《增长和扩散现象的数学模式和应用》求取 ⇩

1Introduction1

2Some Basic Frameworks5

2.1Exponential Function7

2.1.1 The Exponential Function and Its Properties7

2.1.2 Doubling Times10

2.1.3 An Illustration:Population of the United States11

2.1.4 Exponential Function with Migration13

2.1.5 Power Law Exponential Function15

2.1.6 An Illustration:Population of the World16

2.1.7 Combinations of Exponential Functions19

2.1.8 Solutions and Properties of the Equations21

2.1.9 An Illustration:Oxygen Distribution in a River24

2.2Logistic Distribution27

2.2.1 The Differential Equation and Its Solution27

2.2.2 Properties of the Logistic Distribution29

2.2.3 An Illustration:Technology Substitution34

2.2.4 An Illustration:Diffusion of Improved Pasture Technology in Uruguay38

2.2.5 An Illustration:Growth of Prime Mover Horsepower in the U.S44

2.3Confined Exponential Distribution52

2.3.1 Scope of Applications of the Distribution52

2.3.2 The Differential Equation with Constant Coefficients53

2.3.3 The Differential Equation with Variable Coefficient？53

2.3.4 Variable Transfer Coefficient55

2.3.5 Variable Equilibrium Value60

2.3.6 An Illustration:Oxygen Transfer Across a Water Surface65

2.3.7 An Illustration:Biochemical Oxygen Demand67

2.3.8 An Illustration:Growth of Humans69

2.3.9 An Illustration:Public Interest in a News Event73

2.4Combination of the Logistic Distribution and the Confined Exponential Distribution76

2.4.1 Comparison of the Two Distributions76

2.4.2 Phenomena in Industrial Technology Transfer77

2.4.3 Phenomena in Social Innovation Diffusion78

2.4.4 Phenomena in Chemical Reaction Kinetics79

2.4.5 Phenomena in the Psychology of Learning80

2.4.6 The Differential Equation,Its Solution and Properties82

2.4.7 An Illustration:Adoption of a Tornado Warning Device86

2.4.8 Combination of the Exponential Function and the Confined Exponential Distribution88

2.4.9 An Illustration:Population of California89

2.5Normal Probability Distribution92

2.5.1 The Normal Probability Function and Its Features92

2.5.2 Relationship Between the Normal Probability Function and the Error Function94

2.5.3 Approximate Expressions for the Normal Function and Its Inverse95

2.5.4 Comparison of the Logistic and Normal Probability Distributions96

2.5.5 An Illustration:Adoption of Herbicides by Mexican Barley Farmers103

2.6Power Law Logistic Distribution105

2.6.1 The Differential Equation and Its Features105

2.6.2 The Growth Curve and Its Properties106

2.6.3 The Richards Function110

2.6.4 An Illustration:Sale of Development Property110

2.6.5 An Illustration:Growth of Pine Trees in New Zealand112

2.6.6 An Illustration:Adoption of Hybrid Corn in the United States114

2.7Logistic Growth with Migration117

2.7.1 Immigration and Emigration117

2.7.2 Logistic Growth with Constant Stocking117

2.7.3 Logistic Growth with Constant Harvesting119

2.7.4 Logistic Growth with Variable Harvesting121

2.7.5 An Illustration:Fish Harvesting122

2.7.6 An Illustration:The Sandhill Crane123

2.8Epidemics and Technology Transfer125

2.8.1 Simple and General Epidemics125

2.8.2 The Differential Equations and Phase Plane Display126

2.8.3 Solutions to the Differential Equations129

2.8.4 Logarithmic Form of the Solutions133

2.8.5 A Technology Transfer Analogy134

2.8.6 An Illustration:Bombay Plague of 1905-1906135

2.9Some Modifications of the Logistic Distribution136

2.9.1 Use of Taylor Series136

2.9.2 First Order Differential Equations137

2.9.3 An Illustration:Growth of Water Fleas and Trees138

2.9.4 An Illustration:Sale of Development Property Revisited140

2.9.5 Second Order Differential Equations141

2.9.6 An Illustration:Multiplier-Accelerator Model of a National Economy144

3Some Additional Frameworks149

3.1Gompertz Distribution149

3.1.1 The Gompertz Distribution and Its Features149

3.1.2 An Illustration:Growth of Plant Leaves151

3.1.3 An Illustration:Dynamics of Tumor Growth155

3.2Weibull Distribution156

3.2.1 The Weibull Distribution and Its Features156

3.2.2 An Illustration:Substitution of Diesel and Electric Locomotives for Steam Locomotives in the United States159

3.3A Generalized Distribution162

3.3.1 Cumulative and Density Distribution Functions162

3.3.2 A Generalized Symmetrical Function163

3.3.3 Extreme Maximum Value Distribution167

3.3.4 Extreme Minimum Value Distribution169

3.3.5 An Illustration:Dose Response Analysis of Beetle Mortality Data172

3.4Hyperlogistic Distribution174

3.4.1 The Differential Equation and Some Examples174

3.4.2 Solution to the Differential Equation176

3.4.3 Some Properties of the Hyperlogistic Equation178

3.4.4 Numerical Examples of the Hyperlogistic Equation180

3.4.5 An Illustration:Adoption of a Tornado Warning Device Revisited182

3.4.6 Coalition and Modified Coalition Growth Models185

3.4.7 An Illustration:Population of the World188

3.4.8 An Illustration:Growth of the Public Debt of the United States191

3.5Various Other Distributions194

3.5.1 Comparison of Distribution Functions194

3.5.2 Arctangent-Exponential Distribution195

3.5.3 Pearson Type VII Distribution196

3.5.4 Arctangent Distribution197

3.5.5 Gamma Distribution199

3.5.6 Generalized Gamma Distribution201

3.5.7 An Illustration:Generation Times of Cells201

3.5.8 An Illustration:Population of Great Britain205

4Phenomena with Variable Growth Coefficients209

4.1Linearly Variable Growth Coefficient210

4.1.1 The Growth Curve and Its Properties210

4.1.2 An Illustration:Growth and Decline of U.S.Sailing Vessels212

4.2Hyperbolically Variable Growth Coefficient214

4.2.1 The Growth Curve and Its Properties214

4.2.2 Relationship to Power Law Exponential Growth215

4.2.3 An Illustration:Population of the Great Plains States216

4.3Exponentially Variable Growth Coefficient218

4.3.1 Extreme Maximum Value Distribution218

4.3.2 Extreme Minimum Value Distribution220

4.3.3 An Illustration:Survival of Rats222

4.4Sinusoidally Variable Growth Coefficient224

4.4.1 Some Examples of Oscillatory Phenomena224

4.4.2 Simple Harmonic Growth Coefficient224

4.4.3 Exponentially Decreasing Growth Coefficient:Type Ⅰ225

4.4.4 An Illustration:Growth of a Species of Land Snails227

4.4.5 Exponentially Decreasing Growth Coefficient:Type Ⅱ230

4.4.6 An Illustration:Growth of Cell Populations231

4.4.7 Sinusoidally Variable Growth Coefficient in a Power Law Exponential Equation237

4.4.8 An Illustration:Number of Patents Issued for Inventions237

5Phenomena with Variable Carrying Capacities241

5.1Exponentially Variable Carrying Capacity242

5.1.1 The Growth Curve and Its Properties242

5.1.2 An Illustration:Farm Population of the United States245

5.2Logistically Variable Carrying Capacity249

5.2.1 Some Previous Studies249

5.2.2 The Growth Curve and Its Properties249

5.2.3 Relative Values of Growth Parameters252

5.2.4 An Illustration:Enrollments in Universities in the United States254

5.3Linearly Variable Carrying Capacity256

5.3.1 The Growth Curve and Its Properties256

5.3.2 An Illustration:Horses and Mules on U.S.Farms258

5.3.3 An Illustration:Steam Locomotives on U.S.Railroads260

5.4Hyperbolically Variable Carrying Capacity262

5.4.1 Linearly Changing Crowding Coefficient262

5.4.2 The Growth Curves and Its Properties263

5.4.3 An Illustration:Growth Rates of Wheat Plant Components263

5.5Sinusoidally Variable Carrying Capacity267

5.5.1 Cyclic Variations in Growth and Transfer Phenomena267

5.5.2 The Growth Curve and Its Properties267

5.5.3 Phase Plane Display270

5.5.4 Exponentially Changing Carrying Capacity275

5.5.5 An Illustration:Railway Mileage in the United States277

5.6Power Law Logistic with a Power Law Logistically Variable Carrying Capacity279

5.6.1 The Power Law Logistic279

5.6.2 The Differential Equation and Its Solution279

5.6.3 An Illustration:Population of the United States281

6Phenomena with Time Delays287

6.0.1 Introduction287

6.0.2 Types and Features of Delay Equations288

6.1Discrete Time Delay in the Exponential Equation289

6.1.1 The Delay Differential Equation and Its Solution289

6.1.2 Roots of the Characteristic Equation292

6.1.3 Behavior of the Solutions294

6.1.4 An Illustration:Tinbergen's Shipbuilding Cycle295

6.2Discrete Time Delay in the Logistic Equation297

6.2.1 Introduction to the Delay Differential Equation297

6.2.2 Solution to the Discrete Delay Logistic Function298

6.2.3 Numerical Example300

6.2.4 An Illustration:Nicholson's Blowflies302

6.3Distributed Time Delay:Delay Integral in the Crowding Term304

6.3.1 The Integro-differential Equation304

6.3.2 Solution to the Integro-differential Equation307

6.3.3 An Illustration:Growth and Decline of the Populations of Northeast and East North Central American Cities310

6.4Distributed Time Delay:Delay Integral in a Pollution Term317

6.4.1 The Integro-differential Equation and Its Solution317

6.4.2 An Approximate Sech-squared Solution319

6.4.3 Numerical Examples320

6.4.4 An Illustration:Growth and Self-Contamination of Bacteria322

7Phenomena with Spatial Diffusion327

7.0.1 Introduction327

7.0.2 The Diffusion Equation327

7.1Diffusion from Instantaneous Sources329

7.1.1 Plane Source,Line Source and Point Source329

7.1.2 Rectilinear Diffusion with Convection331

7.1.3 An Illustration:Dispersion in Pipelines332

7.1.4 Radial Diffusion with Exponential Growth335

7.1.5 An Illustration:Biological Dispersion336

7.2Diffusion from Continuous Source341

7.2.1 Rectilinear Diffusion with Constant Boundary Condition341

7.2.2 An Illustration:Bacterial Motility343

7.2.3 Rectilinear Diffusion with Variable Boundary Condition344

7.2.4 An Illustration:Temperature Distribution in the Soil345

7.2.5 Radial Diffusion346

7.2.6 An Illustration:Unsteady Fluid Flow in an Aquifer349

7.2.7 Rectilinear Diffusion with Convection352

7.3Diffusion with Reaction in a Finite Region354

7.3.1 Dimensional Analysis354

7.3.2 Exponential Growth in a Finite Region356

7.3.3 Power Law Exponential Growth in a Finite Region361

7.3.4 Logistic Growth in a Finite Region362

7.3.5 An Illustration:Zone of Regulated Fishing366

7.4Diffusion with Convection and Reaction367

7.4.1 The Differential Equation and Its Solution367

7.4.2 Exponential Growth with Convection and Diffusion371

7.4.3 Exponential Decay with Convection and Diffusion372

7.4.4 Convection and Diffusion with Interphase Transfer373

7.4.5 An Illustration:Chemical Solute Removal by Adsorption378

7.5Diffusion with Confined Exponential Growth379

7.5.1 Rectilinear Diffusion379

7.5.2 An Illustration:Heat Transfer from a River380

7.5.3 Radial Diffusion383

7.5.4 An Illustration:Population in Cities385

7.5.5 Temporal-Spatial Diffusion390

7.5.6 An Illustration:Population of London395

7.6Diffusion with Logistic Growth402

7.6.1 Traveling Wave Solutions402

7.6.2 A Power Law Traveling Wave Solution408

7.6.3 An Illustration:Diffusion of Tractor Utilization410

7.6.4 An Illustration:Adoption of Hybrid Corn Revisited414

8Conclusion419

References429

Author Index445

Subject Index451