Tung K. K. - Topics in Mathematical Modeling

TOPICS IN MATHEMATICAL MODELING

TOPICS IN MATHEMATICAL MODELING

K. K. TUNG

PRINCETON UNIVERSITY PRESS

Princeton and Oxford

Copyright 2007 by Ka-Kit Tung

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Library of Congress Control Number: 2006939425

ISBN-13: 978-0-691-11642-6 (cloth)
ISBN-10: 0-691-11642-3 (cloth)

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10 9 8 7 6 5 4 3 2 1

To my parents,
David and Lily Tung

Preface

This book is based on my lecture notes for a college sophomore/junior-level course on mathematical modeling. I designed this course with partial funding from a VIGRE grant by the National Science Foundation to the University of Washington and further developed it into a book when I taught it six times to undergraduate students in our Applied and Computational Mathematical Sciences program, for which this course was a required overview of applied mathematics. The style of the book is that of an organized collection of modeling examples with a mathematical theme. The textbook is aimed at students who have had calculus and some ordinary differential equations. Students we surveyed liked seeing how mathematics is applied to real-life situations and were surprised that the simple mathematics they learned could already be useful. However, they overwhelmingly expressed the opinion that they were not interested in the traditional modeling topics such as pendulums and springs, or how long it takes to cook a turkey, a sentiment that I partly share. I also agree that a deductive subject such as classical mechanics probably should not belong to a course on modeling. To me an important process of modeling is to distill from empirical data a conceptual model and to quantify the conceptual model with a set of governing mathematical equations. To model classical mechanics, we probably have to pretend that we are back in the 16th or 17th century. If, however, the topics are broadened from physical sciences to other emerging areas, interesting applied mathematical problems exist that (a) can be captured and solved using simple mathematics and (b) are current research problems whose solutions have important societal impacts. The difficulty is finding such modeling problems and presenting them in a coherent context, while at the same time retaining some historical perspective. In this endeavor I have benefited enormously from my colleagues, who pointed me to interesting research papers in various sources in fields far from my own. I especially would like to thank Professor James D. Murray, FRS, whose encyclopedic knowledge of historical cases and recent developments in mathematical biology is a valuable source of information. So is his book Mathematical Biology. Professor Mark Kot taught a similar course, and his lecture notes helped my teaching greatly. Had his spouse agreed to let him write another book! he would probably have been my coauthor on this project. His recent monograph, Elements of Mathematical Ecology, provides more interesting examples. My former colleague and chairman, Professor Frederic Wan, started the tradition of teaching a modeling course in our department. I used his textbook, Mathematical Models and Their Analysis, for a graduate-level course with the same title. That book has also been used by others at the undergraduate level. At the start of my academic career as an assistant professor, I cotaught a modeling course with University Professor C. C. Lin to Honors sophomores at MIT, using the book by C. C. Lin and L. A. Segal, Mathematics Applied to Deterministic Problems in the Natural Sciences, which, in my opinion, was ahead of its time when it was published in 1974. I benefited greatly from all of my colleagues and teachers writings and their philosophies. To them I am deeply indebted.

A wonderfully refreshing new book, Modeling Differential Equations in Biology , by C. H. Taubes of Harvard, provides a compendium of current research articles in the biology literature and a brief commentary on each. I have found it useful, in teaching our course, to supplement my lecture notes with reading material from Taubess book when the topics covered are related to biology. On more traditional topics, the 1983 text by Braun, Coleman, and Drew, Differential Equation Models , is still a classic. Richard Habermans Mathematical Models: Mechanical Vibrations, Population Dynamics and Traffic Flow became a Classic in the SIAM Classics series. Recently I was delighted to come across a new book, Mathematical Modeling with Case Studies, A Differential Equation Approach Using Maple, by B. Barnes and G. Fulford. It contains many more nice examples on interacting populations than I have included in the present book.

A comment on pedagogyalthough each chapter is independent and can be taught out of order, there is a progression of mathematical themes: from linear to nonlinear problems, from difference to differential equations, from single to coupled equations, and then to partial differential equations. Some chapters are not essential to this progression of mathematical techniques. They can be skipped by an instructor if the focus of the course is more on the mathematical tools. These chapters, however, present interesting phenomena to be modeled and are actually my favorites. Calculus is a prerequisite for this course. If students taking this course are unfamiliar with ordinary differential equations, an instructor may need to spend more time going over ) can be skipped. Students in my class are required to write a term paper to be turned in at the end of the term on a topic of their choice. The paper can be on modeling a phenomenon or a critical review of existing work in the literature. Since this is likely to be the first time a student is writing a scientific report, it is helpful if the instructor gives one lecture on what is expected of them.

This book is based on my lecture notes for a course dealing with continuous modeling. Although I have tried to add some discrete modeling topics, I probably have not done justice to the diversity of methods and phenomena in that area. As it stands, the book contains more than enough material for a one-semester course on mathematical modeling and gives some introduction to both discrete and continuous topics. The book is written in such a way that there is little cross-referencing of topics across different chapters. Each chapter is independent. An instructor can pick and choose the topics of interest for his/her class to suit the preparation of the students and to fit into a quarter or semester schedule.

I am deeply indebted to Frances Chen, who patiently and expertly typed countless versions of the lecture notes that eventually became this textbook. I am grateful to the students in my modeling class, to the instructors who have taught this class using my lecture notes for correcting my mistakes and typos and giving me feedback, and to William Dickenson for drafting some of the figures. Last but not least are the friendly and professional people I would like to thank at Princeton University Press: senior editor Vickie Kearn, who first contacted me in December 2001 about writing this book and stood by me for the ensuing five years, for her great patience, enthusiasm, and constant encouragement, and her assistant, Sarah Pachner, for her valuable help in obtaining copyright permissions on the figures; and Linny Schenck, my production editor, and Beth Gallagher, my copy editor, for their great work and flexibility. Thank you!