A Concrete Approach to Mathematical Modelling

Critical praise for A Concrete Approach to Mathematical Modelling "...a treasure house of material for students and teachers alike...can be dipped into regularly for inspiration and ideas. It deserves to become a classic."--London Times Higher Education Supplement "The author succeeds in his goal of serving the needs of the undergraduate population who want to see mathematics in action, and the mathematics used is extensive and provoking."--SIAM Review "Each chapter discusses a wealth of examples ranging from old standards...to novelty ... Each model is developed critically, analyzed critically, and assessed critically."--Mathematical Reviews Mike Mesterton-Gibbons has done what no author before him could: he has written an in-depth, systematic guide to the art and science of mathematical modelling that's a great read from first page to last. With an abundance of both wit and common sense, he shows readers exactly how the modelling process works, using fascinating real-life examples from virtually every realm of human, machine, natural, and cosmic activity. You'll find models for determining how fast cars drive through a tunnel; how many workers industry should employ; the length of a supermarket checkout line; how birds should select worms; the best methods for avoiding an automobile accident; and when a barber should hire an assistant; just to name a few. Offering more examples, more detailed explanations, and by far, more sheer enjoyment than any other book on the subject, A Concrete Approach to Mathematical Modelling is the ultimate how-to guide for students and professionals in the hard sciences, social sciences, engineering, computers, statistics, economics, politics, business management, and every other discipline in which mathematical modelling plays a role. An Instructor's Manual presenting detailed solutions to all the problems in the book is available upon request from the Wiley editorial department. Cover Design / Illustration: Keithley Associates, Inc.

Table of Contents:
An ABC of modelling xix I The Deterministic View 1 Growth and decay. Dynamical systems 3 1.1 Decay of pollution. Lake purification 5 1.2 Radioactive decay 7 1.3 Plant growth 7 1.4 A simple ecosystem 8 1.5 A second simple ecosystem 11 1.6 Economic growth 13 1.7 Metered growth (or decay) models 21 1.8 Salmon dynamics 23 1.9 A model of U.S. population growth 26 1.10 Chemical dynamics 29 1.11 More chemical dynamics 30 1.12 Rowing dynamics 32 1.13 Traffic dynamics 34 1.14 Dimensionality, scaling, and units 35 Exercises 40 2 Equilibrium 46 2.1 The equilibrium concentration of contaminant in a lake 52 2.2 Rowing in equilibrium 53 2.3 How fast do cars drive through a tunnel? 57 2.4 Salmon equilibrium and limit cycles 58 2.5 How much heat loss can double-glazing prevent? 63 2.6 Why are pipes circular ? 66 2.7 Equilibrium shifts 71 2.8 How quickly must driver s react to preserve an equilibrium ? 76 Exercises 83 3 Optimal control and utility 91 3.1 How fast should a bird fly when migrating? 93 3.2 How big a pay increase should a professor receive? 95 3.3 How many worker s should industry employ? 103 3.4 When should a forest be cut? 104 3.5 How dense should traffic be in a tunnel? 109 3.6 How much pesticide should a crop grower use.an d when? 111 3.7 How many boats in a fishing fleet should be operational? 115 Exercises 119 II Validating a Model 4 Validation: accept, improve, or reject 127 4.1 A model of U.S. population growth 127 4.2 Cleaning Lake Ontario 128 4.3 Plant growth 129 4.4 The speed of a boat 130 4.5 The extent of bird migration 132 4.6 The speed of cars in a tunnel 136 4.7 The stability of cars in a tunnel 138 4.8 The forest rotation time 142 4.9 Crop spraying 146 4.10 How right was Poiseuille? 148 4.11 Competing species 151 4.12 Predator-prey oscillations 154 4.13 Sockeye swings, paradigms, and complexity 157 4.14 Optimal fleet size and higher paradigms 159 4.15 On the advantages of flexibility in prescriptive models 161 Exercises 163 III The Probabilistic View 5 Birth and death. Probabilistic dynamics 175 5.1 When will an old man die? The exponential distribution 180 5.2 When will Í men die? A pure death process 183 5.3 Forming a queue. A pure birth process 185 5.4 How busy must a road be to require a pedestrian crossing control? 187 5.5 The rise and fall of the company executive 189 5.6 Discrete models of a day in the life of an elevator 193 5.7 Birds in a cage. A birth and death chain 198 5.8 Trees in a forest. An absorbing birth and death chain 200 Exercises 202 6 Stationary distributions 208 6.1 The certainty of death 210 6.2 Elevator stationarity. The stationary birth and death process 213 6.3 How long is the queue at the checkout? A first look 215 6.4 How long is the queue at the checkout? A second look 217 6.5 How long must someone wait at the checkout? Another view 219 6.6 The structure of the work force 225 6.7 When does a T-junction require a left-turn lane? 227 Exercises 234 7 Optimal decision and reward 237 7.1 How much should a buyer buy? A first look 237 7.2 How many roses for Valentine’s Day? 243 7.3 How much should a buyer buy? A second look 245 7.4 How much should a retailer spend on advertising? 247 7.5 How much should a buyer buy? A third look 253 7.6 Why don’t fast-food restaurants guarantee service times anymore? 258 7.7 When should one barber employ another? Comparing alternatives 263 7.8 On the subjectiveness of decision making 267 Exercises 268 IV The Art of Application 8 Using a model: choice and estimation 275 8.1 Protecting the cargo boat. A message in a bottle 276 8.2 Oil extraction. Choosing an optimal harvesting model 279 8.3 Models within models. Choosing a behavioral response function 281 8.4 Estimating parameters for fitted curves: an error control problem 285 8.5 Assigning probabilities: a brief overview 291 8.6 Empirical probability assignment 293 8.7 Choosing theoretical distribution s and estimating their parameters 304 8.8 Choosing a utility function. Cautious attitudes to risk 316 Exercises 322 9 Building a model: adapting, extending, and combining 327 9.1 How many papers should a news vendor buy? An adaptation 328 9.2 Which trees in a forest should be felled? A combination 329 9.3 Cleaning Lake Ontario. An adaptation 334 9.4 Cleaning Lake Ontario. An extension 337 9.5 Pure diffusion of pollutants. A combination 345 9.6 Modelling a population’s age structure. A first attempt 350 9.7 Modelling a population’s age structure. A second attempt 360 Exercises 373 V Toward More Advance d Model s 10 Further dynamical systems 383 10.1 How does a fetus get glucose from its mother? 383 10.2 A limit-cycle ecosystem model 389 10.3 Does increasing the money supply raise or lower interest rates? 393 10.4 Linearizing time: The semi-Markov process. An extension 398 10.5 A more general semi-Markov process. A further extension 406 10.6 Who wil l govern Britain in the twenty-first century? A combination 409 Exercises 412 11 Further flow and diffusion 416 11.1 Unsteady heat conduction. An adaptation 417 11.2 How does traffic move after the train has gone by? A first look 421 11.3 How does traffic move after the train has gone by? A second look 423 11.4 Avoiding a crash at the other end. A combination 429 11.5 Spreading canal pollution. An adaptation 433 11.6 Flow and diffusion in a tube: a generic model 436 11.7 River cleaning. The Streeter-Phelps model 440 11.8 Why does a stopped organ pipe sound an octave lower than an open one? 446 Exercises 454 12 Further optimization 458 12.1 Finding an optimal policy by dynamic programming 458 12.2 The interviewer’s dilemma. An optimal stopping problem 465 12.3 A faculty hiring model 470 12.4 The motorist’s dilemma. Choosing the optimal parking space 475 12.5 How should a bird select worms? An adaptation 479 12.6 Where should an insect lay eggs? A combination 496 Exercises 507 Epilogue 514 Appendix 1: A review of probability and statistics 516 Appendix 2: Models, sources, and further reading arranged by discipline 531 Solutions to selected exercises 539 References 583 Index 591