Topics in Mathematical Modeling

Topics in Mathematical Modeling is an introductory textbook on mathematical modeling. The book teaches how simple mathematics can help formulate and solve real problems of current research interest in a wide range of fields, including biology, ecology, computer science, geophysics, engineering, and the social sciences. Yet the prerequisites are minimal: calculus and elementary differential equations. Among the many topics addressed are HIV; plant phyllotaxis; global warming; the World Wide Web; plant and animal vascular networks; social networks; chaos and fractals; marriage and divorce; and El Niño. Traditional modeling topics such as predator-prey interaction, harvesting, and wars of attrition are also included. Most chapters begin with the history of a problem, follow with a demonstration of how it can be modeled using various mathematical tools, and close with a discussion of its remaining unsolved aspects. Designed for a one-semester course, the book progresses from problems that can be solved with relatively simple mathematics to ones that require more sophisticated methods. The math techniques are taught as needed to solve the problem being addressed, and each chapter is designed to be largely independent to give teachers flexibility. The book, which can be used as an overview and introduction to applied mathematics, is particularly suitable for sophomore, junior, and senior students in math, science, and engineering.

K. K. Tung

Preface xiii
Chapter 1: Fibonacci Numbers, the Golden Ratio, and Laws of Nature?
1.1 Leonardo Fibonacci 1
1.2 The Golden Ratio 7
1.3 The Golden Rectangle and Self-Similarity 10
1.4 Phyllotaxis 12
1.5 Pinecones, Sunflowers, and Other Seed Heads 15
1.6 The Hofmeister Rule 17
1.7 A DynamicalModel 20
1.8 Concluding Remarks 21
1.9 Exercises 22
Chapter 2: Scaling Laws of Life, the Internet, and Social Networks
2.1 Introduction 27
2.2 Law of Quarter Powers 27
2.3 A Model of Branching Vascular Networks 30
2.4 Predictions of theModel 35
2.5 Complications andModifications 36
2.6 The Fourth Fractal Dimension of Life 38
2.7 Zipf's Law of Human Language, of the Size of Cities, and Email 39
2.8 TheWorldWideWeb and the Actor's Network 42
2.9 MathematicalModeling of Citation Network and theWeb 44
2.10 Exercises 47
Chapter 3: Modeling Change One Step at a Time
3.1 Introduction 54
3.2 Compound Interest and Mortgage Payments 54
Your Bank Account 54
Your Mortgage Payments,Monthly Interest Compounding 56
Your Mortgage Payments, Daily Interest Compounding 57
3.3 Some Examples 58
3.4 Compounding Continuously 58
Continuous Compounding 59
Double My Money: "Rule of 72," or Is It "Rule of 69"? 60
3.5 Rate of Change 62
Continuous Change 63
3.6 Chaotic Bank Balances 63
3.7 Exercises 65
Chapter 4: Differential Equation Models: Carbon Dating, Age of the Universe, HIV Modeling
4.1 Introduction 68
4.2 Radiometric Dating 68
4.3 The Age of Uranium in Our Solar System 70
4.4 The Age of the Universe 71
4.5 Carbon Dating 74
4.6 HIV Modeling 77
4.7 Exercises 79
Chapter 5: Modeling in the Physical Sciences, Kepler, Newton, and Calculus
5.1 Introduction 84
5.2 Calculus, Newton, and Leibniz 87
5.3 Vector Calculus Needed 88
5.4 Rewriting Kepler's Laws Mathematically 90
5.5 Generalizations 93
5.6 Newton and the Elliptical Orbit 95
5.7 Exercises 96
Chapter 6: Nonlinear Population Models: An Introduction to Qualitative Analysis Using Phase Planes
6.1 Introduction 98
6.2 PopulationModels 98
6.3 Qualitative Analysis 100
6.4 HarvestingModels 101
6.5 Economic Considerations 103
6.6 Depensation Growth Models 104
6.7 Comments 108
6.8 Exercises 108
Chapter 7: Discrete Time Logistic Map, Periodic and Chaotic Solutions
7.1 Introduction 113
Logistic Growth for Nonoverlapping Generations 114
7.2 DiscreteMap 115
7.3 Nonlinear Solution 117
7.4 Sensitivity to Initial Conditions 120
7.5 Order Out of Chaos 121
7.6 Chaos Is Not Random 122
7.7 Exercises 122
Chapter 8: Snowball Earth and Global Warming
8.1 Introduction 126
8.2 Simple ClimateModels 128
Incoming Solar Radiation 129
Albedo 130
Outward Radiation 130
Ice Dynamics 132
Transport 132
TheModel Equation 133
8.3 The Equilibrium Solutions 134
Ice-Free Globe 135
Ice-Covered Globe 136
Partially Ice-Covered Globe 137
Multiple Equilibria 138
8.4 Stability 139
The Slope-Stability Theorem 140
The Stability of the Ice-Free and Ice-Covered Globes 141
Stability and Instability of the Partially Ice-Covered Globe 141
How Does a Snowball Earth End? 143
8.5 Evidence of a Snowball Earth and Its Fiery End 144
8.6 The GlobalWarming Controversy 146
8.7 A Simple Equation for Climate Perturbation 150
8.8 Solutions 153
Equilibrium GlobalWarming 153
Time-Dependent GlobalWarming 154
Thermal Inertia of the Atmosphere-Ocean System 155
8.9 Exercises 157
Chapter 9: Interactions: Predator-Prey, Spraying of Pests, Carnivores in Australia
9.1 Introduction 161
9.2 The Nonlinear System and Its Linear Stability 162
9.3 Lotka-Volterra Predator-Prey Model 165
Linear Analysis 167
Nonlinear Analysis 170
9.4 Harvesting of Predator and Prey 172
Indiscriminate Spraying of Insects 173
9.5 The Case of theMissing Large Mammalian Carnivores 173
9.6 Comment 176
9.7 More Examples of Interactions 178
9.8 Exercises 182
Chapter 10: Marriage and Divorce
10.1 Introduction 191
10.2 Mathematical Modeling 195
Self-interaction 196
Marital Interactions 197
10.3 Data 198
10.4 An Example of a Validating Couple 199
10.5 Why Avoiding Conflicts Is an Effective Strategy in Marriage 201
10.6 Terminology 202
10.7 General Equilibrium Solutions 203
10.8 Conclusion 206
10.9 Assignment 206
10.10 Exercises 210
Chapter 11: Chaos in Deterministic Continuous Systems, Poincar and Lorenz
11.1 Introduction 212
11.2 Henri Poincaré 212
11.3 Edward Lorenz 214
11.4 The Lorenz Equations 216
11.5 Comments on Lorenz Equations as aModel of Convection 224
11.6 ChaoticWaterwheel 225
11.7 Exercises 226
Chapter 12: El Niño and the Southern Oscillation
12.1 Introduction 229
12.2 Bjerknes' Hypothesis 231
12.3 A SimpleMathematicalModel of El Niño 233
The Atmosphere 233
Air-Sea Interaction 234
Ocean Temperature Advection 235
12.4 OtherModels of El Niño 239
12.5 Appendix: The Advection Equation 240
12.6 Exercises 241
Chapter 13: Age of the Earth: Lord Kelvin's Model
13.1 Introduction 243
13.2 The Heat Conduction Problem 245
13.3 Numbers 250
13.4 Exercises 251
Chapter 14: Collapsing Bridges: Broughton and Tacoma Narrows
14.1 Introduction 254
14.2 Marching Soldiers on a Bridge: A SimpleModel 254
Resonance 259
A Different Forcing Function 260
14.3 Tacoma Narrows Bridge 261
Assignment 262
14.4 Exercises 262
APPENDIX A: Differential Equations and Their Solutions
A.1 First- and Second-Order Equations 267
A.2 Nonhomogeneous Ordinary Differential Equations 273
First-Order Equations 273
Second-Order Equations 275
A.3 Summary of ODE Solutions 277
A.4 Exercises 278
A.5 Solutions to Exercises 279
APPENDIX B: MATLAB Codes
B.1 MATLAB Codes for Lorenz Equations 282
B.2 MATLAB Codes for Solving Vallis's Equations 284
Bibliography 287
Index 293