Topics in Mathematical Modeling

✍ Scribed by K. K. Tung

• Publisher: Princeton University Press
• Year: 2016
• Tongue: English
• Leaves: 318
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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