How many mathematics books can you think of that evoke images of Wile E. Coyote in pursuit of Road Runner? Chases and Escapes: The Mathematics of Pursuits and Evasion manages this with a certain élan. The author, a retired professor of electrical engineering, suggests that he wrote this book at least in part to answer the “What’s this stuff good for?” question about differential equations. Whatever the motivation, this is an entertaining and amusing look at chasing and getting caught or getting away.
It’s clear from the introduction that the author loves pursuit-and-evasion movies. (He is particularly fond of the 1941 film Man Hunt and The Naked Prey from 1961.) The pursuits described in his book are a bit more formalized than chases in the movies. The author looks at three general classes of pursuit problems: the classical pursuit (exemplified by a pirate ship chasing a merchant vessel), pursuit of maneuvering targets, and cyclic pursuit. The pirate ship pursuit problem, posed and solved by Pierre Bouguer in 1732, was one of the earliest mathematical investigations of chase and capture. Other classical intercept problems (e.g., a torpedo pursuing a submarine) also fall in this category.
Pursuit of maneuvering targets adds some new wrinkles. A typical problem is due to Hathaway, writing in the American Mathematical Monthly in 1920:
A dog at the center of a circular pond makes straight for a duck which is swimming counterclockwise along the edge of the pond. If the rate of the swimming of the dog is to the rate of the swimming of the duck as n:1, determine the equation of the curve of pursuit and the distance the dog swims to catch the duck.
It’s not especially easy to derive the differential equations governing this pursuit, and those equations don’t have a closed form solution once you do find them. The author presents MATLAB-generated numerical solutions for several values of n and displays pursuit curves for the various configurations. The collection of these trajectories hints at a valuable general conclusion about limiting trajectories and limit cycles.
The best known example of cyclic pursuit is the 4-bugs problem. Martin Gardner is responsible for popularizing this one. Here four bugs start on the vertices of a square and then, moving a constant speed, chase each other. What path do they follow, and how long does the pursuit last?
The author concludes with discussions of seven classic evasion problems, a couple of which use some elementary game theory and bit of probability. This is a highly readable book that offers several colorful applications of differential equations and good examples of non-trivial integrals for calculus students. It would be a good source of examples for the classroom and or a starting point for an independent project.
Bill Satzer (firstname.lastname@example.org) 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.