Fractals and Multifractals in Ecology and Aquatic Science

366 Pages·2009·7.29 MB·English

by  Laurent Seuront

FRACTALS
AND
MULTIFRACTALS
ECOLOGY
IN AND
AQUATIC SCIENCE

FRACTALS
AND
MULTIFRACTALS
ECOLOGY
IN AND
AQUATIC SCIENCE
LAURENT SEURONT
CRC Press
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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
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Dedication
To my sweetheart, for the past, present, and future.

Contents
Preface ...........................................................................................................................................xiii
About the Author............................................................................................................................xv
1 Introduction ...............................................................................................................................1
2 About Geometries and Dimensions .......................................................................................11
2.. From Euclidean to Fractal Geometry .....................................................................................
2.2 Dimensions .............................................................................................................................6
2.2.. Euclidean, Topological, and Embedding Dimensions ..............................................6
2.2.... Euclidean Dimension ..............................................................................6
2.2...2 Topological Dimension ...........................................................................6
2.2.... Embedding Dimension ...........................................................................7
2.2.2 Fractal Dimension ....................................................................................................8
2.2.2.. Fractal Codimension ..............................................................................22
2.2.2.2 Sampling Dimension ..............................................................................2.
3 Self-Similar Fractals ..............................................................................................................25
... Self-Similarity, Power Laws, and the Fractal Dimension .....................................................25
..2 Methods for Self-Similar Fractals .........................................................................................28
..2.. Divider Dimension, D ............................................................................................29
d
..2.... Theory ....................................................................................................29
..2...2 Case Study: Movement Patterns of the Ocean Sunish, Mola Mola .......2
..2.... Methodological Considerations ..............................................................5
..2.2 Box Dimension, D ..................................................................................................46
b
..2.2.. Theory ....................................................................................................46
..2.2.2 Case Study: Burrow Morphology of the Grapsid Crab,
Helograpsus Haswellianus ....................................................................47
..2.2.. Methodological Considerations .............................................................5.
..2.2.4 Theoretical Considerations ....................................................................52
..2.. Cluster Dimension, D .............................................................................................56
c
..2.... Theory ....................................................................................................56
..2...2 Case Study: The Microscale Distribution of the Amphipod
Corophium Arenarium ...........................................................................57
..2.... Methodological Considerations: Constant Numbers or Constant
Radius? ...................................................................................................59
..2.4 Mass Dimension, D ...............................................................................................60
m
..2.4.. Theory ....................................................................................................60
..2.4.2 Case Study: Microscale Distribution of Microphytobenthos Biomass .....6.
..2.4.. Comparing the Mass Dimension D to Other Fractal Dimensions .......65
m
..2.5 Information Dimension, D .......................................................................................66
i
..2.5.. Theory ....................................................................................................66
..2.5.2 Comparing the Information Dimension D
i
to Other Fractal Dimensions ..................................................................67
vii
viii Contents
..2.6 Correlation Dimension, D ..................................................................................68
cor
..2.6.. Theory ....................................................................................................68
..2.6.2 Comparing the Correlation Dimension D to Other Fractal
cor
Dimensions ............................................................................................69
..2.7 Area-Perimeter Dimensions ....................................................................................69
..2.7.. Perimeter Dimension, D .......................................................................70
p
..2.7.2 Area Dimension, D ...............................................................................72
a
..2.7.. Landscape/Seascape Dimension, D ......................................................72
s
..2.7.4 Fractal Dimensions, Areas, and Perimeters...........................................7.
..2.8 Ramiication Dimension, D ....................................................................................87
r
..2.8.. Theory ....................................................................................................87
..2.8.2 Fractal Nature of Growth Patterns.........................................................87
..2.9 Surface Dimensions .................................................................................................92
..2.9.. Transect Dimension, D ........................................................................9.
t
..2.9.2 Contour Dimension, D .......................................................................94
co
..2.9.. Geostatistical Dimension, D .................................................................95
g
..2.9.4 Elevation Dimension, D ........................................................................96
e
4 Self-Afine Fractals .................................................................................................................99
4.. Several Steps toward Self-Afinity ........................................................................................99
4.... Deinitions ...............................................................................................................99
4...2 Fractional Brownian Motion ...................................................................................99
4.... Dimension of Self-Afine Fractals ..........................................................................00
4...4 ./f Noise, Self-Afinity, and Fractal Dimensions ...................................................02
4...5 Fractional Brownian Motion, Fractional Gaussian Noise, and
Fractal Analysis ......................................................................................................0.
4.2 Methods for Self-Afine Fractals .........................................................................................06
4.2.. Power Spectrum Analysis .......................................................................................06
4.2.... Theory ...................................................................................................06
4.2...2 Spectral Analysis in Aquatic Sciences .................................................08
4.2.... Case Study: Eulerian and Lagrangian Scalar Fluctuations in
Turbulent Flows ....................................................................................09
4.2.2 Detrended Fluctuation Analysis ..............................................................................7
4.2.2.. Theory ....................................................................................................7
4.2.2.2 Case Study: Assessing Stress in Interacting Bird Species .....................9
4.2.. Scaled Windowed Variance Analysis .....................................................................24
4.2.... Theory ...................................................................................................24
4.2...2 Case Study: Temporal Distribution of the Calanoid Copepod,
Temora Longicornis ..............................................................................25
4.2.4 Signal Summation Conversion Method ..................................................................28
4.2.5 Dispersion Analysis ................................................................................................28
4.2.6 Rescaled Range Analysis and the Hurst Dimension, D .......................................28
H
4.2.6.. Theory ...................................................................................................28
4.2.6.2 Example: R/S Analysis and River Flushing Rates..................................
4.2.7 Autocorrelation Analysis ..........................................................................................
4.2.8 Semivariogram Analysis ..........................................................................................
4.2.8.. Theory .....................................................................................................
4.2.8.2 Case Study: Vertical Distribution of Phytoplankton in Tidally
Mixed Coastal Waters ............................................................................4
4.2.9 Wavelet Analysis .....................................................................................................9
Contents ix
4.2..0 Assessment of Self-Afine Methods .......................................................................40
4.2..0.. Comparing Self-Afine Methods ..........................................................40
4.2..0.2 From Self-Afinity to Intermittent Self-Afinity ...................................4.
5 Frequency Distribution Dimensions ...................................................................................147
5.. Cumulative Distribution Functions and Probability Density Functions ..............................47
5.... Theory ....................................................................................................................47
5...2 Case Study: Motion Behavior of the Intertidal Gastropod,
Littorina Littorea ....................................................................................................47
5...2.. The Study Organism .............................................................................47
5...2.2 Experimental Procedures and Data Analysis .......................................48
5...2.. Results ...................................................................................................49
5...2.4 Ecological Interpretation ......................................................................50
5.2 The Patch-Intensity Dimension, D ....................................................................................5.
pi
5.. The Korcak Dimension, D .................................................................................................5.
K
5.4 Fragmentation and Mass-Size Dimensions, D and D .....................................................54
fr ms
5.5 Rank-Frequency Dimension, D .........................................................................................55
rf
5.5.. Zipf’s Law, Human Communication, and the Principle
of Least Effort ........................................................................................................55
5.5.2 Zipf’s Law, Information, and Entropy ....................................................................56
5.5.. From the Zipf Law to the Generalized Zipf Law ...................................................58
5.5.4 Generalized Rank-Frequency Diagram for Ecologists ..........................................60
5.5.5 Practical Applications of Rank-Frequency Diagrams for Ecologists .....................6.
5.5.5.. Zipf’s Law as a Diagnostic Tool to
Assess Ecosystem Complexity ..............................................................6.
5.5.5.2 Case Study: Zipf Laws of Two-Dimensional Patterns ..........................77
5.5.5.. Distance between Zipf’s Laws ..............................................................88
5.5.6 Beyond Zipf’s Law and Entropy .............................................................................89
5.5.6.. n-Tuple Zipf’s Law ................................................................................89
5.5.6.2 n-Gram Entropy and n-Gram Redundancy ..........................................9.
6 Fractal-Related Concepts: Some Clariications .................................................................201
6.. Fractals and Deterministic Chaos .......................................................................................20.
6.... Chaos Theory ........................................................................................................20.
6...2 Feigenbaum Universal Numbers ...........................................................................205
6.... Attractors ...............................................................................................................205
6...... Visualizing Attractors: Packard-Takens Method .................................206
6.....2 Quantifying Attractors: Diagnostic Methods for Deterministic
Chaos....................................................................................................209
6...... Case Study: Plankton Distribution in
Turbulent Coastal Waters .....................................................................2..
6.....4 Chaos, Attractors, and Fractals ............................................................224
6...4 Chaos in Ecological Sciences ................................................................................224
6...5 A Few Misconceptions about Chaos .....................................................................225
6...6 Then, What Is Chaos?............................................................................................225
6.2 Fractals and Self-Organization ...........................................................................................226
6.. Fractals and Self-Organized Criticality ..............................................................................226
6.... Deining Self-Organized Criticality ......................................................................226
6...2 Self-Organized Criticality in Ecology and Aquatic Sciences ...............................229