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 Nino. 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.

Table of Contents:
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 Poincare 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 Nino and the Southern Oscillation 12.1 Introduction 229 12.2 Bjerknes' Hypothesis 231 12.3 A SimpleMathematicalModel of El Nino 233 The Atmosphere 233 Air-Sea Interaction 234 Ocean Temperature Advection 235 12.4 OtherModels of El Nino 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