Fractals and multifractals in ecology and aquatic science

366 Pages·2010·10.91 MB·English

by  Laurent Seuront

FRACTALS
AND
MULTIFRACTALS
ECOLOGY
IN AND
AQUATIC SCIENCE
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FRACTALS
AND
MULTIFRACTALS
ECOLOGY
IN 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
http://www.crcpress.com
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Dedication
To my sweetheart, for the past, present, and future.
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Contents
Preface ...........................................................................................................................................xiii
About the Author............................................................................................................................xv
1 Introduction ...............................................................................................................................1
2 About Geometries and Dimensions .......................................................................................11
2.1. From Euclidean to Fractal Geometry ...................................................................................1.1.
2.2 Dimensions ............................................................................................................................1.6
2.2.1. Euclidean, Topological, and Embedding Dimensions .............................................1.6
2.2.1..1. Euclidean Dimension .............................................................................1.6
2.2.1..2 Topological Dimension ..........................................................................1.6
2.2.1..3. Embedding Dimension ..........................................................................1.7
2.2.2 Fractal Dimension ...................................................................................................1.8
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 ............................................................................................29
d
3..2.1..1. Theory ....................................................................................................29
3..2.1..2 Case Study: Movement Patterns of the Ocean Sunfish, Mola Mola ......3.2
3..2.1..3. Methodological Considerations .............................................................3.5
3..2.2 Box Dimension, D ..................................................................................................46
b
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 .............................................................................................56
c
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 ...............................................................................................60
m
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 to Other Fractal Dimensions .......65
m
3..2.5 Information Dimension, D .......................................................................................66
i
3..2.5.1. Theory ....................................................................................................66
3..2.5.2 Comparing the Information Dimension D
i
to Other Fractal Dimensions ..................................................................67
vii
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viii Contents
3..2.6 Correlation Dimension, D ..................................................................................68
cor
3..2.6.1. Theory ....................................................................................................68
3..2.6.2 Comparing the Correlation Dimension D to Other Fractal
cor
Dimensions ............................................................................................69
3..2.7 Area-Perimeter Dimensions ....................................................................................69
3..2.7.1. Perimeter Dimension, D .......................................................................70
p
3..2.7.2 Area Dimension, D ...............................................................................72
a
3..2.7.3. Landscape/Seascape Dimension, D ......................................................72
s
3..2.7.4 Fractal Dimensions, Areas, and Perimeters...........................................73.
3..2.8 Ramification Dimension, D ....................................................................................87
r
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 ........................................................................93.
t
3..2.9.2 Contour Dimension, D .......................................................................94
co
3..2.9.3. Geostatistical Dimension, D .................................................................95
g
3..2.9.4 Elevation Dimension, D ........................................................................96
e
4 Self-Affine Fractals .................................................................................................................99
4.1. Several Steps toward Self-Affinity ........................................................................................99
4.1..1. Definitions ...............................................................................................................99
4.1..2 Fractional Brownian Motion ...................................................................................99
4.1..3. Dimension of Self-Affine Fractals .........................................................................1.00
4.1..4 1./f Noise, Self-Affinity, and Fractal Dimensions ..................................................1.02
4.1..5 Fractional Brownian Motion, Fractional Gaussian Noise, and
Fractal Analysis .....................................................................................................1.03.
4.2 Methods for Self-Affine Fractals ........................................................................................1.06
4.2.1. Power Spectrum Analysis ......................................................................................1.06
4.2.1..1. Theory ..................................................................................................1.06
4.2.1..2 Spectral Analysis in Aquatic Sciences ................................................1.08
4.2.1..3. Case Study: Eulerian and Lagrangian Scalar Fluctuations in
Turbulent Flows ...................................................................................1.09
4.2.2 Detrended Fluctuation Analysis ............................................................................1.1.7
4.2.2.1. Theory ..................................................................................................1.1.7
4.2.2.2 Case Study: Assessing Stress in Interacting Bird Species ...................1.1.9
4.2.3. Scaled Windowed Variance Analysis ....................................................................1.24
4.2.3..1. Theory ..................................................................................................1.24
4.2.3..2 Case Study: Temporal Distribution of the Calanoid Copepod,
Temora Longicornis .............................................................................1.25
4.2.4 Signal Summation Conversion Method .................................................................1.28
4.2.5 Dispersion Analysis ...............................................................................................1.28
4.2.6 Rescaled Range Analysis and the Hurst Dimension, D ......................................1.28
H
4.2.6.1. Theory ..................................................................................................1.28
4.2.6.2 Example: R/S Analysis and River Flushing Rates...............................1.3.1.
4.2.7 Autocorrelation Analysis .......................................................................................1.3.1.
4.2.8 Semivariogram Analysis .......................................................................................1.3.3.
4.2.8.1. Theory ..................................................................................................1.3.3.
4.2.8.2 Case Study: Vertical Distribution of Phytoplankton in Tidally
Mixed Coastal Waters ..........................................................................1.3.4
4.2.9 Wavelet Analysis ...................................................................................................1.3.9
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Contents ix
4.2.1.0 Assessment of Self-Affine Methods ......................................................................1.40
4.2.1.0.1. Comparing Self-Affine Methods .........................................................1.40
4.2.1.0.2 From Self-Affinity to Intermittent Self-Affinity ..................................1.43.
5 Frequency Distribution Dimensions ...................................................................................147
5.1. Cumulative Distribution Functions and Probability Density Functions .............................1.47
5.1..1. Theory ...................................................................................................................1.47
5.1..2 Case Study: Motion Behavior of the Intertidal Gastropod,
Littorina Littorea ...................................................................................................1.47
5.1..2.1. The Study Organism ............................................................................1.47
5.1..2.2 Experimental Procedures and Data Analysis ......................................1.48
5.1..2.3. Results ..................................................................................................1.49
5.1..2.4 Ecological Interpretation .....................................................................1.50
5.2 The Patch-Intensity Dimension, D ...................................................................................1.51.
pi
5.3. The Korcak Dimension, D ................................................................................................1.53.
K
5.4 Fragmentation and Mass-Size Dimensions, D and D ....................................................1.54
fr ms
5.5 Rank-Frequency Dimension, D ........................................................................................1.55
rf
5.5.1. Zipf’s Law, Human Communication, and the Principle
of Least Effort .......................................................................................................1.55
5.5.2 Zipf’s Law, Information, and Entropy ...................................................................1.56
5.5.3. From the Zipf Law to the Generalized Zipf Law ..................................................1.58
5.5.4 Generalized Rank-Frequency Diagram for Ecologists .........................................1.60
5.5.5 Practical Applications of Rank-Frequency Diagrams for Ecologists ....................1.61.
5.5.5.1. Zipf’s Law as a Diagnostic Tool to
Assess Ecosystem Complexity .............................................................1.61.
5.5.5.2 Case Study: Zipf Laws of Two-Dimensional Patterns .........................1.77
5.5.5.3. Distance between Zipf’s Laws .............................................................1.88
5.5.6 Beyond Zipf’s Law and Entropy ............................................................................1.89
5.5.6.1. n-Tuple Zipf’s Law ...............................................................................1.89
5.5.6.2 n-Gram Entropy and n-Gram Redundancy .........................................1.93.
6 Fractal-Related Concepts: Some Clarifications .................................................................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 .....................................................................21.3.
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. Defining Self-Organized Criticality ......................................................................226
6.3..2 Self-Organized Criticality in Ecology and Aquatic Sciences ...............................229
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