Fractals and Multifractals in Ecology and Aquatic Science

by Seuront; Laurent

• ISBN13: 9780849327827
• ISBN10: 0849327822
• Format: Hardcover
• Copyright: 2009-10-12
• Publisher: CRC Press

Summary

Provides an Intuitive View of Various Ecological Patterns and Processes

Table of Contents

Preface	p. xiii
About the Author	p. xv
Introduction	p. 1
About Geometries and Dimensions	p. 11
From Euclidean to Fractal Geometry	p. 11
Dimensions	p. 16
Euclidean, Topological, and Embedding Dimensions	p. 16
Euclidean Dimension	p. 16
Topological Dimension	p. 16
Embedding Dimension	p. 17
Fractal Dimension	p. 18
Fractal Codimension	p. 22
Sampling Dimension	p. 23
Self-Similar Fractals	p. 25
Self-Similarity, Power Laws, and the Fractal Dimension	p. 25
Methods for Self-Similar Fractals	p. 28
Divider Dimension, Dd	p. 29
Theory	p. 29
Case Study: Movement Patterns of the Ocean Sunfish, Mola Mola	p. 32
Methodological Considerations	p. 35
Box Dimension, Db	p. 46
Theory	p. 46
Case Study: Burrow Morphology of the Grapsid Crab, Helograpsus Haswellianus	p. 47
Methodological Considerations	p. 51
Theoretical Considerations	p. 52
Cluster Dimension, Dc	p. 56
Theory	p. 56
Case Study: The Microscale Distribution of the Amphipod Corophium Arenarium	p. 57
Methodological Considerations: Constant Numbers or Constant Radius?	p. 59
Mass Dimension, Dm	p. 60
Theory	p. 60
Case Study: Microscale Distribution of Microphytobenthos Biomass	p. 61
Comparing the Mass Dimension Dm to Other Fractal Dimensions	p. 65
Information Dimension, Di	p. 66
Theory	p. 66
Comparing the Information Dimension Di to Other Fractal Dimensions	p. 67
Correlation Dimension, Dcor	p. 68
Theory	p. 68
Comparing the Correlation Dimension Dcor to Other Fractal Dimensions	p. 69
Area-Perimeter Dimensions	p. 69
Perimeter Dimension, Dp	p. 70
Area Dimension, Da	p. 72
Landscape/Seascape Dimension, Ds	p. 72
Fractal Dimensions, Areas, and Perimeters	p. 73
Ramification Dimension, Dr	p. 87
Theory	p. 87
Fractal Nature of Growth Patterns	p. 87
Surface Dimensions	p. 92
Transect Dimension, Dt	p. 93
Contour Dimension, Dco	p. 94
Geostatistical Dimension, Dg	p. 95
Elevation Dimension, De	p. 96
Self-Affine Fractals	p. 99
Several Steps toward Self-Affinity	p. 99
Definitions	p. 99
Fractional Brownian Motion	p. 99
Dimension of Self-Affine Fractals	p. 100
1/f Noise, Self-Affinity, and Fractal Dimensions	p. 102
Fractional Brownian Motion, Fractional Gaussian Noise, and Fractal Analysis	p. 103
Methods for Self-Affine Fractals	p. 106
Power Spectrum Analysis	p. 106
Theory	p. 106
Spectral Analysis in Aquatic Sciences	p. 108
Case Study: Eulerian and Lagrangian Scalar Fluctuations in Turbulent Flows	p. 109
Detrended Fluctuation Analysis	p. 117
Theory	p. 117
Case Study: Assessing Stress in Interacting Bird Species	p. 119
Scaled Windowed Variance Analysis	p. 124
Theory	p. 124
Case Study: Temporal Distribution of the Calanoid Copepod, Temora Longicornis	p. 125
Signal Summation Conversion Method	p. 128
Dispersion Analysis	p. 128
Rescaled Range Analysis and the Hurst Dimension, DH	p. 128
Theory	p. 128
Example: R/S Analysis and River Flushing Rates	p. 131
Autocorrelation Analysis	p. 131
Semivariogram Analysis	p. 133
Theory	p. 133
Case Study: Vertical Distribution of Phytoplankton in Tidally Mixed Coastal Waters	p. 134
Wavelet Analysis	p. 139
Assessment of Self-Affine Methods	p. 140
Comparing Self-Affine Methods	p. 140
From Self-Affinity to Intermittent Self-Affinity	p. 143
Frequency Distribution Dimensions	p. 147
Cumulative Distribution Functions and Probability Density Functions	p. 147
Theory	p. 147
Case Study: Motion Behavior of the Intertidal Gastropod, Littorina Littorea	p. 147
The Study Organism	p. 147
Experimental Procedures and Data Analysis	p. 148
Results	p. 149
Ecological Interpretation	p. 150
The Patch-Intensity Dimension, Dpi	p. 151
The Korcak Dimension, DK	p. 153
Fragmentation and Mass-Size Dimensions, Dfr and Dms	p. 154
Rank-Frequency Dimension, Drf	p. 155
Zipf's Law, Human Communication, and the Principle of Least Effort	p. 155
Zipf's Law, Information, and Entropy	p. 156
From the Zipf Law to the Generalized Zipf Law	p. 158
Generalized Rank-Frequency Diagram for Ecologists	p. 160
Practical Applications of Rank-Frequency Diagrams for Ecologists	p. 161
Zipf's Law as a Diagnostic Tool to Assess Ecosystem Complexity	p. 161
Case Study: Zipf Laws of Two-Dimensional Patterns	p. 177
Distance between Zipf's Laws	p. 188
Beyond Zipf's Law and Entropy	p. 189
n-Tuple Zipf's Law	p. 189
n-Gram Entropy and n-Gram Redundancy	p. 193
Fractal-Related Concepts: Some Clarifications	p. 201
Fractals and Deterministic Chaos	p. 201
Chaos Theory	p. 201
Feigenbaum Universal Numbers	p. 205
Attractors	p. 205
Visualizing Attractors: Packard-Takens Method	p. 206
Quantifying Attractors: Diagnostic Methods for Deterministic Chaos	p. 209
Case Study: Plankton Distribution in Turbulent Coastal Waters	p. 213
Chaos, Attractors, and Fractals	p. 224
Chaos in Ecological Sciences	p. 224
A Few Misconceptions about Chaos	p. 225
Then, What Is Chaos?	p. 225
Fractals and Self-Organization	p. 226
Fractals and Self-Organized Criticality	p. 226
Defining Self-Organized Criticality	p. 226
Self-Organized Criticality in Ecology and Aquatic Sciences	p. 229
Estimating Dimensions with Confidence	p. 231
Scaling or Not Scaling? That Is the Question	p. 231
Identifying Scaling Properties	p. 232
Procedure 1: R2 - SSR Procedure	p. 233
Procedure 2: Zero-Slope Procedure	p. 234
Procedure 3: Compensated-Slope Procedure	p. 238
Scaling, Multiple Scaling, and Multiscaling: Demixing Apples and Oranges	p. 239
Errors Affecting Fractal Dimension Estimates	p. 241
Geometrical Constraint, Shape Topology, and Digitization Biases	p. 241
Isotropy	p. 243
Stationarity	p. 243
Statistical Stationarity	p. 243
Fractal Stationarity	p. 244
Defining the Confidence Limits of Fractal Dimension Estimates	p. 246
Performing a Correct Analysis	p. 246
Self-Similar Case	p. 247
Self-Affine Case	p. 247
From Fractals to Multifractals	p. 249
A Random Walk toward Multifractality	p. 249
A Qualitative Approach to Multifractality	p. 249
Multifractality So Far	p. 250
From Fractality to Multifractality: Intermittency	p. 253
A Bit of History	p. 253
Intermittency in Ecology and Aquatic Sciences	p. 253
Defining Intermittency	p. 253
Variability, Inhomogeneity, and Heterogeneity: Terminological Considerations	p. 255
Intuitive Multifractals for Ecologists	p. 257
Methods for Multifractals	p. 260
Generalized Correlation Dimension Function D(q) and the Mass Exponents ¿(q)	p. 260
Theory	p. 260
Application: Salinity Stress in the Cladoceran Daphniopsis Australis	p. 262
Multifractal Spectrum f(¿)	p. 262
Theory	p. 262
Application: Temperature Stress in the Calanoid Copepod Temora Longicornis	p. 265
Codimension Function c(¿) and Scaling Moment Function K(q)	p. 265
Structure Function Exponents ¿(q)	p. 268
Theory	p. 268
Eulerian and Lagrangian Multiscaling Relations for Turbulent Velocity and Passive Scalars	p. 271
Cascade Models for Intermittency	p. 276
Historical Background	p. 276
Cascade Models for Turbulence	p. 278
Lognormal Model	p. 278
The Log-Lévy Model	p. 279
Log-Poisson Model	p. 280
Assessment of Cascade Models for Passive Scalars in a Turbulent Flow	p. 280
Multifractals: Misconceptions and Ambiguities	p. 282
Spikes, Intermittency, and Power Spectral Analysis	p. 282
Frequency Distributions and Multifractality	p. 284
Joint Multifractals	p. 285
Joint Multifractal Measures	p. 285
The Generalized Correlation Functions	p. 287
Definition	p. 287
Applications	p. 290
Intermittency and Multifractals: Biological and Ecological Implications	p. 293
Intermittency, Local Dissipation Rates, and Zooplankton Swimming Abilities	p. 294
Intermittency, Local Dissipation Rates, and Biological Fluxes in the Ocean	p. 296
Intermittency, Turbulence, and Nutrient Fluxes toward Phytoplankton Cells	p. 297
Intermittency, Turbulence, and Physical Coagulation of Phytoplankton Cells	p. 298
Intermittency, Turbulence, and Encounter Rates in the Plankton	p. 299
Conclusion	p. 301
References	p. 303
Index	p. 337
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