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. Contents Preface 1 Fibonacci Numbers, the Golden Ratio, and Laws of Nature? 1.1 Leonardo Fibonacci 1.2 The Golden Ratio 1.3 The Golden Rectangle and Self-Similarity 1.4 Phyllotaxis 1.5 Pinecones, Sunflowers, and Other Seed Heads 1.6 The Hofmeister Rule 1.7 A Dynamical Model 1.8 Concluding Remarks 1.9 Exercises 2 Scaling Laws of Life, the Internet, and Social Networks 2.1 Introduction 2.2 Law of Quarter Powers 2.3 A Model of Branching Vascular Networks 2.4 Predictions of the Model 2.5 Complications and Modifications 2.6 The Fourth Fractal Dimension of Life 2.7 Zipf’s Law of Human Language, of the Size of Cities, and Email 2.8 The World Wide Web and the Actor’s Network 2.9 Mathematical Modeling of Citation Network and the Web 2.10 Exercises 3 Modeling Change One Step at a Time 3.1 Introduction 3.2 Compound Interest and Mortgage Payments Your Bank Account Your Mortgage Payments, Monthly Interest Compounding Your Mortgage Payments, Daily Interest Compounding 3.3 Some Examples 3.4 Compounding Continuously Continuous Compounding Double My Money: “Rule of 72,” or Is It “Rule of 69”? 3.5 Rate of Change Continuous Change 3.6 Chaotic Bank Balances 3.7 Exercises 4 Differential Equation Models: Carbon Dating, Age of the Universe, HIV Modeling 4.1 Introduction 4.2 Radiometric Dating 4.3 The Age of Uranium in Our Solar System 4.4 The Age of the Universe 4.5 Carbon Dating 4.6 HIV Modeling 4.7 Exercises 5 Modeling in the Physical Sciences, Kepler, Newton, and Calculus 5.1 Introduction 5.2 Calculus, Newton, and Leibniz 5.3 Vector Calculus Needed 5.4 Rewriting Kepler’s Laws Mathematically 5.5 Generalizations 5.6 Newton and the Elliptical Orbit 5.7 Exercises 6 Nonlinear Population Models: An Introduction to Qualitative Analysis Using Phase Planes 6.1 Introduction 6.2 Population Models 6.3 Qualitative Analysis 6.4 Harvesting Models 6.5 Economic Considerations 6.6 Depensation Growth Models 6.7 Comments 6.8 Exercises 7 Discrete Time Logistic Map, Periodic and Chaotic Solutions 7.1 Introduction Logistic Growth for Nonoverlapping Generations 7.2 Discrete Map 7.3 Nonlinear Solution 7.4 Sensitivity to Initial Conditions 7.5 Order Out of Chaos 7.6 Chaos Is Not Random 7.7 Exercises 8 Snowball Earth and Global Warming 8.1 Introduction 8.2 Simple Climate Models Incoming Solar Radiation Albedo Outward Radiation Ice Dynamics Transport The Model Equation 8.3 The Equilibrium Solutions Ice-Free Globe Ice-Covered Globe Partially Ice-Covered Globe Multiple Equilibria 8.4 Stability The Slope-Stability Theorem The Stability of the Ice-Free and Ice-Covered Globes Stability and Instability of the Partially Ice-Covered Globe How Does a Snowball Earth End? 8.5 Evidence of a Snowball Earth and Its Fiery End 8.6 The Global Warming Controversy 8.7 A Simple Equation for Climate Perturbation 8.8 Solutions Equilibrium Global Warming Time-Dependent Global Warming Thermal Inertia of the Atmosphere–Ocean System 8.9 Exercises 9 Interactions: Predator–Prey, Spraying of Pests, Carnivores in Australia 9.1 Introduction 9.2 The Nonlinear System and Its Linear Stability 9.3 Lotka–Volterra Predator–Prey Model Linear Analysis Nonlinear Analysis 9.4 Harvesting of Predator and Prey Indiscriminate Spraying of Insects 9.5 The Case of the Missing Large Mammalian Carnivores 9.6 Comment 9.7 More Examples of Interactions 9.8 Exercises 10 Marriage and Divorce 10.1 Introduction 10.2 Mathematical Modeling Self-interaction Marital Interactions 10.3 Data 10.4 An Example of a Validating Couple 10.5 Why Avoiding Conflicts Is an Effective Strategy in Marriage 10.6 Terminology 10.7 General Equilibrium Solutions 10.8 Conclusion 10.9 Assignment 10.10 Exercises 11 Chaos in Deterministic Continuous Systems, Poincaré and Lorenz 11.1 Introduction 11.2 Henri Poincaré 11.3 Edward Lorenz 11.4 The Lorenz Equations 11.5 Comments on Lorenz Equations as a Model of Convection 11.6 Chaotic Waterwheel 11.7 Exercises 12 El Niño and the Southern Oscillation 12.1 Introduction 12.2 Bjerknes’ Hypothesis 12.3 A Simple Mathematical Model of El Niño The Atmosphere Air–Sea Interaction Ocean Temperature Advection 12.4 Other Models of El Niño 12.5 Appendix: The Advection Equation 12.6 Exercises 13 Age of the Earth: Lord Kelvin’s Model 13.1 Introduction 13.2 The Heat Conduction Problem 13.3 Numbers 13.4 Exercises 14 Collapsing Bridges: Broughton and Tacoma Narrows 14.1 Introduction 14.2 Marching Soldiers on a Bridge: A Simple Model Resonance A Different Forcing Function 14.3 Tacoma Narrows Bridge Assignment 14.4 Exercises APPENDIX A: Differential Equations and Their Solutions A.1 First- and Second-Order Equations A.2 Nonhomogeneous Ordinary Differential Equations First-Order Equations Second-Order Equations A.3 Summary of ODE Solutions A.4 Exercises A.5 Solutions to Exercises APPENDIX B: MATLAB Codes B.1 MATLAB Codes for Lorenz Equations B.2 MATLAB Codes for Solving Vallis’s Equations Bibliography Index Topics In Mathematical Modeling 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.--publisher. Fibonacci Numbers, The Golden Ratio, And The Laws Of Nature? -- Scaling Laws Of Life, The Internet, And Social Networks -- Modeling Change One Step At A Time -- Differential Equation Models : Carbon Dating, Age Of The Universe, Hiv Modeling -- Modeling In The Physical Sciences, Kepler, Newton, And Calculus -- Nonlinear Population Models : An Introduction To Qualitative Analysis Using Phase Planes -- Discrete Time Logistic Map, Periodic And Chaotic Solutions -- Snowball Earth And Global Warming -- Interations : Predator-prey, Spraying Of Pests, Carnivores In Australia -- Marriage And Divorce -- Chaos In Deterministic Continuous Systems, Poincaré And Lorenz -- El Niño And The Southern Oscillation -- Age Of The Earth : Lord Kelvin's Model -- Collapsing Bridges : Broughton And Tacoma Narrows -- Appendix A : Differential Equations And Their Solutions -- Appendix B : Matlab Codes. K.k. Tung. Includes Bibliographical References (p. [287]-291) And Index.