describe how any two arbitrarly chosen Cantor sets intersect as they are translated across each other.
Robert Howe (an ecologist) and I have been collaborating since 1988 in a study of mathematical BIDE (birth, immigration, death, and emigration) metapopulation models. Our research explores the dynamics of subdivided populations which are connected via dispersal of individuals; such a group of populations, when taken as a unit, is a metapopulation. Mathematical analysis together with computer simulations have been used to describe the metapopulation dynamics. Our present goal is to modify our existing model in ways that it will be more realistic. Applications of our work include endangered species conservation, design of refuges and preserves, exotic species control, and habitat management in multiple use landscapes. We have been able to involve a number of masters level students in projects associated with this line of research.
- Davis, J.H. and G.J. Davis, 1998. Determining Chi-Square Cell Significance Using Adjusted Residuals: A Data Analysis Technique for Ecologists, submitted to American Midland Naturalist.
- Davis, G.J., 1998. A Graphical Analysis of Function Composition, submitted to Teaching Mathematics and its Applications.
- Davis, G.J., 1998. Intersections of Middle Alpha Cantor Sets, in press Missouri Journal of Mathematical Sciences.
- Davis, G.J. and T. Hu, 1995. On the structure of the intersection of two middle third Canter sets. Publicacions Matematiques 39:43-60.
- Wolf, A.T., R.W. Howe and G.J. Davis, 1994. Detectability of forest birds from stationary points in northern Wisconsin. in Monitoring bird populations by point counts; Ralph, C.J., J. Sauer and S. Droege, Editors. US Forest Service.
- Davis, G.J. and R.W. Howe, 1992. Juvenile dispersal, limited breeding sites, and metapopulation dynamics. Theoretical Population Biology 42(1): 184-207.
- Howe, R.W., G.J. Davis and V. Mosca, 1991. The demographic significance of 'sink' populations. Biological Conservation 57: 239-255.
- Davis, G.J., 1991. Infinitely many co-existing sinks from degenerate homoclinic tangencies. AMS Transactions 323(2): 727-748.
- Davis, G.J., 1990. Hyperbolicity in a class of one-dimensional maps. Publicacions Matematiques 34: 93-105.
- PAPERS IN PROGRESS
- Davis, G.J. and R.W. Howe, The dynamics of declining BIDE metapopulations.
- Davis, G.J. and J.H. Davis, Induced age structure in a BIDE metapopulation model.
- Davis, G.J. and J.H. Davis, Nestling mortality in a Cliff Swallow colony due to House Sparrows.
- Davis, G.J. Antimonotonicity and degenerate homoclinic tangencies.
- Davis, G.J. Hopf Bifurcation in a three-level food chain.
- The Dynamics of Declining Metapopulations. June 1997. Conference on Age-Structured Populations and Dynamical Systems, North Carolina State University, Raleigh, North Carolina.
- Intersections of Middle Alpha Cantor Sets. March 1996. Northeastern Wisconsin Mathematical Seminar Series, St. Norbert College, DePere, Wisconsin.
- A Graphical Approach to Function Composition. April 1995. MAA-Wisconsin Section Annual Meeting, University of Wisconsin-Green Bay.
- Chaos on a Donut. March 1995. Northeastern Wisconsin Mathematics Seminar Series, St. Norbert College, Depere, Wisconsin.
- Neotropical migrant birds in managed forest landscapes: sources, sinks, and complex population dynamics. September 1994, Presenter: R.W. Howe. Wildlife Society First Annual Conference. Published Abstract: Howe, R.W. and G.J. Davis.
- Intersecting Middle Third Cantor Sets. April 1993. MAA-Wisconsin Section Annual Meeting, University of Wisconsin-Fox Center, Menasha, Wisconsin.
- Homoclinic tangles and some implications in computing. October 1992. Natural and Applied Sciences Seminar Series, University of Wisconsin-Green Bay.
- Metapopulation Dynamics. April 1991. MAA-Wisconsin Section Annual Meeting, University of Wisconsin-Oshkosh, Oshkosh, Wisconsin.
- Dynamics of Simple Maps. October 1990. Chancellor's Colloquium on Research, University of Wisconsin-Green Bay.
- Juvenile dispersal, limited breeding sites, and metapopulation dynamics in a class of BIDE models. July 1990. SIAM Annual Meeting, Chicago, Illinois.
- Is Henon's strange attractor really strange? July 1988. SIAM Annual Meeting, Minneapolis, Minnesota.
- UNIVERSITY SERVICE
- Chair, Research Council (1994-1997); Member (1992-1998)
- Member, Cofrin Arboretum Committee (1995-present)
- Member, Faculty Senate (1995, 1992-1993, 1990-1991)
- Member, Affirmative Action Subcommittee (1991, 1992, 1994)
- Chair, Student Conduct Policy Committee (1993-1994)
- University representative for MAA, Wisconsin Section (1994-present)
- Member, NAS Curriculum Development Committee (1993-1994)
- Vice-Chancellor's Assessment Committee for Student Academics (1990-1991)
- Organizer, Putnam Exam (1987-1990)
- Involvement with local high school Academic Competition (1988-present