Seuront L. Fractals and Multifractals in Ecology and Aquatic Science

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174 Fractals and Multifractals in Ecology and Aquatic Science

a log-log plot, exhibits the different characteristic features previously identied: a power-law behav-

ior not signicantly different from the original one (that is, a′ ≈ a = 0.24) followed by a roll-off

toward low Y

values (Figure 5.20). As stated above, successive positive random uctuations might

lead to local increasing trends slightly diverging from a power law (gray arrow) but should not be

associated with a step function.

Next, consider a situation where the positive and negative uctuations are purely randomly

driven as:

= X

± kX

(5.33)

and

10r

= X

± eX

(5.34)

where k is a constant (0 ≤ k ≤ 1) and e is a random-noise process, that is, e ∈ [0, 1], whose amplitude

is dened as being a given percentage of the maximum value of X

; k and e are randomly chosen as

being positive or negative. The resulting Zipf signatures of the functions Y

and Y

10r

are shown in

Figure 5.21 as log-log plots. The positive and negative components of Equation (5.33) clearly appear

as separated by a step function (Figure 5.21A). The positive components lead to power laws that are

not signicantly different from the original. A separate analysis of the range of values separated

from the power laws by step functions (arrows; Figure 5.21A) did not show any power-law behavior.

Alternatively, the effects of Equation (5.34) on the initial power-law behavior are the characteristic

downward roll-off signature related to randomness and the uctuations around a power-law behav-

ior that are not signicantly different from the original (Figure 5.21B). To ensure the relevance of

Zipf analysis, as introduced by Equations (5.17) and (5.22), the few data points diverging upward,

or attening, toward the rst-rank values must not be included in the regression analysis aimed at

estimating a. Indeed, the former case describes a distribution marked by the dominance of a few

(ultimately one) “hotspots” that is likely to be chronically undersampled (Seuront et al. 1999), while

the latter case refers to a distribution that has been systematically oversampled. These issues have

100

1 10010 1000

Figure 5.20 Log-log plot signature of the Zipf behavior expected in case of a power law X

(open dots)

competing with random growth and mortality components (Y

= X

+ (e

− e

) X

), where the random processes

and e

have been chosen as e

= 0.75 and e

= 0.25, e

= 0.50 and e

= 0.25, and and e

= 0.75 (from bottom

to top). The dashed lines indicate the power-law behavior of the initial values X

. (Modied from Seuront and

Mitchell, 2008.)

2782.indb 174 9/11/09 12:11:38 PM

Frequency Distribution Dimensions 175

been specically detailed elsewhere in the framework of information theory (Mandelbrot 1953) and

led to the modied version of the generalized Zipf law presented above; see Equation (5.19).

It is now known that the distributions of nutrients, phytoplankton, and zooplankton exhibit differ-

ent levels of persistence (Tsuda 1995; Seuront et al. 1996a, 1996b, 1999, 2002; Seuront and Lagadeuc

2001). In addition, the interplay between the biotic properties of individuals and populations and

abiotic processes produce space-time structures characterized by long-range correlation (that is,

persistence) (Kendall et al. 2000). We consider, nally, a situation where the local concentration of

a phytoplankton population initially driven by a power law (X

∝ r

−0.24

) could be inuenced by a

fractional Brownian motion resulting from the combined effects of local biological (nutrient uptake,

100

1 10010 1000

100

1 10010 1000

10r

Figure 5.21 Log-log plot signature of the Zipf behavior expected in case of a power law X

(open dots)

competing with combined constant random and growth components (Y

= X

± kX

; (A) and combined random

growth and mortality components (Y

= X

± eX

; (B), where the constant k and the noise e have been chosen

as k = 0.25, 0.50, and 0.75, and e = 0.25, 0.50, 0.75, and 100 (from bottom to top). The arrows indicate a step

function (A) and the beginning of a local departure from a pure power law due to successively increasing

random increments (B). The dashed lines indicate the power-law behavior of the initial values X

. (Modied

from Seuront and Mitchell, 2008.)

2782.indb 175 9/11/09 12:11:40 PM

176 Fractals and Multifractals in Ecology and Aquatic Science

inter- and intraspecic competition, grazing pressure, infection) and physical (advection, diffusion,

turbulence) processes following

11r

= X

± fBmX

(5.35)

where fBm is a persistent fractional Brownian motion (see Figure 5.9b) whose amplitude is dened

as being a given percentage of the maximum value of X

, and randomly chosen as being positive

or negative. The resulting Zipf signature (Figure 5.22) exhibits the downward roll-off characteris-

tic of randomness for high rank values, and long-range correlations around a power-law behavior

11r

∝ r

−a

′, with a′ = 0.240 ± 0.005;

x ±SD

) that is signicantly different from the original power

law (X

∝ r

−0.24

, p > 0.05). These long-range correlations exist whatever the values of r but are more

clearly visible for the low values of r, that is, high values of X

in Equation (5.35).

5.5.5.1.6 On the Relevance of Zipf’s Law to Diagnose Ecosystem Complexity

The previous sections illustrate the potential for the seldom-used Zipf’s law to be a powerful tool in

the analysis and the classication of marine and terrestrial ecosystems in the presence of random-

ness, monotonic and periodic trends, internal and external noise, and both biotic and abiotic forc-

ings. Specically, Zipf analysis can be directly and easily applied to any data set without intensive

computational, mathematical, or statistical analysis, and with a minimum amount of calculation.

It can be conducted in a few minutes with most standard software packages, even for a data set

of several thousand data points. The results of Zipf analysis should not, however, be used without

a preliminary visual inspection of the data (an absolute prerequisite in data analysis that is often

neglected, especially by undergraduate students), as they could erroneously be used as a direct index

of patchiness. For instance, a distribution characterized by a patch of 10 high-density data points,

10 randomly or regularly spaced hotspots, or 10 ranked hotspots will return exactly the same Zipf

shape. This issue has also been raised in the framework of power spectrum analysis in marine ecol-

ogy (Franks 2005).

The one-to-one correspondence between Zipf and Pareto distributions analytically derived here

(see Box 5.1) could be regarded as a way to reconcile previous and future works using one or

1.5

1.2

1.3

1.4

1.1

1.0

0.9

0.8

0.0 0.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1.5

1.1

1.2

1.3

1.4

2.01.0 1.5 2.5

Log r

Log Y

11r

Log Y

11r

Log r

Figure 5.22 Log-log plot signature of the Zipf behavior expected in case of a power law X

(open dots) com-

peting with a persistent fractional Brownian motion. (Modied from Seuront and Mitchell, 2008.)

2782.indb 176 9/11/09 12:11:44 PM

Frequency Distribution Dimensions 177

the other technique. The strength of the Zipf framework also has the desirable properties of not

requiring any assumptions about the statistical distribution, regularity of sampling intervals, and

stationarity of the data set that are sometimes absolute prerequisites to some statistical data analy-

sis techniques.

Finally, the Zipf framework can be conveniently used as a tool to identify and classify structures

in marine ecosystems and also to infer the underlying processes that generate the observed pat-

terns. The characteristic shapes introduced above, and most importantly their potential changes in

time and space, make it possible to hypothesize the origin and the ecological implications of such

modications, as well as providing useful insights on what further analysis to conduct and how to

design sampling schemes. For instance, a transect study providing a step function in a Zipf plot (see

Figure 5.9d) will indicate different levels of organization within the same populations, and maybe

different subpopulations, that would require separate analysis or additional sampling. In the specic

case of phytoplankton distribution, it is easy to imagine that mixing or changing nutrient and/or

zooplankton concentrations will alter distribution and intensity to the extent that the characteristic

exponents a and b for a set of data will vary due to natural processes. Thus, a phytoplankton popu-

lation exhibiting a single power-law behavior before wind stratication and investigated temporally

from a xed point might exhibit successive changes (see, for example, Figure 5.17 and Figure 5.18),

and the identication and the classication of the Zipf shapes will then allow one to infer the nature

of the observed changes. In turn, a study mainly focusing on phytoplankton distributions that results

in transitions such as those shown in Figure 5.13 and Figure 5.15 in spring and autumn respectively

may well be modied and adapted to investigate the potential differences in the grazer community.

More generally, if phytoplankton properties such as growth or distribution follow a power law, then

mortality processes such as grazing and lysis may well follow a similar but competing power law, as

hypothesized and illustrated above. Thus, if such power-law behavior can be shown in phytoplank-

ton, the removal of the rst ranks (large values) could be interpreted as an indication of predation.

However, the ubiquity of power laws is not an absolute requirement as many non-power-law pro-

cesses could be involved in the modication of the pure power-law behavior as well as the removal

of the last ranks (low values).

5.5.5.2 case study: zipf laws of two-dimensional Patterns

5.5.5.2.1 Ecological Framework

An extensive amount of work has still to be done to cover spatial gaps between remote sensing,

which provides low-resolution data over large areas, and experimental approaches, which give infor-

mation about local processes but are unable to provide continuous, spatially explicit data over large

areas. This is a particularly salient issue considering the increasing awareness of the heterogeneous

nature of the microscale distribution of nutrient, bacteria, phytoplankton, and microphytobenthos

(Seymour et al. 2000, 2004, 2007, 2008; Seuront and Spilmont 2002; Seuront and Leterme 2006;

Seuront et al. 2002; Waters and Mitchell 2002; Waters et al. 2003). Large-scale and small-scale pat-

terns and processes could thus be reconciled by achieving a full understanding of how the effects

of small-scale and microscale processes on the biology and the ecology of individual organisms

propagate toward larger scales, for example, at the population level.

In this framework, the objective of the present case study is to demonstrate the applicability of

the Zipf method described above (Section 5.5.5.1) to characterize two-dimensional patterns. This

method, investigated theoretically by Seuront and Mitchell (2008) and applied to one-dimensional

data sets (Mitchell and Seuront 2008), does not require any assumptions about the distribution of

the data set or regular sampling intervals, and presents the desirable feature of being extremely

easy to implement. To ensure the generality and the relevance of this work, original data sets of

centimeter-scale spatial distributions of bacterioplankton, phytoplankton, and microphytobenthos

are considered.

2782.indb 177 9/11/09 12:11:44 PM

178 Fractals and Multifractals in Ecology and Aquatic Science

5.5.5.2.2 Experimental Procedures and Data Analysis

5.5.5.2.2.1 Centimeter-Scale (1.2 cm) Bacterioplankton Distribution

Sampling sites. The distribution of bacterial marine populations has been investigated from two

coastal sites, Port Noarlunga and Port River, in the metropolitan area of Adelaide, South Australia.

These sites have been chosen, rst, because their bacterial concentrations have been shown to dif-

fer by one order of magnitude and exhibit different degrees of variability in their total abundance

(Seymour et al. 2000, 2004) and, second, because they are characterized by two different hydrody-

namic and hydrological regimes. Port Noarlunga is an oligotrophic environment, exposed to turbu-

lence induced by waves breaking over a reef, while Port River is located in a eutrophic, sheltered

estuary inuenced by high levels of urban and industrial waste and intermittent ows of freshwa-

ter. Samples were taken from subsurface waters from the end of a coastal pier at Port Noarlunga

(35°09′S, 138°28′E) on March 21, 2002, and from a oating pontoon platform at Port River (34°49′S,

138°30′E) on May 9, 2002.

Microscale sampling. The two-dimensional distribution of bacterial distributions has been

investigated using a sampling device conceptually similar to the millimeter-scale resolution system

extensively described elsewhere (Seymour et al. 2000). The system consists of a 10×10 array of 1 ml

syringes, each separated by a distance of 1.2 cm and set to sample volumes of 100 µl. A messenger

weight would release the sampling mechanism and 100 subsamples would simultaneously be taken

across an area of 116 cm

(Seymour et al. 2004). Subsamples were subsequently transferred to 1

ml cryovials and immediately incubated with 2.5% paraformaldehyde for 20 minutes, before being

quick-frozen in liquid nitrogen and subsequently stored at –80°C.

Enumeration of bacterioplankton. Prior to ow cytometric analysis, frozen samples were quick-

thawed and transferred to 5 ml cytometry tubes. Samples were then stained with SYBR-I Green

solution (1:10000 dilution; Molecular Probes), and incubated in the dark for 15 minutes (Marie et al.

1997, 1999). Fluorescent beads of 1 µm diameter (Molecular Probes) were added to samples in a

nal concentration of ca. 10

beads ml

-1

(Gasol and del Giorgio 2000), and all measured cytometry

parameters were normalized to bead concentration and uorescence. After each cytometry session,

working bead solutions were enumerated using epiuorescent microscopy to ensure consistency

of the bead concentration (Gasol and del Giorgio 2000). Samples were analyzed using a Becton

Dickinson FACScan ow cytometer, with phosphate buffered saline (PBS) solution employed as a

sheath uid. For each sample, forward-angle light scatter (FALS), right-angle light scatter (RALS),

green (SYBR-I) uorescence, red uorescence, and orange uorescence were acquired. Acquisition

was run until at least 50 to 100 µl of the sample was analyzed at an approximate rate of 40 µl mn

−1

To avoid coincidence of particles, it was ensured that the rate of analysis was kept below 1000

events sec

−1

by diluting samples with 0.2 µm ltered seawater collected from the study site at time

of sampling when necessary (Gasol and del Giorgio 2000). Data were analyzed and bacterial popu-

lations were identied and enumerated using WinMDI (Scripps Research Institute) and CYTOWIN

(Vaulot 1989) ow cytometry analysis software.

5.5.5.2.2.2 Centimeter-Scale (1.2 cm) Phytoplankton Distribution

Sampling sites. Sampling was conducted on 9 December 2003, from a oating pontoon platform, in

the above described Port River estuary, Adelaide, South Australia (34°49′S, 138°30′E).

Microscale sampling. Two-dimensional samples were collected using the above-mentioned spring-

loaded 10×10 syringe array sampler, set up to simultaneously collect 100 samples of 200 µl each.

During sample collection, the array sampler was oriented vertically, 10 cm below the water surface

with the syringe inlets facing upstream. Sampling consisted of four sets of 100 samples, collected in

succession, at a time interval of 10 minutes, and referred to as P1, P2, P3, and P4 hereafter. At the

completion of sampling, syringe contents were subsampled (150 µl), transferred to cryovials with 2%

nal concentration paraformaldehyde, frozen in liquid nitrogen, and subsequently stored at −80°C.

2782.indb 178 9/11/09 12:11:45 PM

Frequency Distribution Dimensions 179

Enumeration of phytoplankton. Total phytoplankton cell concentrations were estimated using

a FACScan ow cytometer (Becton Dickinson) at the Flinders Medical Centre of South Australia.

The nozzle diameter of the ow cytometer was 70 µm, which was taken to be the maximum size of

cells enumerated. Samples were quick-thawed and analyzed at a rate of approximately 20 µl min

−1

employing PBS as sheath uid. For each sample, natural orange uorescence (from phycoerythrin)

and red uorescence (from chlorophyll), together with FALS and RALS parameters, were recorded

on three decade logarithmic scales, sorted in list mode, and analyzed using CYTOWIN custom-

designed software (Vaulot 1989). All parameters were normalized to a known concentration of 1

µm uorescent marker beads (Molecular Probes), which were added to the sample prior to analysis

at a nal concentration of ca. 10

beads ml

−1

5.5.5.2.2.3 Centimeter-Scale (6.6 cm) Microphytobenthos Distribution

Sampling sites. The two study sites, located on the French coast of the eastern English Channel, were

chosen because of their intrinsic sharp differences in terms of hydrodynamic exposure, sediment

nature, and biotic properties.

The rst study site, an intertidal at of sand in Wimereux (50°45′896 N, 1°36′364 E), is typi-

cal of the hydrodynamically sandy beach habitats that dominate the littoral zone along the French

coast of the eastern English Channel. Measurements were performed on a at area located in the

upper intertidal zone, without sharp topographical features such as ripple marks, high pinnacles, or

deep surge channels. The substrate was homogeneous, medium-size sand (200 to 250 µm, modal

size), typical of the surrounding sandy habitat. Because of the substrate homogeneity and the weak

biomass, productivity and production of phyto- and zoobenthic organisms, the microphytobenthos

biomass distribution is a priori expected to be rather homogeneous (Seuront and Spilmont 2002). In

addition, due to the highly dynamic environment, microphytobenthos is resuspended, and surface

concentrations at low tide are low.

The second study site is located in the Bay of Somme, at Le Crotoy (50°13′524 N, 1°36′506 E),

which is the second-largest estuarine system in France, after the Seine estuary, and the largest

sandy-muddy (72 km²) intertidal area on the French coasts of the eastern English Channel. The

sampling site was chosen in a topographically homogeneous area, where the substrate grain size

typically varied between 125 and 250 µm (modal size), and is characterized by higher phyto-

and zoobenthos biomass, activity, and spatial heterogeneity when compared to the Wimereux site

(Seuront and Leterme 2006). Because of the weak hydrodynamic conditions, the microphytoben-

thos biomass is only weakly inuenced by resuspension processes, and surface concentrations at

low tide are high.

Microscale sampling. All measurements were performed at low tide, on October 9 and 10, 2003,

at the Wimereux and Bay of Somme study sites, respectively. The two-dimensional spatial distri-

bution of microphytobenthos was investigated for scales smaller than 1 m

, which is usually the

nest grain considered in both landscape ecology (He et al. 1994) and intertidal benthic ecology

(MacIntyre et al. 1996; Blanchard and Bourget 1999). A rigid 1 m

aluminum quadrant was used,

and 225 sediment samples were collected every 6.67 cm using 1.9 cm

plastic cores. The cores were

pushed into the sediment down to a depth of 1 cm, where the most of the active cells are concen-

trated, carefully removed, mixed with 5 ml of methanol, and then stored in an insulated, cool box,

brought back to the laboratory, and stored in the dark at –20°C.

Chlorophyll content analysis. Five ml of methanol were added directly to the sampled sediment

sections, and the chlorophyll content was assayed in a Turner 450 uorometer previously calibrated

with a pure chlorophyll a solution (Anacystis nidulans extract, Sigma Chemicals) after an extraction

time of 4 hours. Chlorophyll a concentrations in the sediment sections were then converted in terms

of Chl.a m

−2

, taking into account the surface (1.9 cm

) of the sampling units.

2782.indb 179 9/11/09 12:11:46 PM

180 Fractals and Multifractals in Ecology and Aquatic Science

5.5.5.2.3 Results

5.5.5.2.3.1 Centimeter-Scale Bacterioplankton Distribution

Zipf analyses clearly show that the two-dimensional bacterial distributions (Figure 5.23) are not

uniformly distributed (Figure 5.24). The Zipf plots estimated for bacterial populations sampled at

Port Noarlunga and Port River exhibit a linear behavior with a = 0.05 (r

= 0.98) for bacterial

concentrations ranging from 3.1 × 10

to 3.7 × 10

cell ml

−1

(Figure 5.24A), and with a = 0.03

= 0.98) for bacterial concentrations ranging from 37.7 × 10

to 39.8 × 10

cell ml

−1

(Figure 5.24B).

Respectively, 33% and 28% of the values observed in Port Noarlunga and Port River are included

in the identied power-law behaviors. While the Port Noarlunga Zipf plot exhibits a power-law

behavior up to the highest bacterial concentrations (Figure 5.24A), note that the three highest con-

centrations (that is, the three rst ranks) observed in Port River were not included in the regression

analysis (Figure 5.24B). As discussed on the basis of Zipf analyses of simulated data (Seuront and

Mitchell 2008), such a local increasing trend is intrinsically caused by random uctuations and

should be regarded as a source of contamination of the observed power law resulting in a distribu-

tion dominated by a few hotspots rather than a breakpoint indicative of structural discontinuities

(see Section 5.5.5.1). A proper normalization further shows that for concentrations lower than 0.37

× 10

cell ml

−1

and 3.98 × 10

cell ml

–1

respectively at Port Noarlunga and Port River, the Zipf plots

are extremely similar (Figure 5.25A), continuously diverging from a power-law behavior as a step

function toward the lowest concentrations (Figure 5.24A,B and Figure 5.25A). Because a step func-

tion might be indicative of the presence of structural discontinuities within the distributions, we

Figure 5.23 Two-dimensional distributions of the bacterioplankton abundance sampled in (A) Port

Noarlunga (×10

cell ml

−

) and (B) Port River (×10

cell ml

−

) with a spatial resolution of 1.2 cm.

2782.indb 180 9/11/09 12:11:49 PM

Frequency Distribution Dimensions 181

performed separate Zipf analyses for the ranges of concentrations separated by identied break-

points (gray arrows in Figure 5.24A,B). The resulting Zipf plots (Figure 5.25B), shown here for con-

centrations lower than 0.37 × 10

−6

cell ml

−1

at Port Noarlunga and 3.98 × 10

−6

cell ml

−1

at Port River,

do not exhibit any power-law behavior, but instead produce a continuous roll-off from a horizontal

line (that is, a → 0) to a vertical line (that is, a → ∞). This is representative of the fact that no value

is more likely to be more common than any other one, a characteristic of uniformity.

5.5.5.2.3.2 Centimeter-Scale Phytoplankton Distribution

The 1.2-cm resolution, two-dimensional phytoplankton distributions obtained from the four “replicate”

experiments conducted in this study (hereafter referred to as P1, P2, P3, and P4) exhibit specic

5.25

0.0 0.3 0.6

α = 0.027; (r

= 0.982)

α = 0.049; (r

= 0.983)

0.9

Log r

1.2 1.5 1.8 2.1

5.30

5.35

5.40

5.45

Log X

5.50

5.55

5.60

6.35

0.0 0.3 0.6 0.9

Log r

1.2 1.5 1.8 2.1

6.40

6.45

6.50

Log X

6.55

6.60

6.65

Figure 5.24 Zipf plots of the bacterioplankton abundance in Port Noarlunga (A) and Port River (B), shown

in log-log plots. The black diamonds correspond to the range of abundance values exhibiting a power-law

behavior, and used to estimate the exponent a as the slope of the linear t maximizing the coefcient of

determination and minimizing the total sum of the residuals in the regression (dotted lines). The continuous

black and gray lines correspond to the Zipf plots obtained from 100 simulated uniform distributions with the

same minimum and maximum values as the empirical ones and from 100 simulated normal distributions with

the same mean and variance as the empirical ones, respectively. The gray arrows indicate the structural break

points in the Zipf distributions.

2782.indb 181 9/11/09 12:11:52 PM

182 Fractals and Multifractals in Ecology and Aquatic Science

–0.6

Log r

Log X

0.0

–0.1

–0.2

–0.3

–0.4

–0.5

–0.6

0.51.0 1.52.0 2.0

0.0 0.5 1.0

α = 0.027; (r

= 0.982)

α = 0.049; (r

= 0.983)

1.5

Log r

2.0 2.5

–0.5

–0.4

–0.3

Log X

–0.2

–0.1

0.0

Figure 5.25 Zipf plots of (A) the normalized bacterioplankton abundance in Port Noarlunga (open rhombs)

and Port River (black rhombs) showing the similarity of their overall shape and (B) the bacterioplankton

abundance that did not exhibit a power-law behavior in (A).

350

340

330

320

300

280

260

240

220

200

180

160

140

120

100

Figure 5.26 Replicate two-dimensional distributions of the phytoplankton abundance (×10

cell ml

−

) sam-

pled in Port River with a spatial resolution of 1.2 cm.

2782.indb 182 9/11/09 12:11:56 PM

Frequency Distribution Dimensions 183

features such as localized gradients, hotspots, and “coldspots” (Figure 5.26) that are not compatible

with a homogeneous or a normal distribution. As previously observed for bacterioplankton distribu-

tions, the Zipf analysis of two-dimensional phytoplankton patterns shows that phytoplankton cells

are not uniformly distributed (p < 0.01) and exhibit two different types of organization (Figure 5.27).

P2 and P4 thus exhibit a linear behavior starting from the highest values, while distributions P1

and P3 present local increasing trends characterizing a distribution dominated by a few hotspots.

Phytoplankton patterns thus exhibit a power-law behavior for cell concentrations ranging from 38.2

× 10

to 31.8 × 10

cell ml

−1

with a = 0.12 (r

= 0.99) for P1 (Figure 5.27A), 35.9 × 10

to 24.3 × 10

cell ml

–1

with a = 0.13 (r

= 0.99) for P2 (Figure 5.27B), 26.7 × 10

to 22.8 × 10

cell ml

−1

with a

= 0.09 (r

= 0.97) for P3 (Figure 5.27C), and 29.8 × 10

to 25.6 × 10

cell ml

−1

with a = 0.06 (r

0.97) for P4 (Figure 5.27D). The percentage of values contributing to the power laws are 20% for

P1, 23% for P2, 36% for P3, and 11% for P4. As stated above, separate analyses were performed for

the ranges of concentrations separated by breakpoints (arrows in Figure 5.27). Except in the case of

the distribution P4 that shows a power law for concentrations ranging from 21.4 × 10

to 25.1 × 10

cell

−1

with a = 0.05 (r

= 0.97), no power laws were observed. The corresponding Zipf plots instead

exhibit a continuous roll-off from a horizontal line (that is, a → 0) to a vertical line (that is, a → ∞),

representative of uniformity (Seuront and Mitchell 2008).

4.0

0.0 0.3 0.6

α = 0.119 (r

= 0.988)

1.2 1.5

0.9

Log r

1.8 2.1

4.2

4.4

4.6

Log X

4.8

5.0

5.2

4.0

3.9

3.8

0.0 0.3 0.6

α = 0.125 (r

= 0.991)

1.2 1.5

0.9

Log r

1.8 2.1

4.1

4.2

4.3

Log X

4.4

4.5

4.6

4.0

0.0 0.3 0.6

α = 0.089 (r

= 0.969)

1.2 1.5

0.9

Log r

1.8 2.1

4.1

4.2

4.3

Log X

4.4

4.5

4.6

4.7C

4.0

0.0 0.3 0.6

α = 0.059 (r

= 0.968)

1.2 1.5

0.9

Log r

1.8 2.1

4.1

4.2

4.3

Log X

4.4

4.5D

Figure 5.27 Zipf plots of the phytoplankton abundance sampled in Port River, shown in log-log plots. The

black diamonds correspond to the range of abundance values exhibiting a power-law behavior, and used to

estimate the exponent a as the slope of the linear t maximizing the coefcient of determination and minimiz-

ing the total sum of the residuals in the regression (dotted lines). The continuous black lines correspond to the

Zipf plots obtained from 100 simulated uniform distributions with the same minimum and maximum values

as the empirical ones. The gray arrows indicate the structural breakpoints in the Zipf distributions.

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