Introduction

1. Linear Models
2. Logistic Growth
3. Nonlinear Dynamical Systems
4. U. S. Population

Mathematical modeling is a broad area of Applied Mathematics that is interpreted in many different ways. My expertise is in Mathematical Biology and Dynamical Systems, particularly delay differential equations. This will clearly color my perspective of the subject. However, there are many general principles that follow in the modeling of any system, independent of the area of study or the mathematical technique employed.

A mathematical model is a representation of a real system. The essence of a good mathematical model is that it is simple in design and exhibits the basic properties of the real system that we are attempting to understand. The model should be testable against empirical data. The comparisons of the model to the real system should ideally lead to improved mathematical models. The model may suggest improved experiments to highlight a particular aspect of the problem, which in turn may improve the collection of data. Thus, modeling itself is an evolutionary process, which continues toward learning more about certain processes rather than finding an absolute reality. This use of mathematics is quite different from K-12 training in mathematics, where mathematics is treated as an absolute with exact answers.

This course will present a number of studies from a variety of areas, though the emphasis is likely to be more on Biological systems because of my expertise. Whenever possible, we want to closely examine actual data or a real physical system in order to better understand the process of creating an abstract mathematical model from observations. It is also very important to learn how to connect the model back to the physical system to provide a better understanding of what is happening, which means finding the actual parameters in the model. The course begins with studies of discrete dynamical systems, which in the simplest case is the classic principle and interest problem. This is followed by some relatively simple population models. Other modeling methods, including curve fitting, optimization, stochastic processes, and differential equations, will be applied to a variety of problems from several fields of science.

Before computers, scientists worked on clever methods to simplify the modeling problem to make it more tractable, usually using some approximations. You will be expected to perform numerical experiments in this class. I will be using Excel, Maple, and possibly MatLab though you may choose to use a number of other computational methods. Maple and MatLab are taught in Math 241 and are invaluable tools for any modern mathematician. I do have some additional notes for Maple, but I'll try to present most of the programming skills that you need as they are encountered in the class.