This is a very unusual book about mathematical modeling. It offers no broad overview of modeling principles and practice; the chapters are essentially independent of one another; there are whole pages without equations; and the author spends a lot of time telling stories. Nonetheless, this is a very good introduction to mathematical modeling, one of the best that I have seen.

Authors of texts on mathematical modeling sometimes have a tendency to get a bit over-formal about their subject. This can lead to extended and often rather boring chapters on the general principles of modeling. Modeling texts can also give the impression that their subject begins and ends with mathematics, which rather gets the whole thing upside down. The author of this book gets it right. It starts with the story and has to get back to the story if it’s to be of any value at all.

Take the first chapter of this book. It begins with a story of the life of hermit crabs and how they migrate from smaller shells to larger shells as they grow. Of course, the abandonment of the smaller shell gives smaller crabs an opportunity for new housing, and the shells the smaller crabs abandon are open to yet smaller crabs. If one considers a whole population of crabs at various stages of growth and search for housing, one might well start thinking about this process as transitions between states. It’s not too far from there to Markov chains.

But the discussion doesn’t end there. A study of hermit crab movements in a tidal pool off Long Island Sound provided data from an experiment where an empty shell was dropped into the water 500 times to initiate a chain of vacancies. Half of this data is used to estimate the frequency of transitions between states and the other half to get empirical estimates of quantities like average chain length that could be compared with theoretical results from the model. And even then we’re still not quite finished. The last part of the chapter takes the Markov chain model and applies it to investigate criminal behavior and recidivism.

As this develops, students see concepts such as transition matrices, absorbing states, and probability distributions arise more or less naturally from the context of the story. They also observe that the author pays close attention to data and model validation.

The author insists that this is not a textbook, and indeed it has no exercises. Instead he presents his book as a reference or a tool for self-study by mathematically prepared undergraduates who wish to see interesting and unusual applications in the biological and social sciences. He expects readers to have a background that includes multivariate calculus and some experience with matrices, probability and differential equations. Anything more advanced than this is explained in the text.

The topics in the book include a mix of older and newer applications. Even the older ones (such as modeling of epidemics or population fluctuations of fish) come with a story and usually at least one new twist. The only chapter that is driven by a mathematical topic is one on Bayesian analysis, and even that one builds upon four different stories.

This is a delightful book. By design it is not a textbook, but it would be a wonderful accompaniment to any course in modeling.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.