增长和扩散现象的数学模式和应用PDF电子书下载

1 Introduction 1

2 Some Basic Frameworks 5

2.1 Exponential Function 7

2.1.1 The Exponential Function and Its Properties 7

2.1.2 Doubling Times 10

2.1.3 An Illustration:Population of the United States 11

2.1.4 Exponential Function with Migration 13

2.1.5 Power Law Exponential Function 15

2.1.6 An Illustration:Population of the World 16

2.1.7 Combinations of Exponential Functions 19

2.1.8 Solutions and Properties of the Equations 21

2.1.9 An Illustration:Oxygen Distribution in a River 24

2.2 Logistic Distribution 27

2.2.1 The Differential Equation and Its Solution 27

2.2.2 Properties of the Logistic Distribution 29

2.2.3 An Illustration:Technology Substitution 34

2.2.4 An Illustration:Diffusion of Improved Pasture Technology in Uruguay 38

2.2.5 An Illustration:Growth of Prime Mover Horsepower in the U.S 44

2.3 Confined Exponential Distribution 52

2.3.1 Scope of Applications of the Distribution 52

2.3.2 The Differential Equation with Constant Coefficients 53

2.3.3 The Differential Equation with Variable Coefficient？ 53

2.3.4 Variable Transfer Coefficient 55

2.3.5 Variable Equilibrium Value 60

2.3.6 An Illustration:Oxygen Transfer Across a Water Surface 65

2.3.7 An Illustration:Biochemical Oxygen Demand 67

2.3.8 An Illustration:Growth of Humans 69

2.3.9 An Illustration:Public Interest in a News Event 73

2.4 Combination of the Logistic Distribution and the Confined Exponential Distribution 76

2.4.1 Comparison of the Two Distributions 76

2.4.2 Phenomena in Industrial Technology Transfer 77

2.4.3 Phenomena in Social Innovation Diffusion 78

2.4.4 Phenomena in Chemical Reaction Kinetics 79

2.4.5 Phenomena in the Psychology of Learning 80

2.4.6 The Differential Equation,Its Solution and Properties 82

2.4.7 An Illustration:Adoption of a Tornado Warning Device 86

2.4.8 Combination of the Exponential Function and the Confined Exponential Distribution 88

2.4.9 An Illustration:Population of California 89

2.5 Normal Probability Distribution 92

2.5.1 The Normal Probability Function and Its Features 92

2.5.2 Relationship Between the Normal Probability Function and the Error Function 94

2.5.3 Approximate Expressions for the Normal Function and Its Inverse 95

2.5.4 Comparison of the Logistic and Normal Probability Distributions 96

2.5.5 An Illustration:Adoption of Herbicides by Mexican Barley Farmers 103

2.6 Power Law Logistic Distribution 105

2.6.1 The Differential Equation and Its Features 105

2.6.2 The Growth Curve and Its Properties 106

2.6.3 The Richards Function 110

2.6.4 An Illustration:Sale of Development Property 110

2.6.5 An Illustration:Growth of Pine Trees in New Zealand 112

2.6.6 An Illustration:Adoption of Hybrid Corn in the United States 114

2.7 Logistic Growth with Migration 117

2.7.1 Immigration and Emigration 117

2.7.2 Logistic Growth with Constant Stocking 117

2.7.3 Logistic Growth with Constant Harvesting 119

2.7.4 Logistic Growth with Variable Harvesting 121

2.7.5 An Illustration:Fish Harvesting 122

2.7.6 An Illustration:The Sandhill Crane 123

2.8 Epidemics and Technology Transfer 125

2.8.1 Simple and General Epidemics 125

2.8.2 The Differential Equations and Phase Plane Display 126

2.8.3 Solutions to the Differential Equations 129

2.8.4 Logarithmic Form of the Solutions 133

2.8.5 A Technology Transfer Analogy 134

2.8.6 An Illustration:Bombay Plague of 1905-1906 135

2.9 Some Modifications of the Logistic Distribution 136

2.9.1 Use of Taylor Series 136

2.9.2 First Order Differential Equations 137

2.9.3 An Illustration:Growth of Water Fleas and Trees 138

2.9.4 An Illustration:Sale of Development Property Revisited 140

2.9.5 Second Order Differential Equations 141

2.9.6 An Illustration:Multiplier-Accelerator Model of a National Economy 144

3 Some Additional Frameworks 149

3.1 Gompertz Distribution 149

3.1.1 The Gompertz Distribution and Its Features 149

3.1.2 An Illustration:Growth of Plant Leaves 151

3.1.3 An Illustration:Dynamics of Tumor Growth 155

3.2 Weibull Distribution 156

3.2.1 The Weibull Distribution and Its Features 156

3.2.2 An Illustration:Substitution of Diesel and Electric Locomotives for Steam Locomotives in the United States 159

3.3 A Generalized Distribution 162

3.3.1 Cumulative and Density Distribution Functions 162

3.3.2 A Generalized Symmetrical Function 163

3.3.3 Extreme Maximum Value Distribution 167

3.3.4 Extreme Minimum Value Distribution 169

3.3.5 An Illustration:Dose Response Analysis of Beetle Mortality Data 172

3.4 Hyperlogistic Distribution 174

3.4.1 The Differential Equation and Some Examples 174

3.4.2 Solution to the Differential Equation 176

3.4.3 Some Properties of the Hyperlogistic Equation 178

3.4.4 Numerical Examples of the Hyperlogistic Equation 180

3.4.5 An Illustration:Adoption of a Tornado Warning Device Revisited 182

3.4.6 Coalition and Modified Coalition Growth Models 185

3.4.7 An Illustration:Population of the World 188

3.4.8 An Illustration:Growth of the Public Debt of the United States 191

3.5 Various Other Distributions 194

3.5.1 Comparison of Distribution Functions 194

3.5.2 Arctangent-Exponential Distribution 195

3.5.3 Pearson Type VII Distribution 196

3.5.4 Arctangent Distribution 197

3.5.5 Gamma Distribution 199

3.5.6 Generalized Gamma Distribution 201

3.5.7 An Illustration:Generation Times of Cells 201

3.5.8 An Illustration:Population of Great Britain 205

4 Phenomena with Variable Growth Coefficients 209

4.1 Linearly Variable Growth Coefficient 210

4.1.1 The Growth Curve and Its Properties 210

4.1.2 An Illustration:Growth and Decline of U.S.Sailing Vessels 212

4.2 Hyperbolically Variable Growth Coefficient 214

4.2.1 The Growth Curve and Its Properties 214

4.2.2 Relationship to Power Law Exponential Growth 215

4.2.3 An Illustration:Population of the Great Plains States 216

4.3 Exponentially Variable Growth Coefficient 218

4.3.1 Extreme Maximum Value Distribution 218

4.3.2 Extreme Minimum Value Distribution 220

4.3.3 An Illustration:Survival of Rats 222

4.4 Sinusoidally Variable Growth Coefficient 224

4.4.1 Some Examples of Oscillatory Phenomena 224

4.4.2 Simple Harmonic Growth Coefficient 224

4.4.3 Exponentially Decreasing Growth Coefficient:Type Ⅰ 225

4.4.4 An Illustration:Growth of a Species of Land Snails 227

4.4.5 Exponentially Decreasing Growth Coefficient:Type Ⅱ 230

4.4.6 An Illustration:Growth of Cell Populations 231

4.4.7 Sinusoidally Variable Growth Coefficient in a Power Law Exponential Equation 237

4.4.8 An Illustration:Number of Patents Issued for Inventions 237

5 Phenomena with Variable Carrying Capacities 241

5.1 Exponentially Variable Carrying Capacity 242

5.1.1 The Growth Curve and Its Properties 242

5.1.2 An Illustration:Farm Population of the United States 245

5.2 Logistically Variable Carrying Capacity 249

5.2.1 Some Previous Studies 249

5.2.2 The Growth Curve and Its Properties 249

5.2.3 Relative Values of Growth Parameters 252

5.2.4 An Illustration:Enrollments in Universities in the United States 254

5.3 Linearly Variable Carrying Capacity 256

5.3.1 The Growth Curve and Its Properties 256

5.3.2 An Illustration:Horses and Mules on U.S.Farms 258

5.3.3 An Illustration:Steam Locomotives on U.S.Railroads 260

5.4 Hyperbolically Variable Carrying Capacity 262

5.4.1 Linearly Changing Crowding Coefficient 262

5.4.2 The Growth Curves and Its Properties 263

5.4.3 An Illustration:Growth Rates of Wheat Plant Components 263

5.5 Sinusoidally Variable Carrying Capacity 267

5.5.1 Cyclic Variations in Growth and Transfer Phenomena 267

5.5.2 The Growth Curve and Its Properties 267

5.5.3 Phase Plane Display 270

5.5.4 Exponentially Changing Carrying Capacity 275

5.5.5 An Illustration:Railway Mileage in the United States 277

5.6 Power Law Logistic with a Power Law Logistically Variable Carrying Capacity 279

5.6.1 The Power Law Logistic 279

5.6.2 The Differential Equation and Its Solution 279

5.6.3 An Illustration:Population of the United States 281

6 Phenomena with Time Delays 287

6.0.1 Introduction 287

6.0.2 Types and Features of Delay Equations 288

6.1 Discrete Time Delay in the Exponential Equation 289

6.1.1 The Delay Differential Equation and Its Solution 289

6.1.2 Roots of the Characteristic Equation 292

6.1.3 Behavior of the Solutions 294

6.1.4 An Illustration:Tinbergen's Shipbuilding Cycle 295

6.2 Discrete Time Delay in the Logistic Equation 297

6.2.1 Introduction to the Delay Differential Equation 297

6.2.2 Solution to the Discrete Delay Logistic Function 298

6.2.3 Numerical Example 300

6.2.4 An Illustration:Nicholson's Blowflies 302

6.3 Distributed Time Delay:Delay Integral in the Crowding Term 304

6.3.1 The Integro-differential Equation 304

6.3.2 Solution to the Integro-differential Equation 307

6.3.3 An Illustration:Growth and Decline of the Populations of Northeast and East North Central American Cities 310

6.4 Distributed Time Delay:Delay Integral in a Pollution Term 317

6.4.1 The Integro-differential Equation and Its Solution 317

6.4.2 An Approximate Sech-squared Solution 319

6.4.3 Numerical Examples 320

6.4.4 An Illustration:Growth and Self-Contamination of Bacteria 322

7 Phenomena with Spatial Diffusion 327

7.0.1 Introduction 327

7.0.2 The Diffusion Equation 327

7.1 Diffusion from Instantaneous Sources 329

7.1.1 Plane Source,Line Source and Point Source 329

7.1.2 Rectilinear Diffusion with Convection 331

7.1.3 An Illustration:Dispersion in Pipelines 332

7.1.4 Radial Diffusion with Exponential Growth 335

7.1.5 An Illustration:Biological Dispersion 336

7.2 Diffusion from Continuous Source 341

7.2.1 Rectilinear Diffusion with Constant Boundary Condition 341

7.2.2 An Illustration:Bacterial Motility 343

7.2.3 Rectilinear Diffusion with Variable Boundary Condition 344

7.2.4 An Illustration:Temperature Distribution in the Soil 345

7.2.5 Radial Diffusion 346

7.2.6 An Illustration:Unsteady Fluid Flow in an Aquifer 349

7.2.7 Rectilinear Diffusion with Convection 352

7.3 Diffusion with Reaction in a Finite Region 354

7.3.1 Dimensional Analysis 354

7.3.2 Exponential Growth in a Finite Region 356

7.3.3 Power Law Exponential Growth in a Finite Region 361

7.3.4 Logistic Growth in a Finite Region 362

7.3.5 An Illustration:Zone of Regulated Fishing 366

7.4 Diffusion with Convection and Reaction 367

7.4.1 The Differential Equation and Its Solution 367

7.4.2 Exponential Growth with Convection and Diffusion 371

7.4.3 Exponential Decay with Convection and Diffusion 372

7.4.4 Convection and Diffusion with Interphase Transfer 373

7.4.5 An Illustration:Chemical Solute Removal by Adsorption 378

7.5 Diffusion with Confined Exponential Growth 379

7.5.1 Rectilinear Diffusion 379

7.5.2 An Illustration:Heat Transfer from a River 380

7.5.3 Radial Diffusion 383

7.5.4 An Illustration:Population in Cities 385

7.5.5 Temporal-Spatial Diffusion 390

7.5.6 An Illustration:Population of London 395

7.6 Diffusion with Logistic Growth 402

7.6.1 Traveling Wave Solutions 402

7.6.2 A Power Law Traveling Wave Solution 408

7.6.3 An Illustration:Diffusion of Tractor Utilization 410

7.6.4 An Illustration:Adoption of Hybrid Corn Revisited 414

8 Conclusion 419

References 429

Author Index 445

Subject Index 451