Description
Book SynopsisThe goal of this book is to encourage the teaching and learning of mathematical model building relatively early in the undergraduate program. The text introduces the student to a number of important mathematical topics and to a variety of models in the social sciences, life sciences, and humanities.
Table of ContentsPreface viii
Acknowledgements xiii
1 Mathematical Models 1
I. Mathematical Systems and Models 1
II. An Example: Modeling Free Fall 4
III. Discrete Examples: Credit Cards and Populations 10
IV. Classification of Mathematical Models 16
V. Uses and Limitations of Mathematical Models 18
Exercises 19
Suggested Projects 21
2 Stable and Unstable Arms Races 23
I. The Real-World Setting 23
II. Constructing a Deterministic Model 25
III. A Simple Model for an Arms Race 25
IV. The Richardson Model 28
V. Interpreting and Testing the Richardson Model 45
VI. Obtaining an Exact Solution 53
Exercises 59
Suggested Projects 63
3 Ecological Models: Single Species 65
I. Introduction 65
II. The Pure Birth Process 65
III. Exponential Decay 71
IV. Logistic Population Growth 72
V. The Discrete Model of Logistic Growth and Chaos 80
VI. The Allee Effect 87
VII. Historical and Biographical Notes 89
Exercises 100
Suggested Projects 104
Biographical References 105
4 Ecological Models: Interacting Species 106
I. Introduction 106
II. Two Real-World Situations 106
III. Autonomous Systems 108
IV. The Competitive Hunters Model 116
V. The Predator-Prey Model 123
VI. Concluding Remarks on Simple Models in Population Dynamics 131
VII. Biographical Sketches 133
Exercises 137
Suggested Projects 139
5 Tumor Growth Models 141
I. Introduction 141
II. A General Tumor Growth Model 142
III. The Gompertz Model 145
IV. Modeling Colorectal Cancer 155
V. Historical and Biographical Notes 167
Exercises 176
Suggested Projects 177
6 Social Choice and Voting Procedures 179
I. Three Voting Situations 179
II. Two Voting Mechanisms 180
III. An Axiomatic Approach 185
IV. Arrow’s Impossibility Theorem 187
V. The Liberal Paradox and the Theorem of the Gloomy Alternatives 191
VI. Instant Runoff Voting 197
VII. Approval Voting 203
VIII. Topological Social Choice 207
IX. Historical and Biographical Notes 212
Exercises 224
Suggested Projects 229
7 Foundations of Measurement Theory 232
I. The Registrar’s Problem 232
II. What Is Measurement? 233
III. Simple Measures on Finite Sets 238
IV. Perception of Differences 240
V. An Alternative Approach 242
VI. Some Historical Notes 245
Exercises 245
Suggested Projects 247
8 Introduction to Utility Theory 249
I. Introduction 249
II. Gambles 250
III. Axioms of Utility Theory 251
IV. Existence and Uniqueness of Utility 254
V. Classification of Scales 257
VI. Interpersonal Comparison of Utility 259
VII. Historical and Biographical Notes 261
Exercises 265
Suggested Projects 266
9 Equilibrium in an Exchange Economy 268
I. Introduction 268
II. A Two-Person Economy with Two Commodities 268
III. An m-Person Economy 276
IV. Existence of Economic Equilibrium 283
V. Some Remaining Questions 293
VI. Historical and Biographical Notes 294
Exercises 298
Suggested Projects 301
VII. Additional Historical and Biographical Notes 302
10 Elementary Probability 303
I. The Need for Probability Models 303
II. What Is Probability? 304
III. A Probabilistic Model 322
IV. Stochastic Processes 325
Exercises 331
Suggested Projects 335
11 Markov Processes 336
I. Markov Chains 336
II. Matrix Operations and Markov Chains 341
III. Regular Markov Chains 347
IV. Absorbing Markov Chains 357
V. Historical and Biographical Notes 369
Exercises 371
Suggested Projects 374
12 Two Models of Cultural Stability 375
I. Introduction 375
II. The Gadaa System 375
III. A Deterministic Model 378
IV. A Probabilistic Model 381
V. Criticisms of the Models 383
VI. Hans Hoffmann 384
Exercises 386
Suggested Projects 387
13 Paired-Associate Learning 388
I. The Learning Problem 388
II. The Model 389
III. Testing the Model 397
IV. Historical and Biographical Notes 401
Exercises 404
Suggested Projects 406
14 Epidemics 407
I. Introduction 407
II. Deterministic Models 411
III. A Probabilistic Approach 449
IV. Historical and Biographical Notes 455
Exercises 459
Suggested Projects 463
15 Roulette Wheels and Hospital Beds: A Computer Simulation of Operating and Recovery Room Usage 464
I. Introduction 464
II. The Problems of Interest 468
III. Projecting the Number of Surgical Procedures 468
IV. Estimating Operating Room Demands 469
V. The Simulation Model 474
VI. Other Examples of Simulation 480
VII. Historical and Biographical Notes 484
Exercises 487
Suggested Projects 488
16 Game Theory 490
I. Two Difficult Decisions 490
II. Game Theory Basics 492
III. The Binding of Isaac 502
IV. Tosca and the Prisoners’ Dilemma 507
V. Nash Equilibrium 511
VI. Dynamic Solutions 515
VII. Historical and Biographical Notes 519
Exercises 522
Suggested Projects 526
Appendices
Appendix I: Sets 613
Appendix II: Matrices 617
Appendix III: Solving Systems of Equations 631
Appendix IV: Functions of Two Variables 645
Appendix V: Differential Equations 648
Index 657
