# Nebraska REU Project Areas for Summer 2004

## Differential/Difference Equations

Prerequisite: 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.

## Elemental Dynamics in Biology

Prerequisites: well developed calculus techniques together with a thorough understanding of introductory differential equations and basic linear algebra. Some experience with computational software will be helpful. Students will learn and use Mathematica. Some background in ecology is preferred, but most important is a strong interest in ecology and mathematical modeling.

Such diverse issues as interactions among competing populations, effects of globally rising carbon dioxide on plants, or molecular mechanisms of protein synthesis in cells, can be studied by using techniques of ecological stoichiometry - that is modeling mass balance, ratios and flows of chemical elements such as carbon, nitrogen and phosphorus. These elements are fundamental to all life forms.

The REU students will be studying ecological problems and will construct mathematical models that reflect the essence of the underlying biological dynamics. Students will carry out analytical analyses of the models searching for stable equilibria, limit cycles, or chaos, and running numerical simulations using Mathematica. All these mathematical techniques will not be used for their own sake, but instead will be geared toward understanding the biology of the problem.