Concrete Approach to Mathematical Modelling

Mike Mesterton-Gibbons, PhD, is Professor of Mathematics at Florida State University.

• An ABC of modelling xixI The Deterministic View1 Growth and decay. Dynamical systems 31.1 Decay of pollution. Lake purification 51.2 Radioactive decay 71.3 Plant growth 71.4 A simple ecosystem 81.5 A second simple ecosystem 111.6 Economic growth 131.7 Metered growth (or decay) models 211.8 Salmon dynamics 231.9 A model of U.S. population growth 261.10 Chemical dynamics 291.11 More chemical dynamics 301.12 Rowing dynamics 321.13 Traffic dynamics 341.14 Dimensionality, scaling, and units 35Exercises 402 Equilibrium 462.1 The equilibrium concentration of contaminant in a lake 522.2 Rowing in equilibrium 532.3 How fast do cars drive through a tunnel? 572.4 Salmon equilibrium and limit cycles 582.5 How much heat loss can double-glazing prevent? 632.6 Why are pipes circular ? 662.7 Equilibrium shifts 712.8 How quickly must driver s react to preserve an equilibrium ? 76Exercises 833 Optimal control and utility 913.1 How fast should a bird fly when migrating? 933.2 How big a pay increase should a professor receive? 953.3 How many worker s should industry employ? 1033.4 When should a forest be cut? 1043.5 How dense should traffic be in a tunnel? 1093.6 How much pesticide should a crop grower use.an d when? 1113.7 How many boats in a fishing fleet should be operational? 115Exercises 119II Validating a Model4 Validation: accept, improve, or reject 1274.1 A model of U.S. population growth 1274.2 Cleaning Lake Ontario 1284.3 Plant growth 1294.4 The speed of a boat 1304.5 The extent of bird migration 1324.6 The speed of cars in a tunnel 1364.7 The stability of cars in a tunnel 1384.8 The forest rotation time 1424.9 Crop spraying 1464.10 How right was Poiseuille? 1484.11 Competing species 1514.12 Predator-prey oscillations 1544.13 Sockeye swings, paradigms, and complexity 1574.14 Optimal fleet size and higher paradigms 1594.15 On the advantages of flexibility in prescriptive models 161Exercises 163III The Probabilistic View5 Birth and death. Probabilistic dynamics 1755.1 When will an old man die? The exponential distribution 1805.2 When will Í men die? A pure death process 1835.3 Forming a queue. A pure birth process 1855.4 How busy must a road be to require a pedestrian crossing control? 1875.5 The rise and fall of the company executive 1895.6 Discrete models of a day in the life of an elevator 1935.7 Birds in a cage. A birth and death chain 1985.8 Trees in a forest. An absorbing birth and death chain 200Exercises 2026 Stationary distributions 2086.1 The certainty of death 2106.2 Elevator stationarity. The stationary birth and death process 2136.3 How long is the queue at the checkout? A first look 2156.4 How long is the queue at the checkout? A second look 2176.5 How long must someone wait at the checkout? Another view 2196.6 The structure of the work force 2256.7 When does a T-junction require a left-turn lane? 227Exercises 2347 Optimal decision and reward 2377.1 How much should a buyer buy? A first look 2377.2 How many roses for Valentine’s Day? 2437.3 How much should a buyer buy? A second look 2457.4 How much should a retailer spend on advertising? 2477.5 How much should a buyer buy? A third look 2537.6 Why don’t fast-food restaurants guarantee service times anymore? 2587.7 When should one barber employ another? Comparing alternatives 2637.8 On the subjectiveness of decision making 267Exercises 268IV The Art of Application8 Using a model: choice and estimation 2758.1 Protecting the cargo boat. A message in a bottle 2768.2 Oil extraction. Choosing an optimal harvesting model 2798.3 Models within models. Choosing a behavioral response function 2818.4 Estimating parameters for fitted curves: an error control problem 2858.5 Assigning probabilities: a brief overview 2918.6 Empirical probability assignment 2938.7 Choosing theoretical distribution s and estimating their parameters 3048.8 Choosing a utility function. Cautious attitudes to risk 316Exercises 3229 Building a model: adapting, extending, and combining 3279.1 How many papers should a news vendor buy? An adaptation 3289.2 Which trees in a forest should be felled? A combination 3299.3 Cleaning Lake Ontario. An adaptation 3349.4 Cleaning Lake Ontario. An extension 3379.5 Pure diffusion of pollutants. A combination 3459.6 Modelling a population’s age structure. A first attempt 3509.7 Modelling a population’s age structure. A second attempt 360Exercises 373V Toward More Advance d Model s10 Further dynamical systems 38310.1 How does a fetus get glucose from its mother? 38310.2 A limit-cycle ecosystem model 38910.3 Does increasing the money supply raise or lower interest rates? 39310.4 Linearizing time: The semi-Markov process. An extension 39810.5 A more general semi-Markov process. A further extension 40610.6 Who wil l govern Britain in the twenty-first century? A combination 409Exercises 41211 Further flow and diffusion 41611.1 Unsteady heat conduction. An adaptation 41711.2 How does traffic move after the train has gone by? A first look 42111.3 How does traffic move after the train has gone by? A second look 42311.4 Avoiding a crash at the other end. A combination 42911.5 Spreading canal pollution. An adaptation 43311.6 Flow and diffusion in a tube: a generic model 43611.7 River cleaning. The Streeter-Phelps model 44011.8 Why does a stopped organ pipe sound an octave lower than an open one? 446Exercises 45412 Further optimization 45812.1 Finding an optimal policy by dynamic programming 45812.2 The interviewer’s dilemma. An optimal stopping problem 46512.3 A faculty hiring model 47012.4 The motorist’s dilemma. Choosing the optimal parking space 47512.5 How should a bird select worms? An adaptation 47912.6 Where should an insect lay eggs? A combination 496Exercises 507Epilogue 514Appendix 1: A review of probability and statistics 516Appendix 2: Models, sources, and further reading arranged by discipline 531Solutions to selected exercises 539References 583Index 591