Nebraska REU Project Areas for Summer 2005

Control Theory Techniques Applied to Biological Population Problems

Advisors: Professors Richard Rebarber, Andrew Tyre, and Brigitte Tehnumbuerg

Prerequisites: Introductory course in matrix theory. Some introductory mathematical analysis and complex analysis would also be helpful, but are not strictly required. Familiarity with Matlab, Maple or Mathematica would also be helpful, but are not required.

Project Description: In this project we will apply techniques from the subject of mathematical control theory to problems in population dynamics. When analyzing how the population of an animal or a plant changes with time, the transition rates of survival, growth and reproduction can be incorporated into a population projection matrix. The eigenvalues of this matrix provide predictions for the behavior of the population. For instance, if the largest eigenvalue modulus, called the spectral radius, is less than 1, then the population decreases no matter what the initial conditions are. An important task in modern ecology is to predict what happens to the population dynamics when these transition rates are perturbed (that is, changed); the perturbations can be described by one or more parameters. This could happen due to uncertainty in the model, or these rates could be intentionally changed in order to get a desired result, such as maintaining an endangered species or eradicating a predator. Details

Differential/Difference Equations on Time Scales

Advisor: Professor Allan Peterson

Prerequisites: A first course in Differential Equations

We will be concerned with problems that come up in Differential Equations and Difference Equations. One of our interests will be to see how these two theories can be unified. For example, calculus is very useful in Differential Equations and when one studies Difference Equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general calculus. Consequently, we get a generalization of difference equations and differential equations. We study dynamic equations, where differential equations and difference equations are special cases. This leads to many interesting open questions which we will consider. Any student interested in Differential Equations should enjoy this experience. About a fourth of our time will be spent studying Difference Equations so knowledge of Difference Equations is not a prerequisite. Details.