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Diffusion and growth phenomena abound in the real world surrounding us. Someexamples: growth of the world's population, growth rates of humans, public interest in news events, growth and decline of central city populations, pollution of rivers, adoption of agricultural innovations, and spreading of epidemics and migration of insects. These and numerous other phenomena are illustrations of typical growth and diffusion problems confronted in many branches of the physical, biological and social sciences as well as in various areas of agriculture, business, education, engineering medicine and public health. The book presents a large number of mathematical models to provide frameworks forthe analysis and display of many of these. The models developed and utilizedcommence with relatively simple exponential, logistic and normal distribution functions. Considerable attention is given to time dependent growth coefficients and carrying capacities. The topics of discrete and distributed time delays, spatial-temporal diffusion and diffusion with reaction are examined. Throughout the book there are a great many numerical examples. In addition and most importantly, there are more than 50 in-depth "illustrations" of the application of a particular framework ormodel based on real world problems. These examples provide the reader with an appreciation of the intrinsic nature of the phenomena involved. They address mainly readers from the physical, biological, and social sciences, as the only mathematical background assumed is elementary calculus. Methods are developed as required, and the reader can thus acquire useful tools for planning, analyzing, designing,and evaluating studies of growth transfer and diffusion phenomena. The book draws on the author's own hands-on experience in problems of environmental diffusion and dispersion, as well as in technology transfer and innovation diffusion.

Diffusion and growth phenomena abound in the real world surrounding us. Someexamples: growth of the world's population, growth rates of humans, public interest in news events, growth and decline of central city populations, pollution of rivers, adoption of agricultural innovations, and spreading of epidemics and migration of insects. These and numerous other phenomena are illustrations of typical growth and diffusion problems confronted in many branches of the physical, biological and social sciences as well as in various areas of agriculture, business, education, engineering medicine and public health. The book presents a large number of mathematical models to provide frameworks forthe analysis and display of many of these. The models developed and utilizedcommence with relatively simple exponential, logistic and normal distribution functions. Considerable attention is given to time dependent growth coefficients and carrying capacities. The topics of discrete and distributed time delays, spatial-temporal diffusion and diffusion with reaction are examined. Throughout the book there are a great many numerical examples. In addition and most importantly, there are more than 50 in-depth "illustrations" of the application of a particular framework ormodel based on real world problems. These examples provide the reader with an appreciation of the intrinsic nature of the phenomena involved. They address mainly readers from the physical, biological, and social sciences, as the only mathematical background assumed is elementary calculus. Methods are developed as required, and the reader can thus acquire useful tools for planning, analyzing, designing,and evaluating studies of growth transfer and diffusion phenomena. The book draws on the author's own hands-on experience in problems of environmental diffusion and dispersion, as well as in technology transfer and innovation diffusion.

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CONTENTS
1 Introduction = 1
2 Some Basic Frameworks = 5
 2.1 Exponential Frameworks = 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 Coefficients = 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 Tornade 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 Curve 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 Trnasfer 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 od 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 the Northeast and East North Central American Cities = 310
 6.4 Distributed Time Delay : Delay Integrla 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
 7.6 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