Fractals and multifractals in ecology and aquatic science

Ecologists sometimes have a less-than-rigorous background in quantitative methods, yet research within this broad field is becoming increasingly mathematical. Written in a step-by-step fashion, Fractals and Multifractals in Ecology and Aquatic Science provides scientists with a basic understanding of fractals and multifractals and the techniques for utilizing them when analyzing ecological phenomenon.

With illustrations, tables, and graphs on virtually every page - several in color - this book is a comprehensive source of state-of-the-art ecological scaling and multiscaling methods at temporal and spatial scales, respectfully ranging from seconds to months and from millimeters to thousands of kilometers. It illustrates most of the data analysis techniques with real case studies often based on original findings. It also incorporates descriptions of current and new numerical techniques to analyze and deepen understanding of ecological situations and their solutions.

Includes a Wealth of Applications and Examples

This book also includes nonlinear analysis techniques and the application of concepts from chaos theory to problems of spatial and temporal patterns in ecological systems. Unlike other books on the subject, Fractals and Multifractals in Ecology and Aquatic Science is readily accessible to researchers in a variety of fields, such as microbiology, biology, ecology, hydrology, geology, oceanography, social sciences, and finance, regardless of their mathematical backgrounds. This volume demystifies the mathematical methods, many of which are often regarded as too complex, and allows the reader to access new and promising concepts, procedures, and related results.

Author Notes

Laurent Seuront is a Professor in Biological Oceanography at the Flinders University (Adelaide, Australia) and a Senior Research Scientist at the South Australian Research and Development Institute (West Beach, Australia). Prior to his present position, he was a research fellow of the Japanese Society for the Promotion of Science at the Tokyo University of Fisheries (1999-2000) and a research scientist at the Centre National de la Recherche Scientifique (CNRS) in France (2001-2008). Among multiple awards, he recently received the CNRS Bronze Medal in France (2007) in recognition of his early career achievements, and a prestigious Australian Professorial Fellowship from the Australian Research Council.

Preface	p. xiii
About the Author	p. xv
1 Introduction	p. 1
2 About Geometries and Dimensions	p. 11
2.1 From Euclidean to Fractal Geometry	p. 11
2.2 Dimensions	p. 16
2.2.1 Euclidean, Topological, and Embedding Dimensions	p. 16
2.2.1.1 Euclidean Dimension	p. 16
2.2.1.2 Topological Dimension	p. 16
2.2.1.3 Embedding Dimension	p. 17
2.2.2 Fractal Dimension	p. 18
2.2.2.1 Fractal Codimension	p. 22
2.2.2.2 Sampling Dimension	p. 23
3 Self-Similar Fractals	p. 25
3.1 Self-Similarity, Power Laws, and the Fractal Dimension	p. 25
3.2 Methods for Self-Similar Fractals	p. 28
3.2.3 Divider Dimension, D d	p. 29
3.2.1.1 Theory	p. 29
3.2.1.2 Case Study: Movement Patterns of the Ocean Sunfish, Mola Mola	p. 32
3.2.1.3 Methodological Considerations	p. 35
3.2.2 Box Dimension, D b	p. 46
3.2.2.1 Theory	p. 46
3.2.2.2 Case Study: Burrow Morphology of the Grapsid Crab, Helograpsus Haswellianus	p. 47
3.2.2.3 Methodological Considerations	p. 51
3.2.2.4 Theoretical Considerations	p. 52
3.2.3 Cluster Dimension, D c	p. 56
3.2.3.1 Theory	p. 56
3.2.3.2 Case Study: The Microscale Distribution of the Amphipod Corophium Arenarium	p. 57
3.2.3.3 Methodological Considerations: Constant Numbers or Constant Radius?	p. 59
3.2.4 Mass Dimension, D m	p. 60
3.2.4.1 Theory	p. 60
3.2.4.2 Case Study: Microscale Distribution of Microphytobenthos Biomass	p. 61
3.2.4.3 Comparing the Mass Dimension D m to Other Fractal Dimensions	p. 65
3.2.5 Information Dimension, D i	p. 66
3.2.5.1 Theory	p. 66
3.2.5.2 Comparing the Information Dimension D i to Other Fractal Dimensions	p. 67
3.2.6 Correlation Dimension, D cor	p. 68
3.2.6.1 Theory	p. 68
3.2.6.2 Comparing the Correlation Dimension D cor to Other Fractal Dimensions	p. 69
3.2.7 Area-Perimeter Dimensions	p. 69
3.2.7.1 Perimeter Dimension, D p	p. 70
3.2.7.2 Area Dimension, D a	p. 72
3.2.7.3 Landscape/Seascape Dimension, D s	p. 72
3.2.7.4 Fractal Dimensions, Areas, and Perimeters	p. 73
3.2.8 Ramification Dimension, D r	p. 87
3.2.8.1 Theory	p. 87
3.2.8.2 Fractal Nature of Growth Patterns	p. 87
3.2.9 Surface Dimensions	p. 92
3.2.9.1 Transect Dimension, D t	p. 93
3.2.9.2 Contour Dimension, D co	p. 94
3.2.9.3 Geostatistical Dimension, D g	p. 95
3.2.9.4 Elevation Dimension, D e	p. 96
4 Self-Affine Fractals	p. 99
4.1 Several Steps toward Self-Affinity	p. 99
4.1.1 Definitions	p. 99
4.1.2 Fractional Brownian Motion	p. 99
4.1.3 Dimension of Self-Affine Fractals	p. 100
4.1.4 1/f Noise, Self-Affinity, and Fractal Dimensions	p. 102
4.1.5 Fractional Brownian Motion, Fractional Gaussian Noise, and Fractal Analysis	p. 103
4.2 Methods for Self-Affine Fractals	p. 106
4.2.1 Power Spectrum Analysis	p. 106
4.2.1.1 Theory	p. 106
4.2.1.2 Spectral Analysis in Aquatic Sciences	p. 108
4.2.1.3 Case Study: Eulerian and Lagrangian Scalar Fluctuations in Turbulent Flows	p. 109
4.2.2 Detrended Fluctuation Analysis	p. 117
4.2.2.1 Theory	p. 117
4.2.2.2 Case Study: Assessing Stress in Interacting Bird Species	p. 119
4.2.3 Scaled Windowed Variance Analysis	p. 124
4.2.3.1 Theory	p. 124
4.2.3.2 Case Study: Temporal Distribution of the Calanoid Copepod, Temora Longicornis	p. 125
4.2.4 Signal Summation Conversion Method	p. 128
4.2.5 Dispersion Analysis	p. 128
4.2.6 Rescaled Range Analysis and the Hurst Dimension, D H	p. 128
4.2.6.1 Theory	p. 128
4.2.6.2 Example: R/S Analysis and River Flushing Rates	p. 131
4.2.7 Autocorrelation Analysis	p. 131
4.2.8 Semivariogram Analysis	p. 133
4.2.8.1 Theory	p. 133
4.2.8.2 Case Study: Vertical Distribution of Phytoplankton in Tidally Mixed Coastal Waters	p. 134
4.2.9 Wavelet Analysis	p. 139
4.2.10 Assessment of Self-Affine Methods	p. 140
4.2.10.1 Comparing Self-Affine Methods	p. 140
4.2.10.2 From Self-Affinity to Intermittent Self-Affinity	p. 143
5 Frequency Distribution Dimensions	p. 147
5.1 Cumulative Distribution Functions and Probability Density Functions	p. 147
5.1.1 Theory	p. 147
5.1.2 Case Study: Motion Behavior of the Intertidal Gastropod, Littorina Littorea	p. 147
5.1.2.1 The Study Organism	p. 147
5.1.2.2 Experimental Procedures and Data Analysis	p. 148
5.1.2.3 Results	p. 149
5.1.2.4 Ecological Interpretation	p. 150
5.2 The Patch-Intensity Dimension, D pi	p. 151
5.3 The Korcak Dimension, D K	p. 153
5.4 Fragmentation and Mass-Size Dimensions, D fr and D ms	p. 154
5.5 Rank-Frequency Dimension, D rf	p. 155
5.5.1 Zipf's Law, Human Communication, and the Principle of Least Effort	p. 155
5.5.2 Zipf's Law, Information, and Entropy	p. 156
5.5.3 From the Zipf Law to the Generalized Zipf Law	p. 158
5.5.4 Generalized Rank-Frequency Diagram for Ecologists	p. 160
5.5.5 Practical Applications of Rank-Frequency Diagrams for Ecologists	p. 161
5.5.5.1 Zipf's Law as a Diagnostic Tool to Assess Ecosystem Complexity	p. 161
5.5.5.2 Case Study: Zipf Laws of Two-Dimensional Patterns	p. 177
5.5.5.3 Distance between Zipf's Laws	p. 188
5.5.6 Beyond Zipf's Law and Entropy	p. 189
5.5.6.1 n-Tuple Zipf's Law	p. 189
5.5.6.2 n-Gram Entropy and n-Gram Redundancy	p. 193
6 Fractal-Related Concepts: Some Clarifications	p. 201
6.1 Fractals and Deterministic Chaos	p. 201
6.1.1 Chaos Theory	p. 201
6.1.2 Feigenbaum Universal Numbers	p. 205
6.1.3 Attractors	p. 205
6.1.3.1 Visualizing Attractors: Packard-Takens Method	p. 206
6.1.3.2 Quantifying Attractors: Diagnostic Methods for Deterministic Chaos	p. 209
6.1.3.3 Case Study: Plankton Distribution in Turbulent Coastal Waters	p. 213
6.1.3.4 Chaos, Attractors, and Fractals	p. 224
6.1.4 Chaos in Ecological Sciences	p. 224
6.1.5 A Few Misconceptions about Chaos	p. 225
6.1.6 Then, What Is Chaos?	p. 225
6.2 Fractals and Self-Organization	p. 226
6.3 Fractals and Self-Organized Criticality	p. 226
6.3.1 Defining Self-Organized Criticality	p. 226
6.3.2 Self-Organized Criticality in Ecology and Aquatic Sciences	p. 229
7 Estimating Dimensions with Confidence	p. 231
7.1 Scaling or Not Scaling? That Is the Question	p. 231
7.1.1 Identifying Scaling Properties	p. 232
7.1.1.1 Procedure 1: R 2 - SSR Procedure	p. 233
7.1.1.2 Procedure 2: Zero-Slope Procedure	p. 234
7.1.1.3 Procedure 3: Compensated-Slope Procedure	p. 238
7.1.2 Scaling, Multiple Scaling, and Multiscaling: Demixing Apples and Oranges	p. 239
7.2 Errors Affecting Fractal Dimension Estimates	p. 241
7.2.1 Geometrical Constraint, Shape Topology, and Digitization Biases	p. 241
7.2.2 Isotropy	p. 243
7.2.3 Stationarity	p. 243
7.2.3.1 Statistical Stationarity	p. 243
7.2.3.2 Fractal Stationarity	p. 244
7.3 Defining the Confidence Limits of Fractal Dimension Estimates	p. 246
7.4 Performing a Correct Analysis	p. 246
7.4.1 Self-Similar Case	p. 247
7.4.2 Self-Affine Case	p. 247
8 From Fractals to Multifractals	p. 249
8.1 A Random Walk toward Multifractality	p. 249
8.1.1 A Qualitative Approach to Multifractality	p. 249
8.1.2 Multifractality So Far	p. 250
8.1.3 From Fractality to Multifractality: Intermittency	p. 253
8.1.3.1 A Bit of History	p. 253
8.1.3.2 Intermittency in Ecology and Aquatic Sciences	p. 253
8.1.3.3 Defining Intermittency	p. 253
8.1.4 Variability, Inhomogeneity, and Heterogeneity: Terminological Considerations	p. 255
8.1.5 Intuitive Multifractals for Ecologists	p. 257
8.2 Methods for Multifractals	p. 260
8.2.1 Generalized Correlation Dimension Function D(q) and the Mass Exponents ¿(q)	p. 260
8.2.1.1 Theory	p. 260
8.2.1.2 Application: Salinity Stress in the Cladoceran Daphniopsis Australis	p. 262
8.2.2 Multifractal Spectrum f(¿)	p. 262
8.2.2.1 Theory	p. 262
8.2.2.2 Application: Temperature Stress in the Calanoid Copepod Temora Longicornis	p. 265
8.2.3 Codimension Function c(¿) and Scaling Moment Function K(q)	p. 265
8.2.4 Structure Function Exponents ¿(q)	p. 268
8.2.4.1 Theory	p. 268
8.2.4.2 Eulerian and Lagrangian Multiscaling Relations for Turbulent Velocity and Passive Scalars	p. 271
8.3 Cascade Models for Intermittency	p. 276
8.3.1 Historical Background	p. 276
8.3.2 Cascade Models for Turbulence	p. 278
8.3.2.1 Lognormal Model	p. 278
8.3.2.2 The Log-Lévy Model	p. 279
8.3.2.3 Log-Poisson Model	p. 280
8.3.3 Assessment of Cascade Models for Passive Scalars in a Turbulent Flow	p. 280
8.4 Multifractals: Misconceptions and Ambiguities	p. 282
8.4.1 Spikes, Intermittency, and Power Spectral Analysis	p. 282
8.4.2 Frequency Distributions and Multifractality	p. 284
8.5 Joint Multifractals	p. 285
8.5.1 Joint Multifractal Measures	p. 285
8.5.2 The Generalized Correlation Functions	p. 287
8.5.2.1 Definition	p. 287
8.5.2.2 Applications	p. 290
8.6 Intermittency and Multifractals: Biological and Ecological Implications	p. 293
8.6.1 Intermittency, Local Dissipation Rates, and Zooplankton Swimming Abilities	p. 294
8.6.2 Intermittency, Local Dissipation Rates, and Biological Fluxes in the Ocean	p. 296
8.6.2.1 Intermittency, Turbulence, and Nutrient Fluxes toward Phytoplankton Cells	p. 297
8.6.2.2 Intermittency, Turbulence, and Physical Coagulation of Phytoplankton Cells	p. 298
8.6.2.3 Intermittency, Turbulence, and Encounter Rates in the Plankton	p. 299
9 Conclusion	p. 301
References	p. 303
Index	p. 337

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