## Differential/Difference Equations on Time Scales:

Professor Allan Peterson,

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

## Effects of Cannibalism on Ecosystems:

Professor Glenn Ledder,

Prerequisites: a thorough understanding of the material in an introductory course in differential equations and a basic course in linear algebra. (It would be helpful to have an elementary analysis course and some experience with computational software such as Maple, Matlab, or Mathematica).

Project Description: Simple differential equation models for two interacting species have been known since the 1920's. Recent advances in mathematical techniques and computer software have brought about the opportunity to study more complicated systems of interacting populations. Most of the research to date in this area provides more questions than answers. Many of these questions are fascinating and quite accessible using basic tools in dynamical systems and mathematical modeling.

We will begin by identifying an ecologically realistic scenario in which cannibalism appears to be present. One particularly interesting possibility is provided by bass, which under certain circumstances can subsist entirely on a diet of bass larvae, and whose biological characteristics are well known. There are also some kinds of snails in which the first ones to hatch enjoy rapid initial growth by eating an egg that has not yet hatched.

We'll spend our first week studying mathematical methods for population dynamics and doing a literature survey. After that, we will develop one or more mathematical models based on general biological principles that seem to be valid for our chosen scenario. Questions will arise as we attempt to characterize the behavior of the models. Ultimately, we will try to identify environmental and physiological reasons why cannibalism is viable for some species and not for others.

## The Spreading of Information and Social Interactions:

Professor Steve Dunbar,

Prerequisites: Introductory knowledge of probability and matrix algebra. Some knowledge of differential equations and difference equations would also be helpful, but is not strictly required. Familiarity with Matlab, Maple or Mathematica would also be helpful, but is not required.

Project Description: In this project we will apply techniques from the subject of Markov processes to model social dynamics and associated random models in populations. The emphasis will be on modeling, simulation, visualization and analysis of a stochastic process and quantities associated with the process. The mathematical analysis of the Markov processes will rely on "first-step analysis" and limiting arguments resulting in difference and differential equations. A population of individuals can randomly mix and mingle and exchange "information". Depending on how the members exchange "information", the result can model the transmission of a rumor, a social attitude, or an epidemic. A basic question is to deduce whether there are universal trends, such as whether there is always a fraction of the population which is untouched by the rumor, attitude or disease. Another basic question is to determine the distribution of important characteristics of the process, such as duration. Finally, even simple models contain rich information, and multiple views and statistics of the processes help to explain and enlighten.