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A Course in Mathematical Modeling

Douglas Mooney and Randall Swift
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resource Materials
[Reviewed by
Jan Holly
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Anyone who writes a new mathematical modeling book like this should be praised for providing us with fresh real-life data sets and ideas for student projects. This book illustrates statistical concepts, for example, with data about The X-files television show. It also introduces many new examples from population biology and ecology. In addition, the book contains familiar examples (e.g. drug levels in the body) as well as new problems inspired by others' such as Davis, Porta, and Uhl's (Calculus&Mathematica, Addison-Wesley, 1994) attention-grabbing problem on blood alcohol levels.

This book's broad goal is to teach students who have taken single-variable calculus how to build and analyze mathematical models. Computer simulations play a major role, but analytical techniques are also emphasized. The book is written for a one-semester course (but has more material than that), and is organized by two main pairs of themes: discrete-continuous and deterministic-stochastic. Chapters 1 and 2 are, respectively, "Discrete Dynamical Systems" and "Discrete Stochasticity" (including a discussion of basic statistics). Chapter 3 talks about state diagrams and Markov chains, and includes, in effect, a compressed course on matrix algebra. Chapter 4 covers curve-fitting, even going so far as to introduce multiple regression. Chapter 5, "Continuous Models," is basically a model-oriented mini-course on differential equations and dynamical systems. Chapter 6, the final chapter, brings together the continuous and stochastic themes, discussing queueing theory and birth/death processes.

This book's approach differs from that of many traditional modeling textbooks. For one thing, in keeping with the increasing availability of software, the present book relies much more heavily than do some on students' use of computers; while the book is not software-specific, it gives examples using software such as Mathematica, Stella II, and a spreadsheet program (yes, the secret is out: those simple-seeming spreadsheet programs can be great for mathematical modeling). In addition, the book is balanced between discrete and continuous models, unlike, say, the popular textbook by Giordano and Weir ( A First Course in Mathematical Modeling, Brooks/Cole, 1985) which is also accessible to students with one year of calculus but is focused on continuous models. The present book also takes more of its examples from population biology and ecology than do most; anyone interested in focusing on biology, though, might want to consider the excellent book by Edelstein-Keshet ( Mathematical Models in Biology, McGraw-Hill, 1988) which covers both discrete and continuous models. Finally, the present book is less compartmentalized than some. Chapters are nicely connected by reappearing adventures with, for example, M&Ms candies, alcohol, and The X-files. It would be difficult to jump straight into a chapter like "Continuous Models," though, because of the connections to earlier topics such as curve-fitting and matrix algebra.

To read the book is a bit like driving down a bumpy road. While the exposition is enjoyably lively and informal, the informality sometimes leads to imprecision. On some topics, students might learn just enough to be dangerous, so should be guided by a knowledgeable teacher who intervenes with, for example, "This is the general solution only for the case of distinct eigenvalues," a fact mentioned but easily overlooked in the text. In addition, the required maturity level of a reader seems to lurch between that of the intended post-calculus student and a rather higher level. Although each chapter ends with exercises and projects, the teacher may wish to supply additional exercises to more thoroughly cover the underlying ideas. On a more philosophical level, this book joins the crowd in perpetuating a myth: that the usefulness of a model lies only in its ability to fit the data and make correct predictions. Modeling textbooks, including this one, rarely emphasize the usefulness of models---and their mismatches with the data---in guiding experiments; rarely do they complete the real-life modeling cycle by asking, "According to this model and its mismatches with the data, what additional experiments or data would help us better understand the real-world system?"

Non-nitpickers may not mind, or even notice, the bumpiness, but they will mind the potholes. On one occasion (p. 291) the book applies a theorem about linear systems inappropriately to a nonlinear system. On another occasion, well, here it is (from p. 23, with the notation changed slightly for the web):


Theorem 1.2. (Conditions for stability) If x0 is a fixed point of the first-order recurrence equation x(n) = f(x(n-1)), then x0 is a stable fixed point if and only if |f'(x0)| < 1.


This statement is not completely true. It is possible to have f'(x0) = 1 at a stable fixed point x0, such as at the stable fixed point x0 = 0 for f(x) = x - x3. In general, f'(x0) = 1 implies neither stability nor instability. What is even more surprising about the book is that it contains a (supposed) proof of "Theorem" 1.2. (The reader's heart races in anticipation of a major mathematical breakthrough, a proof of the impossible... Oh, darn, there's an error, an incorrect deduction from a linear approximation.) Finally, anyone desiring additional entertainment on the journey through the book can play a travel game: Find the Problem with the Graph. In most cases the problem is simply an absent axis label or an unusual arrangement of tick marks, but once in a while there is a more serious problem: a misplaced point or misdirected curve.

Should you buy the book? Buy it if you want ideas for student projects, or applications for classes in statistics, linear algebra, or differential equations. Consider it if you like the selection of topics and want to use its approach in your class. Take your driver's license. And check out the Web site. For this book, the last word is not on the last page.

Jan Holly ( is Clare Boothe Luce Assistant Professor of Mathematics at Colby College, where she specializes in applied mathematics. She has done research at Los Alamos National Laboratory, the Robert S. Dow Neurological Sciences Institute in Portland, Oregon, and the Center for Computational Biology at Montana State University.