Preface

What are mathematical models? How are they developed? How much should we trust them: can we be confident in their predictions, and do we know when to act upon them? These questions, routinely addressed by mathematical modeling practitioners, are also of interest to citizens and policy makers. Our modern society needs models that can be relied upon, not only to improve our understanding of a situation, but also to inform policy decisions.

This introduction to mathematical modeling was developed for an audience of college seniors pursuing an undergraduate degree in mathematics with emphasis in applied mathematics, the life sciences, or engineering. The course builds on knowledge of calculus, linear algebra, and differential equations to address the basic techniques and thought processes that are fundamental to mathematical modeling. The style is deliberately casual and the main goal is to explain how mathematics learned in core undergraduate classes may be used to understand simple phenomena that arise in physics and biology, and how the corresponding models are put together, tested, and analyzed.

The text covers all of the standard systems that are normally considered in a modeling class: the nonlinear pendulum, chaotic maps, predator-prey models, competing species, chemical reactions, and, towards the end, diffusion and spatially extended systems. None of these are complicated topics and one could argue that such models are too simple to be useful. They however form the building blocks of mathematical modeling and, in spite of their simplicity, provide the tools to tackle more elaborate and realistic models. Emphasis is placed on developing practice with simple but general methods, such as dimensional analysis, phase plane analysis, basic fixed point theory, and numerical explorations; whenever possible, connections between different systems are built by exploring similarities in the mathematical models that describe them. Although some sections involve randomness, most of the text is concerned with deterministic models based on difference or differential equations. This is a deliberate choice, in order to allow coverage of the material in a one semester course. Finally, because modelers need to be good communicators of science and should understand potential uses and abuses of mathematical models, the first chapter of the text discusses such issues, in the context of a few examples.

Many excellent texts on dynamical systems are available in the literature, some of which motivate the study of nonlinear systems through mathematical models. One may thus question the usefulness of a separate course on mathematical modeling. The point of view presented here is that mathematical modeling is the art of using one’s mathematical knowledge to describe the world in mathematical terms. This requires good reasoning skills and a solid understanding of mathematical methods, as well as a type of mathematical fluency that transcends expertise in differential equations, or in any other core mathematics subject. The purpose of this text is to develop these skills and associated mindset through the practice of mathematical modeling in the context of simple, carefully chosen examples. Appendices are provided to review the basic mathematical tools needed to build and analyze the models. MATLAB GUIs are supplied with the course materials, allowing readers to explore the role of model parameters through graphical user interfaces, without requiring knowledge of numerical methods or of a particular numerical software package.

This book will give readers the background necessary to follow general scientific research articles that use mathematical modeling, such as those found in Science, Nature, and PNAS, to name a few. Successive versions of these notes have been used since 2005 as the main text for a one-semester capstone course at the University of Arizona. Students who take the mathematical modeling course also work in teams on a semester-long project, under the supervision of graduate or post-graduate mentors. Each project is based on understanding and reproducing the results of a research article. Teams write midterm and final reports on their projects and present their work in a poster session held in a public venue at the end of the semester. For the online version of the course, posters are replaced with group video presentations, judged by members of the university community. It is highly recommended that a similar model be used when teaching a class with this text. A list of recent projects and related articles is provided as an appendix.

I would like to thank all of my colleagues in the Department of Mathematics at the University of Arizona who, since 2007, have taught our mathematical modeling course with this text. I am also grateful to all of the graduate and postgraduate mentors, who for almost two decades have guided teams of undergraduates taking this course through their modeling projects. Finally, the initial development of these notes was made possible thanks to a University of Arizona TRIF (Technology and Research Initiative Fund) grant, which is acknowledged with great appreciation.

Joceline Lega
The University of Arizona
Fall 2012 & Summer 2024