Introduction

1. Competition models: Yeast data and Malthusian growth
2. Competition models: Logistic growth
3. Competition models: Two species
4. Lotka Volterra models
5. Spring-Pendulum model

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 will begin with a problem that arises in yeast cultures used in brewing beer. This study will be extended to a classical interpretation of the interaction of predators and prey with interpretations that are important in today's agricultural community, which is so dependent on the petro-chemical industry. We will also study a simple pendulum with a spring and possibly a more complex pendulum consisting of two pendulums connected together. I may obtain information to have you study how defects in the structure of airplanes can be modeled, which will get into analysis of the heat equation and an associated inverse problem. At the end of the course, I plan to introduce some Monte Carlo or probabilistic methods.

The text is centered on continuous models using dynamical systems. Most of the course will be examining models that have their basis in differential equations or more general dynamical systems. Dynamical systems began with the advent of Calculus, which was formulated by Leibniz and Newton in the late1600s. Until recently, the emphasis has been on finding exact solutions to the differential equations which govern the dynamics of some modeling problem. However, early on, scientists discovered that analytical methods failed with problems as simple as the three body problem of gravitational attraction from astronomy. 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 MatLab though you may choose to use a number of other computational methods.

In the late 1800s, Poincaré developed geometric methods to analyze mathematical problems. His perspective introduced the concept of qualitative analysis rather than quantitative analysis. Qualitative analysis looks into the behavior of dynamical systems and uses geometric techniques to predict what are the possible outcomes for certain nonlinear problems. Bifurcation theory developed to shown how behavior changed as certain parameters in the system changed. One of the outcomes of this analysis was the theory of chaos, which has been discussed widely in recent years since the popular book by James Gleick in 1987. The explosive growth in studying dynamical systems has been assisted by the high speed computer, which allows easy simulation of complex nonlinear problems. You will see that understanding the role of the parameters in the underlying dynamical systems is very important to understanding the possible outcomes of a model, and thus, whether it is an appropriate model.

A mathematical model that consists of differential equations provides only one direction of the modeling diagram given above. It is very important to be able to validate that mathematical model, which gets to the inverse problems where we must fit actual data and find the parameters in the model. This is often much more difficult and requires a different collection of mathematical tools. This course will provide you with some of the methods for validating a mathematical model, but these techniques are not covered in the text, so will be presented on this website.

One study presented in the text is the damped spring (which we will analyze). Below is a graphical representation of this dynamical system.

More information on the damped oscillator and its solution in Maple with a movie of the solution curves rotating can be found by clicking on the picture. This movie is part of the advertisement for my Math 241 course on Maple.