A Concrete Approach to Mathematical Modelling

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