Fractals and Multifractals in Ecology and Aquatic Science

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  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2009-10-12
  • Publisher: CRC Press

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Provides an Intuitive View of Various Ecological Patterns and Processes

Table of Contents

Prefacep. xiii
About the Authorp. xv
Introductionp. 1
About Geometries and Dimensionsp. 11
From Euclidean to Fractal Geometryp. 11
Dimensionsp. 16
Euclidean, Topological, and Embedding Dimensionsp. 16
Euclidean Dimensionp. 16
Topological Dimensionp. 16
Embedding Dimensionp. 17
Fractal Dimensionp. 18
Fractal Codimensionp. 22
Sampling Dimensionp. 23
Self-Similar Fractalsp. 25
Self-Similarity, Power Laws, and the Fractal Dimensionp. 25
Methods for Self-Similar Fractalsp. 28
Divider Dimension, Ddp. 29
Theoryp. 29
Case Study: Movement Patterns of the Ocean Sunfish, Mola Molap. 32
Methodological Considerationsp. 35
Box Dimension, Dbp. 46
Theoryp. 46
Case Study: Burrow Morphology of the Grapsid Crab, Helograpsus Haswellianusp. 47
Methodological Considerationsp. 51
Theoretical Considerationsp. 52
Cluster Dimension, Dcp. 56
Theoryp. 56
Case Study: The Microscale Distribution of the Amphipod Corophium Arenariump. 57
Methodological Considerations: Constant Numbers or Constant Radius?p. 59
Mass Dimension, Dmp. 60
Theoryp. 60
Case Study: Microscale Distribution of Microphytobenthos Biomassp. 61
Comparing the Mass Dimension Dm to Other Fractal Dimensionsp. 65
Information Dimension, Dip. 66
Theoryp. 66
Comparing the Information Dimension Di to Other Fractal Dimensionsp. 67
Correlation Dimension, Dcorp. 68
Theoryp. 68
Comparing the Correlation Dimension Dcor to Other Fractal Dimensionsp. 69
Area-Perimeter Dimensionsp. 69
Perimeter Dimension, Dpp. 70
Area Dimension, Dap. 72
Landscape/Seascape Dimension, Dsp. 72
Fractal Dimensions, Areas, and Perimetersp. 73
Ramification Dimension, Drp. 87
Theoryp. 87
Fractal Nature of Growth Patternsp. 87
Surface Dimensionsp. 92
Transect Dimension, Dtp. 93
Contour Dimension, Dcop. 94
Geostatistical Dimension, Dgp. 95
Elevation Dimension, Dep. 96
Self-Affine Fractalsp. 99
Several Steps toward Self-Affinityp. 99
Definitionsp. 99
Fractional Brownian Motionp. 99
Dimension of Self-Affine Fractalsp. 100
1/f Noise, Self-Affinity, and Fractal Dimensionsp. 102
Fractional Brownian Motion, Fractional Gaussian Noise, and Fractal Analysisp. 103
Methods for Self-Affine Fractalsp. 106
Power Spectrum Analysisp. 106
Theoryp. 106
Spectral Analysis in Aquatic Sciencesp. 108
Case Study: Eulerian and Lagrangian Scalar Fluctuations in Turbulent Flowsp. 109
Detrended Fluctuation Analysisp. 117
Theoryp. 117
Case Study: Assessing Stress in Interacting Bird Speciesp. 119
Scaled Windowed Variance Analysisp. 124
Theoryp. 124
Case Study: Temporal Distribution of the Calanoid Copepod, Temora Longicornisp. 125
Signal Summation Conversion Methodp. 128
Dispersion Analysisp. 128
Rescaled Range Analysis and the Hurst Dimension, DHp. 128
Theoryp. 128
Example: R/S Analysis and River Flushing Ratesp. 131
Autocorrelation Analysisp. 131
Semivariogram Analysisp. 133
Theoryp. 133
Case Study: Vertical Distribution of Phytoplankton in Tidally Mixed Coastal Watersp. 134
Wavelet Analysisp. 139
Assessment of Self-Affine Methodsp. 140
Comparing Self-Affine Methodsp. 140
From Self-Affinity to Intermittent Self-Affinityp. 143
Frequency Distribution Dimensionsp. 147
Cumulative Distribution Functions and Probability Density Functionsp. 147
Theoryp. 147
Case Study: Motion Behavior of the Intertidal Gastropod, Littorina Littoreap. 147
The Study Organismp. 147
Experimental Procedures and Data Analysisp. 148
Resultsp. 149
Ecological Interpretationp. 150
The Patch-Intensity Dimension, Dpip. 151
The Korcak Dimension, DKp. 153
Fragmentation and Mass-Size Dimensions, Dfr and Dmsp. 154
Rank-Frequency Dimension, Drfp. 155
Zipf's Law, Human Communication, and the Principle of Least Effortp. 155
Zipf's Law, Information, and Entropyp. 156
From the Zipf Law to the Generalized Zipf Lawp. 158
Generalized Rank-Frequency Diagram for Ecologistsp. 160
Practical Applications of Rank-Frequency Diagrams for Ecologistsp. 161
Zipf's Law as a Diagnostic Tool to Assess Ecosystem Complexityp. 161
Case Study: Zipf Laws of Two-Dimensional Patternsp. 177
Distance between Zipf's Lawsp. 188
Beyond Zipf's Law and Entropyp. 189
n-Tuple Zipf's Lawp. 189
n-Gram Entropy and n-Gram Redundancyp. 193
Fractal-Related Concepts: Some Clarificationsp. 201
Fractals and Deterministic Chaosp. 201
Chaos Theoryp. 201
Feigenbaum Universal Numbersp. 205
Attractorsp. 205
Visualizing Attractors: Packard-Takens Methodp. 206
Quantifying Attractors: Diagnostic Methods for Deterministic Chaosp. 209
Case Study: Plankton Distribution in Turbulent Coastal Watersp. 213
Chaos, Attractors, and Fractalsp. 224
Chaos in Ecological Sciencesp. 224
A Few Misconceptions about Chaosp. 225
Then, What Is Chaos?p. 225
Fractals and Self-Organizationp. 226
Fractals and Self-Organized Criticalityp. 226
Defining Self-Organized Criticalityp. 226
Self-Organized Criticality in Ecology and Aquatic Sciencesp. 229
Estimating Dimensions with Confidencep. 231
Scaling or Not Scaling? That Is the Questionp. 231
Identifying Scaling Propertiesp. 232
Procedure 1: R2 - SSR Procedurep. 233
Procedure 2: Zero-Slope Procedurep. 234
Procedure 3: Compensated-Slope Procedurep. 238
Scaling, Multiple Scaling, and Multiscaling: Demixing Apples and Orangesp. 239
Errors Affecting Fractal Dimension Estimatesp. 241
Geometrical Constraint, Shape Topology, and Digitization Biasesp. 241
Isotropyp. 243
Stationarityp. 243
Statistical Stationarityp. 243
Fractal Stationarityp. 244
Defining the Confidence Limits of Fractal Dimension Estimatesp. 246
Performing a Correct Analysisp. 246
Self-Similar Casep. 247
Self-Affine Casep. 247
From Fractals to Multifractalsp. 249
A Random Walk toward Multifractalityp. 249
A Qualitative Approach to Multifractalityp. 249
Multifractality So Farp. 250
From Fractality to Multifractality: Intermittencyp. 253
A Bit of Historyp. 253
Intermittency in Ecology and Aquatic Sciencesp. 253
Defining Intermittencyp. 253
Variability, Inhomogeneity, and Heterogeneity: Terminological Considerationsp. 255
Intuitive Multifractals for Ecologistsp. 257
Methods for Multifractalsp. 260
Generalized Correlation Dimension Function D(q) and the Mass Exponents ┐(q)p. 260
Theoryp. 260
Application: Salinity Stress in the Cladoceran Daphniopsis Australisp. 262
Multifractal Spectrum f(┐)p. 262
Theoryp. 262
Application: Temperature Stress in the Calanoid Copepod Temora Longicornisp. 265
Codimension Function c(┐) and Scaling Moment Function K(q)p. 265
Structure Function Exponents ┐(q)p. 268
Theoryp. 268
Eulerian and Lagrangian Multiscaling Relations for Turbulent Velocity and Passive Scalarsp. 271
Cascade Models for Intermittencyp. 276
Historical Backgroundp. 276
Cascade Models for Turbulencep. 278
Lognormal Modelp. 278
The Log-LÚvy Modelp. 279
Log-Poisson Modelp. 280
Assessment of Cascade Models for Passive Scalars in a Turbulent Flowp. 280
Multifractals: Misconceptions and Ambiguitiesp. 282
Spikes, Intermittency, and Power Spectral Analysisp. 282
Frequency Distributions and Multifractalityp. 284
Joint Multifractalsp. 285
Joint Multifractal Measuresp. 285
The Generalized Correlation Functionsp. 287
Definitionp. 287
Applicationsp. 290
Intermittency and Multifractals: Biological and Ecological Implicationsp. 293
Intermittency, Local Dissipation Rates, and Zooplankton Swimming Abilitiesp. 294
Intermittency, Local Dissipation Rates, and Biological Fluxes in the Oceanp. 296
Intermittency, Turbulence, and Nutrient Fluxes toward Phytoplankton Cellsp. 297
Intermittency, Turbulence, and Physical Coagulation of Phytoplankton Cellsp. 298
Intermittency, Turbulence, and Encounter Rates in the Planktonp. 299
Conclusionp. 301
Referencesp. 303
Indexp. 337
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