Introduction to Mathematical Modelling

1. What is Modeling 1.1 Models and Reality 1.2 Properties of Models 1.3 Building a Model 1.4 An Example 1.5 Another Example; Problems 1.6 Why Study Modeling? Part I. Elementary methods 2. Arguments from Scale 2.1 Effects of Size; Costs of Packaging; Speed of Racing Shells; Size Effects in Animals; Problems 2.2 Dimensional Analysis; Theoretical Background; The Period of a Perfect Pendulum; Scale Models of Structures; Problems 3. Graphical Methods 3.1 Using Graphs in Modeling 3.2 Comparative Statics; The Nuclear Missile Arms Race; Biogeography: Diversity of Species on Islands; Theory of the Firm; Problems 3.3 Stability Questions; Cobweb Models in Economics; Small Group Dynamics; Problems 4. Basic Optimization 4.1 Optimization by Differentiation; Maintaining Inventories; Geometry of Blood vessels; Fighting forest Fires; Problems 4.2 Graphical Methods; A Bartering Model; Changing Environment and Optimal Phenotype; Problems 5. Basic Probability 5.1 Analytic Models; Sex Preference and Sex Ratio; Making Simple Choices; Problems 5.2 Monte Carlo Simulation; A Doctor's Waiting Room; Sediment Volume; Stream Networks; Problems; A Table of 3000 Random Digits 6. Potpourri; Desert Lizards and Radiant Energy; Are Fair Election Procedures Possible?; Impaired Carbon Dioxide Elimination; Problems Part 2. More Advanced Methods 7. Approaches to Differential Equations 7.1 General Discussion 7.2 Limitations of Analytic Solutions 7.3 Alternative Approaches 7.4 Topics Not Discussed 8. Quantitative Differential Equations 8.1 Analytical Methods; Pollution of the Great Lakes; The Left Turn Squeeze; Long Chain Polymers; Problems 8.2 Numerical Methods; Towing a Water Skier; A Ballistics Problem; Problems; The Heun Method 9. Local Stability Theory 9.1 Autonomous systems 9.2 Differential Equations; Theoretical Background; Frictional Damping of a Pendulum; Species Interaction and Population Size; Keynesian Economics; More Complicated Situations; Problems 9.3 Differential-Difference Equations; The Dynamics of Car Following; Problems 9.4 Comments on Global Methods; Problem 10. More Probability; Radioactive Decay; Optimal Facility Location; Distribution of Particle Sizes; Problems Appendix. Some probabilistic Background A.1 The Notion of Probability A.2 Random Variables A.3 Bernoulli Trials A.4 Infinite Events Sets A.5 The Normal Distribution A.6 Generating Random Numbers A.7 Least Squares A.8 The Poisson and Exponential Distributions References; A Guide to Model Topics; Index