## Time Scales

Advisors: Professors Allan Peterson and Lynn Erbe

,Prerequisite: a thorough understanding of the material in an introductory differential equations course.

The theory of time scales, which has recently received a great deal of attention because it unifies and extends calculus on discrete as well as continuous domains, is an ideal topic for undergraduate research. Many of the ideas of discrete and continuous calculus, and more generally dynamic equations, may be combined and unified via this more general structure. The notion of generalized exponential, logarithmic, trigonometric, and other special functions have already been given in an abstract setting. Consider the case of the exponential function. In the real case e^{k(t-a)} may be regarded as the solution of the initial value problem x' = kx, x(a) = 1. More generally, the exponential function e(t,k,a) may be regarded as the solution of the dynamic initial value problem x^{D} = kx, x(a) = 1, where x^{D} is the so-called delta derivative of x; x^{D} is a generalization of the standard derivative if the time scale is the real numbers, and is the forward difference operator if the time scale is the integers. If the time scale is the integers, then e(t,k,a) = (1+k)^{t-a} is this exponential function. For a certain time scale this exponential function is a binomial coefficient, for another time scale it is e(t,k,a) = t^{(ln(a)-ln(t))/2ln(a)}. There are many other interesting examples to be explored. These exponential functions may arise in the modeling of the growth of certain plant species (e.g., when the time scale is the union of disjoint intervals). Students with a good background in calculus and elementary analysis would be prepared to consider other special cases.

Additional topics for study include derivation of explicit forms of certain well-known differential or integral inequalities, and a study of certain properties of the so-called *Hilger complex plane*. In this context generalized trigonometric functions may be defined and their properties investigated, in analogy with the real case. The chain rule does not hold for a general time scale. It would be interesting to find conditions on the time scale that ensure that some kind of a chain rule holds.

## Chaos Theory in Food-Chain Models

Advisors: Professors Gwendolen Hines and Bo Deng

,Prerequisite: a thorough understanding of the material in an introductory course in differential equations. An elementary analysis course or considerable to modest experience in some computational software (such as Maple, Mathematica, or Matlab), is preferred.

Food-chains involving three or more species are fundamental building blocks for ecosystems. While a particular ecosystem may be intractable, simple chains can be modeled by systems of ordinary differential equations. Chaos theory has been found to be an appropriate paradigm within which food-chain models can be analyzed, thus shedding new light on ecocomplexity in general. We have designed an REU project in which the students will discover chaos theory by exploring food-chain models, thus gaining greater appreciation for its usefulness in biocomplexity studies. In this project students will learn how to handle one-dimensional iterative systems, with an emphasis on the sensitivity of the results to initial conditions.

The REU students will work on specific food-chain chaos generation mechanisms. These models have parameters that are variable. The type of chaos, and the way in which it arises, depends upon the relative values of these parameters. The students will determine which ways of varying the parameters will lead to chaos. One way of measuring chaos and sensitivity is to measure the *Lyapunov exponent* of the system. This cannot be done analytically without analytical forms for the solutions, which are usually unavailable, but it can be done numerically. From a literature search the students will learn existing methods (including ones with which the advisors are currently unfamiliar), and figure out how to adapt these to particular problems. One goal is for the students to write a program that works for a general class of systems.

Since many dynamical systems, including food-chain models, have many different time scales, the students will develop multi-time analysis skills on singularly perturbed systems of equations. These skills are mostly rudimentary and are geometric in nature. Geometric singular perturbation analysis on food-chain models will inevitably lead the students to discover that there are many different types of one-dimensional maps embedded in the models, and that these largely chaotic maps can be analyzed by techniques they have already mastered.

During the project, we will assign readings, including but not limited to the following publications: *Introduction to Dynamical Systems*, by Alligood, Sauer and Yorke, *Complex dynamics and phase synchronization in spatially extended ecological systems*, by Bluaius, Huppert and Stone, Yield and dynamics of tritrophic food chains by De Feo and Rinaldi, and *Slow-fast limit cycles in predator-prey models*, by Rinaldi and Muratori.

## Problems in Mathematical Modeling

Advisor: Professor Glenn Ledder,

Prerequisites: a thorough grounding in the material in a standard first course in differential equations, a basic course in linear algebra (sometimes called matrix theory), and college-level background in at least one of biology, chemistry, engineering, geology, or physics.

We will develop and study a mathematical model for some problem of current interest in natural science, physical science, or engineering. The specific topic for the mathematical modeling project will be made shortly after the students for the project have been selected. This will allow the students to participate in the topic selection process and to have about two months to do some background reading on the topic. Two broad categories of topics will be considered:

- Population models from ecology or epidemiology, and
- Models of processes involving diffusion, including such diverse areas as heat flow, fluid mechanics, flow in porous media, pharmacokinetics, and digestion processes in animals.

Example topics include:

Many aquatic parasites have a complicated life history that could include stages in which they live in snails as well as stages that live in fish and free-swimming stages. Typically, biologists have a qualitative description of the population dynamics, but no quantitative mathematical model. Based on the qualitative description and ecological data, we could build a mathematical model that should assist biologists in better understanding the population dynamics.

Dynamic energy budget models are used in biology to describe the evolution of physiological state variables such as biomass and stored energy for the life cycle of individual members of a population. It is possible to couple such models to ecological models that describe the interactions between populations in order to construct a general population model based on both physiological and ecological principles. In this project, the student will develop and study a mathematical model to predict variations in the production of grasshopper eggs under a range of environmental conditions including weather differences and differences in predator populations.

The National Weather Service has just implemented a new system for measuring the effect of wind speed on the level of chill experienced by people. The new system is based on a complicated model selected from a variety of proposed models. While it is undoubtedly more accurate than the old system, it is still subject to questions about how wind chill effects ought to be defined. It is likely that equally accurate results could be obtained from a relatively simple model, with the advantage that the simple model would make it easier to examine the consequence of the choice of definition of the wind chill effect.