Modules in Applied Mathematics: Volume 1 - Springer LINK

Modules in Applied Mathematics: Volume 1

Edited by William F. Lucas

Modules in Applied Mathematics

Volume I Differential Equation Models Martin Braun, Courtney S. Coleman, and Donald A. Drew, Editors

Volume 2 Political and Related Models Steven J. Brams, William F. Lucas, and Philip D. Straffin, Jr., Editors

Volume 3 Discrete and System Models William F. Lucas, Fred S. Roberts, and Robert M. Thrall, Editors

Volume 4 Life Science Models Helen Marcus-Roberts and Maynard Thompson, Editors

Differential Equation Models

Edited by Martin Braun, Courtney S. Coleman, and Donald A. Drew

With 166 Illustrations

Springer-Verlag New York Heidelberg Berlin

Martin Braun

Department of Mathematics Queens College Flushing, NY 11367 USA

Courtney S. Coleman

Department of Mathematics Harvey Mudd College Claremont, CA 91711 USA

AMS Subject Classifications: OOA69, 34AOI, 35A02

Library of Congress Cataloging in Publication Data

Modules in applied mathematics. Includes bibliographies.

Donald A. Drew

Department of Mathematical Sciences

Rensselaer Polytechnic Institute

Troy, NY 12181 USA

Contents: v. I. Differential equation models / edited by Martin Braun, Courtney S. Coleman, and Donald A. Drew. - v. 2. Political and related models / edited by Steven J. Brams, William F. Lucas, and Philip D. Straffin, Jr.

I. Mathematics-I 961- .2. Mathematical models. I. Lucas, William F., 1933- . QA37.2.M6 1982 510 82-10439

This book was prepared with the support of NSF grants Nos. SED77-07482, SED75-00713, and SED72-07370. However, any opinions, findings, conclusions, and/or recommendations herein are those of the authors and do not necessarily reflect the views of NSF.

© 1978 by the Mathematical Association of America; © 1983 by Springer-Verlag New York Inc. Softcover reprint of the hardcover I st edition 1983

All rights reserved. No part of this book may be translated or reproduced in any form without permission from Springer-Verlag New York, 175 Fifth Avenue, New York, New York 10010, USA.

Typeset by Asco Trade Typesetting Ltd., Hong Kong.

9 8 7 6 5 4 3 2 I

ISBN-13: 978-1-4612-5429-4 DOl: 10.1007/978-1-4612-5427-0

e-ISBN-13: 978-1-4612-5427-0

Preface

The purpose of this four volume series is to make available for college teachers and students samples of important and realistic applications of mathematics which can be covered in undergraduate programs. The goal is to provide illustrations of how modem mathematics is actually employed to solve relevant contemporary problems. Although these independent chapters were prepared primarily for teachers in the general mathematical sciences, they should prove valuable to students, teachers, and research scientists in many of the fields of application as well. Prerequisites for each chapter and suggestions for the teacher are provided. Several of these chapters have been tested in a variety of classroom settings, and all have undergone extensive peer review and revision. Illustrations and exercises are included in most chapters. Some units can be covered in one class, whereas others provide sufficient material for a few weeks of class time.

Volume 1 contains 23 chapters and deals with differential equations and, in the last four chapters, problems leading to partial differential equations. Applications are taken from medicine, biology, traffic systems and several other fields. The 14 chapters in Volume 2 are devoted mostly to problems arising in political science, but they also address questions appearing in sociology and ecology. Topics covered include voting systems, weighted voting, proportional representation, coalitional values, and committees. The 14 chapters in Volume 3 emphasize discrete mathematical methods such as those which arise in graph theory, combinatorics, and networks. These techniques are used to study problems in economics, traffic theory, operations research, decision theory, and other fields. Volume 4 has 12 chapters concerned with mathematical models in the life sciences. These include aspects of population growth and behavior, biomedicine (epidemics, genetics and bio-engineering), and ecology.

VI Preface

These four volumes are the result of two educational projects sponsored by The Mathematical Association of America (MAA) and supported in part by the National Science Foundation (NSF). The objective was to produce needed material for the undergraduate curriculum. The first project was undertaken by the MAA's Committee on the Undergraduate Program in Mathematics (CUPM). It was entitled Case Studies and Resource Materials for the Teaching of Applied Mathematics at the Advanced Undergraduate Level, and it received financial support from NSF grant SED72-07370 between September 1, 1972 and May 31, 1977. This project was completed under the direction of Donald Bushaw. Bushaw and William Lucas served as chairmen of CUPM during this effort, and George Pedrick was involved as the executive director of CUPM. The resulting report, which appeared in late 1976, was entitled Case Studies in Applied Mathematics, and it was edited by Maynard Thompson. It contained nine chapters by eleven authors, plus an introductory chapter and a report on classroom trials of the material.

The second project was initiated by the MAA's Committee on Institutes and Workshops (CIW). It was a summer workshop of four weeks duration entitled Modules in Applied Mathematics which was held at Cornell University in 1976. It was funded in part by NSF grant SED75-00713 and a small supplemental grant SED77-07482 between May 1,1975 and September 30, 1978. William F. Lucas served as chairman of CIW at the time of the workshop and as director of this project. This activity lead to the production of 60 educational modules by 37 authors.

These four volumes contain revised versions of 9 of the II chapters from the report Case Studies in Applied Mathematics, 52 of the 60 modules from the workshop Modules in Applied Mathematics, plus two contributions which were added later (Volume 2, Chapters 7 and 14), for a total of 63 chapters. A preliminary version of the chapter by Steven Brams (Volume 2, Chapter 3), entitled "One Man, N Votes," was written in connection with the 1976 MAA Workshop. The expanded version presented here was prepared in conjunction with the American Political Science Association's project Innovation in Instructional Materials which was supported by NSF grant SED77-18486 under the direction of Sheilah K. Mann. The unit was published originally as a monograph entitled Comparison Voting, and was distributed to teachers and students for classroom field tests. This chapter was copyrighted by the APSA in 1978 and has been reproduced here with its permission.

An ad hoc committee of the MAA consisting of Edwin Beckenbach, Leonard Gillman, William Lucas, David Roselle, and Alfred Willcox was responsible for supervising the arrangements for publication and some of the extensive efforts that were necessary to obtain NSF approval of publication in this format. The significant contribution of Dr. Willcox throughout should be noted. George Springer also intervened in a crucial way at one point. It should be stressed, however, that any opinions or recommendations

Preface Vll

are those of the particular authors, and do not necessarily reflect the views of NSF, MAA, the editors, or any others involved in these project activities.

There are many other individuals who contributed in some way to the realization of these four volumes, and it is impossible to acknowledge all of them here. However, there are two individuals in addition to the authors, editors and people named above who should receive substantial credit for the ultimate appearance of this publication. Katherine B. Magann, who had provided many years of dedicated service to CUPM prior to the closing of the CUPM office, accomplished the production of the report Case Studies in Applied Mathematics. Carolyn D. Lucas assisted in the running of the 1976 MAA Workshop, supervised the production of the resulting sixty modules, and served as managing editor for the publication of these four volumes. Without her efforts and perseverance the final product of this major project might not have been realized.

July 1982 W. F. LUCAS

Preface for Volume 1

Volume 1 consists of twenty-three chapters concerned with mathematical modeling and problem solving using differential equations. The chapters in Part I deal with the very beginning, and often the most important part of the modeling process: how to translate the given problem into a mathematical problem. The first chapter by Henderson West shows how to translate various word problems into differential equations, while Chapter 3 by Frauenthal deals with the special case of population growth models, a subject of much current interest. The second chapter, also by Henderson West, describes how to analyze a differential equation, and how to draw qualitative conclusions from it. These three chapters were written with a clarity and painstaking attention to detail that is not often found in textbooks, and thus are "must reading" for the beginning student of modeling.

The three chapters by Braun in Part II deal with three diverse and important problems that can be modeled, and completely solved, by first order differential equations. It is interesting to note that the work described in these units (and indeed, many of the modules in this volume) was originally done not by mathematicians, but by chemists, biologists and sociologists.

Part III is essentially a continuation of Part II, the difference being that the problems in this section are modeled by higher order linear equations and by solvable systems of first order equations. Systems of differential equations can be used to model very complex and even esoteric problems, and the results obtained are often very exciting, as seen in the three modules by Braun, Coleman, and Powers.

The five chapters by Baker and Drew in Part IV describe some applications of mathematics to problems in traffic theory, another popular source of interesting modeling problems. The results obtained are not

Preface for Volume 1 x

as powerful and spectacular as the results obtained in the previous chapters. Nevertheless, they are still extremely important as they illustrate how mathematical modeling can often aid in our understanding of complex and even uncontrollable and unsolvable problems.

The five chapters by Braun and Coleman in Part V deal with systems of non-linear equations and their application to important problems in biology and ecology. Powerful results are obtained via the methods of the qualitative theory of differential equations. It is interesting to note that the qualitative theory of differential equations evolved, originally, from problems in physics and astronomy. These same techniques have had important applications to problems in the biological sciences. Indeed, the existence of such a powerful and polished theory has motivated many mathematicians to undertake the study of several outstanding problems in biology and ecology.

The chapters in Part VI deal with systems that can be modelled by partial differential equations, one of the more difficult areas of mathematical analysis. The unit by Borrelli is basic in that it carefully describes the theory and the actual modeling. The remaining three chapters by Drew, Meyer and Porsching deal with concrete and important real life applications.

July 1982 MARTIN BRAUN

COURTNEY S. COLEMAN

DONALD A. DREW

Contents

Part I. Differential Equations, Models, and What To Do with Them

Chapter 1. Setting Up First-Order Differential Equations from Word Problems 3 Beverly Henderson West

1. Introduction 2. Guidelines 3. Examples

Exercises Solutions Notes for the Instructor

Chapter 2. Qualitative Solution Sketching for First-Order Differential Equations Beverly Henderson West

1. Introduction 2. Direction Field 3. Relevance of Uniqueness Theorems 4. Sketching of Solutions 5. Discussion of Equilibria and Stability 6. A More Difficult Example

Exercises Solutions to Exercises Notes for the Instructor

3 5 8

IS 18 26

28

28 29 31 32 37 38 42 45 52

xii Contents

Chapter 3. Difference and Differential Equation Population Growth Models 53 James C. Frauenthal

1. Introduction 2. Solution of the Difference Equation 3. Solution of the Differential Equation 4. Linearized Stability Analysis 5. Interpretation of Results 6. Concluding Remarks 7. Appendix: The Basic Model

Exercises Solutions References Notes for the Instructor

Part II. Growth and Decay Models: First-Order Differential Equations

Chapter 4. The Van Meegeren Art Forgeries Martin Braun

Exercises References Notes for the Instructor

Chapter 5. Single Species Population Models Martin Braun


Chapter 6. The Spread of Technological Innovations Martin Braun


Part III. Higher Order Linear Models

Chapter 7. A Model for the Detection of Diabetes Martin Braun


53 54 55 56 60 61 62 64 65 67 68

71

77 79 80

81

88 90 90

91

97 97 97

101

107 108 108

Contents

Chapter 8. Combat Models Courtney S. Coleman

I. Introduction 2. Three Lanchester Combat Models 3. Conventional Combat: The Square Law 4. Guerrilla Combat: The Linear Law 5. Vietnam: The Parabolic Law 6. The Battle of Iwo Jima 7. Mathematical Models of Combat: The Dark Side of Modeling

Exercises Project References Notes for the Instructor

Chapter 9. Modeling Linear Systems by Frequency Response Methods William F. Powers

1. Introduction 2. Modeling the Human Operator 3. Properties and Techniques of Frequency Response Modeling 4. Project Human Operator Experimental Data 5. Further Topics

References Notes for the Instructor

Part IV. Traffic Models

xiii

109

109 109 113 115 117 121 126 127 127 129 130

132

132 134 136 148 151 152 153

Chapter to. How Long Should a Traffic Light Remain Amber? 157 Donald A. Drew

I. The Problem and the Model 157 2. The Solution 158

Exercises 160 References 161 Notes for the Instructor 161

Chapter 11. Queue Length at a Traffic Light via Flow Theory 162 Donald A. Drew

1. The Problem: A Simple Model 162 2. A Better Model 164


XIV

Chapter 12. Car-Following Models Robert L. Baker, Jr.

I. Introduction 2. Complexity of Actual Car-Following: Simplicity of Car-Following

Models 3. Pipes' Model 4. Linear Car-Following with Delays 5. Incorporation of Pipes' Model into Linear Car-Following 6. Stability of Linear Car-Following Theory 7. Nonlinear Car-Following Laws 8. Steady-State Flow 9. Conclusion

lO. Appendices Exercises References Notes for the Instructor

Chapter 13. Equilibrium Speed Distributions Donald A. Drew

I. Speed Distributions 2. Overtaking 3. Passing 4. Equilibrium Distributions


Chapter 14. Traffic Flow Theory Donald A. Drew

I. Basic Equations 2. Propagation of a Disturbance 3. Shocks


Part V. Interacting Species: Steady States of Nonlinear Systems

Chapter 15. Why the Percentage of Sharks Caught in the Mediterranean Sea Rose Dramatically During World War I Martin Braun


Contents

168

168

169 170 175 178 179 181 182 183 185 193 196 197

198

198 200 202 203 205 205 206

207

207 210 212 215 216 216

222

227 228 228

Contents XV

Chapter 16. Quadratic Population Models: Almost Never Any C~~ m Courtney S. Coleman

1. Quadratic Population Models 229 2. Equilibrium Populations: No Cycles 232 3. The D'Ancona-Volterra Model: All Cycles 235 4. Stability 236 5. Which Model? 238


Chapter 17. The Principle of Competitive Exclusion in Population Biology 243 Martin Braun


Chapter 18. Biological Cycles and the Fivefold Way 251 Courtney S. Coleman

I. Prologue 251 2. The Canadian Lynx Cycle 252 3. The Canadian Lynx-Snowshoe Hare Cycle 255 4. A Two-Species Population Model 256 5. Differential Systems-Solutions, Orbits, Equilibria 258 6. The Fivefold Way 263 7. Predator-Prey Models: Kolmogoro\>'s Theorem 268 8. Epilogue: Do Hares Eat the Lynx? 273

Exercises 274 Annotated Bibliography References 276 Notes for the Instructor 278

Chapter 19. Hilbert's 16th Problem: How Many Cycles? 279 Courtney S. Coleman

I. Introduction 279 2. Cycles 281 3. Hilbert's 16th Problem 286 4. An Ecological Interpretation 289 5. Three Almost Impossible Projects 293


xvi Contents

Part VI. Models Leading to Partial Differential Equations

Chapter 20. Surge Tank Analysis 301 Donald A. Drew

1. Surge Tanks 301 2. Equations of Flow 301 3. Stability of Steady State 303

Appendix 305 Exercises 309 References 309 Notes for the Instructor

Chapter 21. Shaking a Piece of String to Rest 310 Robert L. Borrelli

1. Introduction 310 2. Linear Model for the Vibrating String 311 3. Conditions at the End of the String 313 4. The Mixed Initial/Boundary Value Problem for the Vibrating String 314 5. Characteristics 314 6. The Cauchy Problem for the Wave Equation 315 7. Properties of Solutions of the Wave Equation 318 8. Solution of the Mixed Initial/Boundary Problem 320 9. Shaking a String to Rest 324


Chapter 22. Heat Transfer in Frozen Soil 330 Gunter H. Meyer

I. Description of the Problem 330 2. The Model 332 3. A Solution Algorithm 337 4. An Application 345 5. Convergence 348


Chapter 23. Network Analysis of Steam Generator Flow 352 T. A. Porsching

1. Origin of the Problem 352 2. Some Elements of Network Theory 356 3. Modeling the Flow Problem 364 4. Numerical Solution of the Network Equations 371 5. Illustrative Problem 375

References 376 Notes for the Instructor 377

Contents for the Companion Volumes

VOLUME 2. POLITICAL AND RELATED MODELS

1 The Process of Applied Mathematics Maynard Thompson

2 Proportional Representation Edward Bolger

3 Comparison Voting Steven J. Brams

4 Modeling Coalitional Values William F. Lucas and Louis J. Billera

5 Urban Wastewater Management Planning James P. Heaney

6 An Everyday Approach to Matrix Operations Robert M. Thrall and E. L. Perry

7 Sources of Applications of Mathematics in Ecological and Environmental Subject Areas, Suitable for Classroom Use Homer T. Hayslett, Jr.

8 How To Ask Sensitive Questions without Getting Punched in the Nose John C. Maceli

9 Measuring Power in Weig4ted Voting Systems William F. Lucas

10 To the (Minimal Winning) Victors Go the (Equally Divided) Spoils: A New Power Index for Simple n-Person Games John Deegan, Jr. and Edward W. Packel

11 Power Indices in Politics Philip D. Straffin, Jr.

xviii Contents for the Companion Volumes

12 Committee Decision Making Peter Rice

13 Stochastic Difference Equations with Sociological Applications Loren Cobb

14 The Apportionment Problem William F. Lucas

VOLUME 3. DISCRETE AND SYSTEM MODELS

Foresight-Insight-Hindsight James C. Frauenthal and Thomas L. Saaty

2 Five Nautical Models Robert L. Baker, Jr.

3 An Optimal Inventory Policy Model Oswaldo Marrero

4 An Arithmetic Model of Gravity Donald Greenspan

5 Four-Way Stop or Traffic Lights? An Illustration of the Modeling Process Edward E. Packel

6 A Model for Municipal Street Sweeping Operations A. C. Tucker and L. Bodin

7 Finite Covering Problems Ronald E. Prather

8 A Pulse Process Model of Athletic Financing E. L. Perry

9 Traffic Equilibria on a Roadway Network Elmor L. Peterson

10 An Optimal Mix Problem: Capacity Expansion in a Power Field Jacob Zahavi

11 Hierarchies, Reciprocal Matrices, and Ratio Scales Thomas L. Saaty

12 Multiple Choice Testing Robert J. Weber

13 Computing Fixed Points, with Applications to Economic Equilibrium Models Michael T. Todd

14 Production Costing and Reliability Assessment of Electrical Power Generation Systems under Supply and Demand Uncertainty Jacob Zahavi

Contents for the Companion Volumes

VOLUME 4. LIFE SCIENCE MODELS

Part I. Population Models

1 Population Mathematics Frank Hoppensteadt

2 Population Growth-An Age Structure Model Robert Fennell

3 A Comparison of Some Deterministic and Stochastic Models of Population Growth Helen Marcus-Roberts

xix

4 Some Examples of Mathematical Models for the Dynamics of SeveralSpecies Ecosystems H. R. van der Vaart

Part /I. Biomedicine: Epidemics, Genetics, and Bioengineering

5 Malaria Helen Marcus-Roberts and Fred Roberts

6 MacDonald's Work on Helminth Infections Donald Ludwig and Benjamin Haytock

7 A Model for the Spread of Gonorrhea Martin Braun

8 DNA, RNA, and Random Mating: Simple Applications of the Multiplication Rule Helen Marcus-Roberts

9 Cigarette Filtration Donald Drew

Part III. Ecology

10 Efficiency of Energy Use in Obtaining Food, I: Humans Fred Roberts

II Efficiency of Energy Use in Obtaining Food, II: Animals Fred Roberts and Helen Marcus-Roberts

12 The Spatial Distribution of Cabbage Butterfly Eggs Daniel Solomon

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