## Differential/Difference Equations

Advisor: Professor Allan Peterson

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

Advisor: Professor Irakli Loladze

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.

## Game Theory and Population Dynamics

Advisors: Professors Wendy Hines

, and Jamie RadcliffePrerequisites: Standard courses in matrix theory and differential equations.

Since John Maynard Smith's first introduction of the notion of an "evolutionary stable strategy" in 1973, there has been a productive synthesis of game theory and problems of population dynamics. Conflict between individuals in a population can be modeled as a game in which each individual has a preferred strategy that it follows over repeated plays of the game. The individuals' relative success at playing the game affects the makeup of the population in future generations. Sometimes there is a single winning strategy leading to a shared species characteristic, while other times the winning strategy involves a mixture of plays. The latter case leads one to consider dynamical systems based on payoff matrices of games.

In this project, the students will investigage the dynamics associated with a variety of non-zero sum games (the classic example being, of course, Prisoner's Dilemma). The students will need to develop realistic but tractable models of potential strategies for repeated play of these games and then investigage the mathematical and biological consequences.