Table of Contents
for student research projects in computational
mathematics and science may be available through the William & Mary
program and other sources. Interested students should contact
myself or one of the
other CSUMS faculty members for more information. Here are just a
few of the possible project topics: CSUMS
Students past and present
Project Title: Global Dynamics of Pulse-Coupled Oscillators
Project Abstract: Networks of pulse-coupled oscillators can be used to model systems from firing neurons to blinking fireflies. Many past studies have focused on numerical simulations and locating the synchronous state of such systems. In this project, we construct a Poincare map for a system of three pulse-coupled oscillators and use rigorous computational techniques and topological tools to study asynchronous dynamics.
Basins of Attraction in Stage Structured Populations (honors paper) electronic appendix coming soon
Abstract: The interaction between invasive and native species can be modeled through a Lefkovitch model using stage structured populations. In this study, we analyze population dynamics of a stage structured population, computing a basin of attraction around the non-trivial attracting equilibrium.This model can be naturally extended to include two or more species in which inter- and intra-specific competition is expressed through a density dependent fertility term. Preliminary results for the two species model will be discussed.
Related publication: Coexistence of competing stage-structured populations, Masami Fujiwara, Georgia Pfeiffer, May Boggess, Sarah Day, and Jay Walton. Scientific Reports 1, doi:10.1038/srep00107 (2011) (electronic) (final version).
Funding: CSUMS, NSF grant: Dynamics at a fixed resolution (Summer 2011)
Awards: Outstanding Poster Award, Joint Meetings 2011, Extinction Equilibria in Stage-Structured Populations
Following graduation: graduate student in Agricultural and Resource Economics, University of Arizona
Project Title: Studying Dynamics at a Fixed Resolution
Project Abstract: In studying discrete time dynamical systems, computer simulations are used to simulate behavior of a particular map. Although to make rigorous conclusions about the behavior of a map an infinite number of iterations must be used, a computer can only produce a simulation with a finite number of iterations. Tools used to compute the Conley Index of a map allow conclusions to be made about the true behavior of a map from a finite simulation of its true behavior. These tools involve using a grid-like, partition structure, called an outer enclosure, which reflects the behavior of the underlying map. This outer enclosure can be made to reflect more of the complicated behavior of a map the smaller the grid boxes are made. The Fixed Resolution problem is, given a certain outer enclosure of a simulation of a map, what is the most amount of information, that is, how much complicate behavior can be detected? In order to solve this problem, we have decided to focus on conclusions that can be drawn about isolating neighborhoods, which are the smallest components of an outer enclosure from which conclusions can be drawn using Conley index theory. In this effort, we have written conjectures about isolating neighborhoods, and are developing code to identify isolating neighborhoods using strongly connected components.
Topological Characterization of Extinction in a Coupled Ricker Patch Model (honors paper) electronic appendix coming soon
Project Abstract: Ecologists use Ricker patch models to study meta-population dynamics for popula- tions undergoing growth and dispersal in a patchy environment. This project uses a modified model in which patch-wise extinction thresholds are used to model local extinction events. Computational homology is used to measure shifts in spatial pat- terns as extinction occurs and to quantify the ways in which dispersal rates affect pattern formation and degradation. Numerical simulations for certain parameters exhibit a decoupling of the system into small regions with periodic dynamics prior to extinction. Lastly, the existence of certain stable periodic orbits which affect population robustness are rigorously proven to exist.
Other: Research description for the SciClone cluster project.
Funding: CSUMS, NSF grant: Dynamics at a fixed resolution (Summer 2010, Summer 2011), UBM
Following graduation: graduate student in Applied Mathematics, University of Arizona
Box-counting dimension and beyond (honors paper) electronic appendix
Abstract: In dynamics, computing the fractal dimension of strange attractors and fractal-like sets that may occur in the study of a dynamical system can give us a way of measuring these sets and sometimes can also have a physical interpretation. In general, computing fractal dimension gives us a scaling factor of the set and a way of telling how much the set fills up space. Since we cannot describe fractals and fractal-like sets using typical geometrical methods, fractal dimension gives us one way of measuring, understanding and studying the geometry of the sets. Box-counting dimension is a one way of measuring fractal dimension. In this project, we explored several properties and characteristics associated with box-counting dimension including: the change in the dimension of the attractor as the parameters of a system change, the dimension of subsets of the attractor and their relationship to the dimension of the whole attractor, the relationship between fractal dimension and Lyapunov exponents, and the dimension of fractals lying in infinite-dimensional space. These topics include many possible open questions for future study.
Funding: CSUMS, NSF grant: Dynamics at a fixed resolution (Summer 2009)
Following graduation: graduate student in Mathematics at Dartmouth
Modeling the effects of harvesting on Virginia's Black Bear population (honors paper) electronic appendix
Abstract: In 1999 the Virginia Department of Game and Inland fisheries developed a long term plan to manage the black bear population in Virginia; in 2001, the VDGIF published a 10 year management plan. Though the plan contained many ideas to manage the bear population - including fertility control, kill permits, regulated hunting, etc., the management proposal lacks any concrete insight as to the ramifications of these options.
The model included in this paper aims to analyze the population dynamics of the black bear population in Virginia by using a non-linear discrete model which separates bears not only by age, but also by gender, and territory. Parameters include survival and harvest (split by age, gender, and territorial status), percentage of breeding females, birthrates, and number of home ranges available. The harvest and survival data are valid only for the state of Virginia, and reflect information from 2002. Thus, the simulations can only give us information concerning the bear population in Virginia, and only general comments can be made concerning the entire black bear population along the Southern Appalachians. The VDGIF states that bear harvest has increased by 7.4% per year over the last decade. This information is taken into account in the model, and the simulations provide further insight as to the black bear population in Virginia.
Funding: CSUMS, UBM
Following graduation: graduate student in the BioMaPs Institute at Rutgers University
Investigating computational aspects of the Coincidence Condition for substitutions of Pisot type (honors paper)
Abstract: Symbolic dynamics provides a discrete way of looking at some of the qualities and characteristics of general dynamical systems. Instead of continuously studying the behavior of a system given some set of rules, symbolic dynamics discretizes everything into a finite set of states, with rules providing for which states can follow which other states. There is also the notion of entropy for symbolic dynamical systems, which measures how chaotic or complicated the system is. While symbolic dynamics is commonly used to model and study dynamical systems, it provides a rich field of research on its own, which is the approach taken by this project.
Following graduation: graduate student in Mathematics at Temple University
Reproduction rate strategies in White-Footed Mice (honors paper)
Abstract: A photoperiod is the measure of the length of daylight each day. This value can potentially determine the behavior and/or biological processes of many species of animals and plants. Peromyscus leucopus (white-footed mouse) responds to changes in photoperiods by altering its reproductive strategies. P. leucopus adjusts reproduction rates due to the high cost of reproduction in the winter and in short photoperiods. In our research we have looked at two groups of mice: Responsive mice (R) which reproduce March through November and Non-responsive mice (NR) which reproduce all year around. Interestingly, in Williamsburg, VA there exists a mixture of the Responsive and Non-responsive mice. The coexistence of these two types of mice suggests some kind of genetic variation. We have created nonlinear discrete population models to better understand the requirements for the co-existence of the two varying phenotypes.
Funding: CSUMS, UBM
Following graduation: graduate student in Nutrition at the University of North Carolina, Chapel Hill
Algorithms for rigorous entropy bounds and symbolic dynamics (with S. Day and R. Trevino) SIAM Journal on Applied Dynamical Systems 7(4), 1477--1506 (2008) (electronic) (final version).
Algorithms for rigorous entropy bounds and symbolic dynamics (with S. Day and R. Frongillo) SIAM Journal on Applied Dynamical Systems 7(4), 1477--1506 (2008) (electronic) (final version).
REU project Computer-Assisted Proofs in Dynamical Systems
Recent developments in the field of rigorous numerical techniques for dynamical systems have allowed for the location and study of dynamical objects from fixed points to sets exhibiting chaotic dynamics. Many of these techniques are based on ideas from topology and analysis. In particular, numerical techniques inspired by a topological tool called Conley index theory have been used to uncover a variety of interesting dynamics for the chaotic Henon map and other discrete dynamical systems. In this project, we will explore extensions of these techniques on both the computational and theoretical levels. This project will include experimenting with specific dynamical systems using existing software and formulating interesting directions for study and extensions of the algorithms. Background: While not required, previous experience with dynamical systems, topology, and programming is helpful. The computational work will be done mainly in Matlab.
Participants and projects:
Rafael Frongillo (Cornell) and Rodrigo Trevino (UT Austin): We developed and expanded existing techniques for computing symbolic dynamics and topological entropy bounds. Our paper on this work Algorithms for rigorous entropy bounds and symbolic dynamics will appear in the SIAM Journal on Applied Dynamical Systems. Rafael and Rodrigo won a prize for a poster that they presented at the Joint Mathematical Meetings of the AMS/MAA in New Orleans in January 2007.
Chris Green (Cornell) and Philipp Meerkamp (Bremen): We chose to study an LPA model describing population dynamics of the flour beetle (genus Tribolium). Here is a project description written by Chris.
Adam Chacon (UT Austin): We studied and tested the limits of existing code and algorithms.