Seuront L. Fractals and Multifractals in Ecology and Aquatic Science
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FRACTALS AND
MULTIFRACTALS
IN ECOLOGY AND
AQUATIC SCIENCE
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FRACTALS AND
MULTIFRACTALS
IN ECOLOGY AND
AQUATIC SCIENCE
LAURENT SEURONT
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CRC Press
Taylor & Francis Group
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Library of Congress Cataloging-in-Publication Data
Seuront, Laurent.
Fractals and multifractals in ecology and aquatic science / author, Laurent Seuront.
p. cm.
“A CRC title.”
Includes bibliographical references and index.
ISBN 978-0-8493-2782-7 (hardcover : alk. paper)
. Biomathematics. 2. Mathematics in nature. 3. Ecology--Mathematics. 4. Aquatic
sciences--Mathematics. 5. Fractals. 6. Multifractals. I. Title.
QH323.5.S458 2010
570.15’1--dc22 2009025699
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
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Dedication
To my sweetheart, for the past, present, and future.
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vii
Contents
Preface ........................................................................................................................................... xiii
About the Author............................................................................................................................xv
1 Introduction ...............................................................................................................................1
2 About Geometries and Dimensions .......................................................................................11
2.1 From Euclidean to Fractal Geometry ................................................................................... 11
2.2 Dimensions ............................................................................................................................ 16
2.2.1 Euclidean, Topological, and Embedding Dimensions ............................................. 16
2.2.1.1 Euclidean Dimension ............................................................................. 16
2.2.1.2 Topological Dimension .......................................................................... 16
2.2.1.3 Embedding Dimension .......................................................................... 17
2.2.2 Fractal Dimension ................................................................................................... 18
2.2.2.1 Fractal Codimension ..............................................................................22
2.2.2.2 Sampling Dimension ..............................................................................23
3 Self-Similar Fractals ..............................................................................................................25
3.1 Self-Similarity, Power Laws, and the Fractal Dimension .....................................................25
3.2 Methods for Self-Similar Fractals .........................................................................................28
3.2.1 Divider Dimension, D
d
............................................................................................29
3.2.1.1 Theory ....................................................................................................29
3.2.1.2 Case Study: Movement Patterns of the Ocean Sunsh, Mola Mola ......32
3.2.1.3 Methodological Considerations .............................................................35
3.2.2 Box Dimension, D
b
..................................................................................................46
3.2.2.1 Theory ....................................................................................................46
3.2.2.2 Case Study: Burrow Morphology of the Grapsid Crab,
Helograpsus Haswellianus ....................................................................47
3.2.2.3 Methodological Considerations ............................................................. 51
3.2.2.4 Theoretical Considerations ....................................................................52
3.2.3 Cluster Dimension, D
c
.............................................................................................56
3.2.3.1 Theory ....................................................................................................56
3.2.3.2 Case Study: The Microscale Distribution of the Amphipod
Corophium Arenarium ........................................................................... 57
3.2.3.3 Methodological Considerations: Constant Numbers or Constant
Radius? ...................................................................................................59
3.2.4 Mass Dimension, D
m
...............................................................................................60
3.2.4.1 Theory ....................................................................................................60
3.2.4.2 Case Study: Microscale Distribution of Microphytobenthos Biomass ..... 61
3.2.4.3 Comparing the Mass Dimension D
m
to Other Fractal Dimensions .......65
3.2.5 Information Dimension, D
i
.......................................................................................66
3.2.5.1 Theory ....................................................................................................66
3.2.5.2 Comparing the Information Dimension D
i
to Other Fractal Dimensions ..................................................................67
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viii Contents
3.2.6 Correlation Dimension, D
cor
..................................................................................68
3.2.6.1 Theory ....................................................................................................68
3.2.6.2 Comparing the Correlation Dimension D
cor
to Other Fractal
Dimensions ............................................................................................69
3.2.7 Area-Perimeter Dimensions ....................................................................................69
3.2.7.1 Perimeter Dimension, D
p
....................................................................... 70
3.2.7.2 Area Dimension, D
a
...............................................................................72
3.2.7.3 Landscape/Seascape Dimension, D
s
......................................................72
3.2.7.4 Fractal Dimensions, Areas, and Perimeters...........................................73
3.2.8 Ramication Dimension, D
r
....................................................................................87
3.2.8.1 Theory ....................................................................................................87
3.2.8.2 Fractal Nature of Growth Patterns.........................................................87
3.2.9 Surface Dimensions .................................................................................................92
3.2.9.1 Transect Dimension, D
t
........................................................................93
3.2.9.2 Contour Dimension, D
co
.......................................................................94
3.2.9.3 Geostatistical Dimension, D
g
.................................................................95
3.2.9.4 Elevation Dimension, D
e
........................................................................96
4 Self-Afne Fractals .................................................................................................................99
4.1 Several Steps toward Self-Afnity ........................................................................................99
4.1.1 Denitions ...............................................................................................................99
4.1.2 Fractional Brownian Motion ...................................................................................99
4.1.3 Dimension of Self-Afne Fractals .........................................................................100
4.1.4 1/f Noise, Self-Afnity, and Fractal Dimensions ..................................................102
4.1.5 Fractional Brownian Motion, Fractional Gaussian Noise, and
Fractal Analysis ..................................................................................................... 103
4.2 Methods for Self-Afne Fractals ........................................................................................106
4.2.1 Power Spectrum Analysis ......................................................................................106
4.2.1.1 Theory .................................................................................................. 106
4.2.1.2 Spectral Analysis in Aquatic Sciences ................................................ 108
4.2.1.3 Case Study: Eulerian and Lagrangian Scalar Fluctuations in
Turbulent Flows ...................................................................................109
4.2.2 Detrended Fluctuation Analysis ............................................................................ 117
4.2.2.1 Theory .................................................................................................. 117
4.2.2.2 Case Study: Assessing Stress in Interacting Bird Species ................... 119
4.2.3 Scaled Windowed Variance Analysis ....................................................................124
4.2.3.1 Theory ..................................................................................................124
4.2.3.2 Case Study: Temporal Distribution of the Calanoid Copepod,
Temora Longicornis .............................................................................125
4.2.4 Signal Summation Conversion Method .................................................................128
4.2.5 Dispersion Analysis ...............................................................................................128
4.2.6 Rescaled Range Analysis and the Hurst Dimension, D
H
......................................128
4.2.6.1 Theory .................................................................................................. 128
4.2.6.2 Example: R/S Analysis and River Flushing Rates............................... 131
4.2.7 Autocorrelation Analysis ....................................................................................... 131
4.2.8 Semivariogram Analysis ....................................................................................... 133
4.2.8.1 Theory .................................................................................................. 133
4.2.8.2 Case Study: Vertical Distribution of Phytoplankton in Tidally
Mixed Coastal Waters ..........................................................................134
4.2.9 Wavelet Analysis ................................................................................................... 139
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Contents ix
4.2.10 Assessment of Self-Afne Methods ...................................................................... 140
4.2.10.1 Comparing Self-Afne Methods ......................................................... 140
4.2.10.2 From Self-Afnity to Intermittent Self-Afnity .................................. 143
5 Frequency Distribution Dimensions ...................................................................................147
5.1 Cumulative Distribution Functions and Probability Density Functions ............................. 147
5.1.1 Theory ................................................................................................................... 147
5.1.2 Case Study: Motion Behavior of the Intertidal Gastropod,
Littorina Littorea ................................................................................................... 147
5.1.2.1 The Study Organism ............................................................................ 147
5.1.2.2 Experimental Procedures and Data Analysis ...................................... 148
5.1.2.3 Results .................................................................................................. 149
5.1.2.4 Ecological Interpretation .....................................................................150
5.2 The Patch-Intensity Dimension, D
pi
................................................................................... 151
5.3 The Korcak Dimension, D
K
................................................................................................ 153
5.4 Fragmentation and Mass-Size Dimensions, D
fr
and D
ms
....................................................154
5.5 Rank-Frequency Dimension, D
rf
........................................................................................ 155
5.5.1 Zipf’s Law, Human Communication, and the Principle
of Least Effort ....................................................................................................... 155
5.5.2 Zipf’s Law, Information, and Entropy ................................................................... 156
5.5.3 From the Zipf Law to the Generalized Zipf Law .................................................. 158
5.5.4 Generalized Rank-Frequency Diagram for Ecologists .........................................160
5.5.5 Practical Applications of Rank-Frequency Diagrams for Ecologists .................... 161
5.5.5.1 Zipf’s Law as a Diagnostic Tool to
Assess Ecosystem Complexity ............................................................. 161
5.5.5.2 Case Study: Zipf Laws of Two-Dimensional Patterns ......................... 177
5.5.5.3 Distance between Zipf’s Laws ............................................................. 188
5.5.6 Beyond Zipf’s Law and Entropy ............................................................................ 189
5.5.6.1 n-Tuple Zipf’s Law ............................................................................... 189
5.5.6.2 n-Gram Entropy and n-Gram Redundancy ......................................... 193
6 Fractal-Related Concepts: Some Clarications .................................................................201
6.1 Fractals and Deterministic Chaos .......................................................................................201
6.1.1 Chaos Theory ........................................................................................................201
6.1.2 Feigenbaum Universal Numbers ...........................................................................205
6.1.3 Attractors ...............................................................................................................205
6.1.3.1 Visualizing Attractors: Packard-Takens Method .................................206
6.1.3.2 Quantifying Attractors: Diagnostic Methods for Deterministic
Chaos....................................................................................................209
6.1.3.3 Case Study: Plankton Distribution in
Turbulent Coastal Waters ..................................................................... 213
6.1.3.4 Chaos, Attractors, and Fractals ............................................................224
6.1.4 Chaos in Ecological Sciences ................................................................................224
6.1.5 A Few Misconceptions about Chaos .....................................................................225
6.1.6 Then, What Is Chaos?............................................................................................225
6.2 Fractals and Self-Organization ...........................................................................................226
6.3 Fractals and Self-Organized Criticality ..............................................................................226
6.3.1 Dening Self-Organized Criticality ......................................................................226
6.3.2 Self-Organized Criticality in Ecology and Aquatic Sciences ...............................229
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