Seuront L. Fractals and Multifractals in Ecology and Aquatic Science

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194 Fractals and Multifractals in Ecology and Aquatic Science

where p(E

…E

) is the probability of the n-tuple and N = l

is the number of different n-tuples, with

l the number of letters (or symbols) in the alphabet. The zero-order entropy H

= log

N (5.39)

is the number of bits of information required to represent a particular sample of different events

N (for example, letters, words, phonemes, musical notes, or behavioral activities). The rst-order

entropy takes into account the probability of occurrence of each event as:

HpEpE

ii1

=−

∑

()log()

(5.40)

where p(E

) is the probability of event E

. Note that Equation (5.40) is equivalent to the Shannon

information index widely used in ecology to describe species diversity (Rosenzweig 1995); see also

Equation (5.20). The second-order entropy introduces conditional probabilities into the structure of

the stream of events (letters, words, phonemes, musical notes, behavioral activities):

HpEE pEE

ij ij2

=−

∑

()log( )

(5.41)

where p(E

) is the probability of event E

given that event E

has occurred. Similarly, the third-

order entropy of an event E

includes the conditional probability that both events E

and E

have

occurred:

HpEEEpEEE

ijk

ijkijk3

111

=−

===

∑

()log( )

(5.42)

As n-gram entropies, H

is a measure of uncertainty and gives the average amount of information

contained in a word of length n; it comes from Equation (5.38) that the conditional entropies h

(Ebeling and Nicolis 1991, 1992):

= H

n+1

−

(5.43)

where h

= H

gives the average amount of information required to predict the (n + 1)

symbol

when the preceding n symbols are known (Ebeling 1997). Note that h

is decreasing as H

n+1

≤

The slope of the conditional entropy h

plotted against n provides information on the complexity

of languages or, more generally, symbolic sequences. In the presence of long-range correlations,

the entropies h

decrease, hence predictability increases. In contrast, a truly random sequential

system would show a slope of zero. As high-order conditional entropies drop from one order to the

next, less statistical information and more organizational complexity are present. The sharper the

decrease in h

, the more sequential organization in the system. It is then suggested that “entropic

slope” can provide a measure of organizational complexity that can be used to evaluate the impor-

tance of sequential order in the communication systems of different species.

5.5.6.2.2 n-Gram Redundancy

Redundancy is another common feature of languages. Letters or even entire words can be omitted

or changed without the text becoming indecipherable; a text with typing errors does not become

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Frequency Distribution Dimensions 195

unintelligible. The concept of redundancy, which can be estimated from entropy, then also stems

back to the seminal work of Shannon (1948). The n-gram redundancy present in a text or a symbolic

sequence is dened as (Almagor 1985; Mantegna et al. 1994):

=−1

(5.44)

where k = log

l, with l being the number of letters (or symbols) in the alphabet. The redundancy is

a manifestation of the exibility of the underlying language. Note that a sequence of random num-

bers will return values of R

0 (Mantegna et al. 1994).

5.5.6.2.3 Case Study: Zooplankton Behavioral Response to Hydrocarbon Contamination

5.5.6.2.3.1 Ecological Framework

Massive crude oil spills such as Torey Canyon (1967), Amoco Cadiz (1978), Ixtoc-I (1979–1980),

Exxon Valdez (1989), Sea Empress (1996), Erika (1999), and Prestige (2002) are a major source

of polycyclic aromatic hydrocarbons (PAHs) in estuarine and coastal waters (Cachot et al. 2006).

However, leakage from ships, petroleum transport, rening, and intentioned spills are more perni-

cious, but equally important, sources of PAHs in the ocean (Fernandes et al. 1997; Cachot et al.

2006), especially in coastal and shelf waters (Doval et al. 2006).

The effects of PAHs contamination on marine planktonic organisms have been studied exten-

sively in the laboratory and in the eld, and a variety of reactive changes have been found in rela-

tion to incidental oil spills for a range of plankton species, manifested as alterations of biomass,

abundance, and ecophysiological effects. To date, the few studies regarding sublethal effects of

hydrocarbons on copepods show, however, a very variable scenario depending on the chemicals

used, their concentrations, and time of exposure, including anomalous metabolism (Samain et al.

1981), decreased or inhibited feeding (Barata et al. 2002), increased mortality (Gajbhiye et al. 1995),

reduction in egg production (Ott et al. 1978), hatching rates (Cowles and Remillard 1983), and clutch

size (Barata et al. 2005).

At low concentration and for short time exposure, hydrocarbons did not have any signicant

effect on feeding and egg production (Calbet et al. 2007). Decreases in egg production are observed,

however, after long exposures to low hydrocarbon concentrations (Ott et al. 1978), indicating det-

rimental cumulative effects unidentiable under short-term incubations. PAHs concentrations can

reach dramatic concentrations. However, the “natural” concentrations of PAHs typically range

between 1 and 100 μg l

–1

(Doval et al. 2006). The ability to assess rapidly any increase in the

background concentration of PAHs related to, for example, incidental, localized oil spills is then

critical to anticipate their pernicious cumulative effects on copepod biology and ecology.

Since swimming and feeding are intertwined in most copepod species, any disruption of copepod

swimming is predicted to have detrimental consequences to copepod biology and ecology. Despite

the few attempts to use the swimming behavior of the freshwater cladoceran Daphnia sp. as an indi-

cator of exposure to toxic chemicals (Piao et al. 2000; Shimizu et al. 2002), similar information for

marine invertebrates is still very limited (Burlinson and Lawrence 2006). In particular, no attempts

have been made to assess the effects of hydrocarbons on copepod behavior, despite an impressive

body of literature devoted to their behavioral ecology; see, for example, Kiørboe (2008).

In this context, the potential for n-gram entropy and redundancy to detect behavioral changes

relates to exposure to “natural” and dramatic concentrations of naphthalene, the most abundant

hydrocarbon dissolved in oil-contaminated waters.

5.5.6.2.3.2 Toxicity Assay

The polycyclic aromatic hydrocarbon tested was naphthalene, as it has been widely used in toxicologi-

cal assays involving copepods (Calbet et al. 2007) and is one of the most abundant hydrocarbons dis-

solved in oil-contaminated waters. Naphthalene (96% purity, Sigma-Aldrich, St. Louis) stock solutions

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196 Fractals and Multifractals in Ecology and Aquatic Science

were prepared using acetone as a carrier (HPLC grade, 0.5 ml l

−1

). Naphthalene stock solutions were

transferred into the behavioral container and diluted with GF/C ltered (porosity 0.45 µm) in situ

seawater at 50, 100, 500, 1,000, 2,500, 5,000, and 10,000 µg 1

−1

to assess the effect of natural (1

to 100 µg l

−1

) and extreme (up to 10,980 µg l

−1

) PAHs concentrations encountered in the ocean. In

addition to the toxicity experiments, two controls were considered with (0.5 ml l

−1

in GF/C ltered

in situ seawater) and without (GF/C ltered in situ seawater) acetone to assess the potential effect of

acetone on C. hamatus swimming behavior (Box 5.3).

Box 5.3 nAPhthALEnE ContAMInAtIon AnD SyMBoLIC

SEquEnCES oF

CEntRoPAGES hAMAtuS

SWIMMInG BEhAvIoR

Copepods were collected in the coastal waters of the eastern English Channel using a WP2 net

(200-μm mesh size) with closed end horizontally towed between 0 and 5 m. Specimens were

gently diluted in 30-liter isotherm tanks using in situ seawater (S = 34 PSU) and transported

to the laboratory, where Centropages hamatus adult females (Figure 5.B3.1) were sorted by

pipette under a dissecting microscope and acclimated for 12 hours in a 2-liter beaker contain-

ing fresh in situ seawater vacuum ltered through Whatman GF/C glass-ber lters (porosity

0.45 μm) prior to the behavioral experiments. All subsequent handling of animals was done at

18°C in a temperature-controlled room.

Behavioral experiments were conducted in a temperature controlled room (18°C), in aer-

ated (100% air saturation) fresh in situ seawater (S = 34 PSU) in the dark and at night to

avoid any behavioral artifact related to endogenous swimming rhythms. Prior to each experi-

ment, 10 females (1.31 ± 0.05 mm cephalotorax length, mean ± SD) were randomly selected

from the female stock, transferred into the experimental container (a cubical glass container,

15 × 15 × 15 cm) lled up with the test solutions, and allowed to acclimatize for 10 minutes

(Seuront 2006). The free-swimming behavior of C. hamatus was recorded in three dimen-

sions at a rate 25 frame s

−1

using two orthogonal, synchronized infrared digital cameras (DV

Sony DCR-PC120E) facing the experimental container. To avoid any bias related to photot-

ropism, the only light source was provided by six arrays of 72 infrared light emitting diodes

(LEDs). For each test condition, 10 individual females were recorded swimming for 20 minutes,

after which valid video clips were identied for analysis. Valid video clips consisted of path-

ways in which the animals were swimming freely, at least two body lengths away from any

chamber’s walls or the surface of the water.

Free-swimming C. hamatus patterns typically exhibit slow- and fast-swimming bouts

separated by breaking events during which they remain motionless or sink with a horizontal

orientation with tail pointed upward. Fast-swimming bouts were identied as movements lon-

ger than 1 body length within 0.08 second. These four behavioral sequences were associated

Figure 5.b3.1 Calanoid copepod Centropages hamatus; scale bar: 0.2 mm.

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Frequency Distribution Dimensions 197

5.5.6.2.3.3 Results

Centropages hamatus adult females consistently swam in helical loops, and no differences were

perceptible between uncontaminated control seawater (Figure 5.36A) and naphthalene-contami-

nated seawater (Figure 5.36B). More specically, the slow-swimming speed (6.32 ± 0.04 mm s

−1

), fast-

swimming speed (27.3 ± 0.11 mm s

−1

), sinking speed (1.52 ± 0.05 mm s

−1

), and time spent motionless

(1.52 ± 0.05 mm s

−1

) were not signicantly different for the control experiments with and with-

out acetone (p > 0.05). No signicant differences in swimming and sinking speeds were found

between C. hamatus considered in uncontaminated and naphthalene-contaminated seawater

(p > 0.05). In contrast, the time spent swimming exhibits a signicant linear increase (p < 0.01)

with naphthalene concentration from 66% to 81%. This indicates an increase in swimming activ-

ity under conditions of naphthalene contamination and is consistent with previous observations

to four letters, that is, S (slow swimming), F (fast swimming), M (motionless), and D (sinking)

for each video frame.

A total of 270,000 swimming sequences were analyzed, corresponding to ca. 30,000

sequences for each experimental conditions, that is, 7 naphthalene solutions and 2 controls

with (0.5 ml l

–1

in GF/C ltered in situ seawater) and without (GF/C ltered in situ seawater)

acetone to assess the potential effect of acetone on C. hamatus swimming behavior.

150

100

150

100

Z (mm)

Y (mm)

X (mm)

150

100

150

100

Z (mm)

Y (mm)

X (mm)

Figure 5.36 Swimming behavior of the calanoid copepod Centropages hamatus in uncontaminated seawa-

ter (A) and in naphthalene contaminated (1000 μg 1

−1

) seawater (B).

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198 Fractals and Multifractals in Ecology and Aquatic Science

showing an increase in the foraging activity of C. hamatus exposed to microscale turbulence,

a ubiquitous natural stressor of the marine environment. Note, however, that no increase in the

occurrence of fast-swimming events, considered as an escape behavior, was recorded.

The n-gram entropies H

(Equation 5.38) were used to estimate the conditional entropies h

(Equation 5.43) and n-gram redundancies R

(Equation 5.44) for n in the range 1 to 5 (Figure 5.37).

The two controls with and without acetone did not exhibit any signicant differences in the result-

ing behavioral sequence organization. The conditional entropies h

(Figure 5.37A) consistently

decrease with increasing sequence length, indicating correlations and long memory in C. hamatus

behavioral activities. However, this decrease is much sharper for uncontaminated than contami-

nated seawater (Figure 5.37A). This shows that the complexity of the behavioral sequences of C.

hamatus decreases with increasing naphthalene concentrations, thus the predictability of the related

symbolic sequences decreases. In addition, the slopes of the function h

vs. n found for naphthalene-

contaminated seawater were not signicantly different from the zero-slope expected for a truly

random sequential system (that is, Markovian sequences) for n ranging from 1 to 4. Similarly, the

n-gram redundancies R

were consistently higher for uncontaminated seawater than for naphtha-

lene treatments (Figure 5.37B). The swimming sequences of the copepod C. hamatus observed

for uncontaminated seawater then possess a larger amount of structure than those observed for

naphthalene-contaminated water, which are closer to random.

5.5.6.2.3.4 Discussion

The previous results, derived from methods originally designed to investigate the complexity of

natural and articial languages, are consistent with the existence of a structured “behavioral lan-

guage” present in the sequential swimming behavior of the copepod C. hamatus. Both the existence

of long-range correlation in this sequential behavior in uncontaminated seawater and disruption of

the complexity of this language under conditions of naphthalene contamination are consistent with

previous results showing that behavioral time series, though they often appear erratic, reveal 1/f -like

spectra (Quenette and Desportes 1992; Alados et al. 1996). Long-range correlation in biological sys-

tems is thus adaptive because it serves as an organizing principle for highly complex, nonlinear pro-

cesses, and it avoids restricting the functional response of an organism to highly periodic behavior

(Buldyrev et al. 1994). For example, 1/f temporal uctuations are found in the heart rate of healthy

individuals (Meesmann et al. 1993), respiratory intervals in animals (Kawahara et al. 1989), and

neuronal discharges during sleep (Yamamoto et al. 1986). The time series of interbeat intervals in

01234 65

2.00

1.95

1.90

1.85

1.80

012345

Figure 5.37 The conditional entropies h

(A) and n-gram redundancies R

(B) estimated for the sequential

behavior of the copepod Centropages hamatus in uncontaminated seawater (black dots), and in seawater con-

taminated with increasing naphthalene concentrations, that is, 50 (open diamonds), 1000 (gray diamonds), and

10,000 μg 1

−1

(black diamonds). The dashed lines in (A) are the best linear ts of h

vs. n.

2782.indb 198 9/11/09 12:12:36 PM

Frequency Distribution Dimensions 199

healthy subjects have more complex uctuations than patients with severe cardiac disease (Stanley

et al. 1992; Buldyrev et al. 1994). Long-range correlations have been also observed in the stride

interval of human gait (Hausdorff et al. 1995, 1997). This present analysis, conducted on marine

invertebrates, then generalizes previous studies conducted on a range of vertebrates (from sh to

primates) and shows that fractal dimension can detect impairments in the behavior sequences of

individuals under stress, indicating its value in early stress assessment (Alados et al. 1996; Alludes

and Huffman 2000).

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201

Fractal-Related Concepts:

Some Clariﬁcations

6.1 Fractals and deterministic chaos

6.1.1 c

h a o s Th E o r y

Chaos theory has been epitomized by the “buttery effect” detailed by Lorenz (1963), and subse-

quently extensively investigated from popular descriptions (Gleick 1987; Kauffman 1995; Smith

2007) to more rigorous mathematical underpinning and specic algorithms (Baker and Gollub

1990; Peitgen et al. 1992; Turcotte 1992; Çambel 1993; Ott 1993, 2002; Hilborn 1994; Strogatz

1994; Kaplan and Glass 1995; Alligood et al. 1996; Williams 1997; Sprott 2003). In his attempt to

simulate numerically a global weather system, Lorenz discovered that minute changes in initial con-

ditions steered subsequent simulations toward radically different nal states. This dependence on

initial conditions is generally exhibited by systems containing multiple elements in nonlinear inter-

actions, particularly when the system is forced or dissipative. A system is said to be forced when its

internal dynamics are driven by externally supplied energy (for example, solar energy driving the

global weather system). A system is considered dissipative when useful energy* is converted into

a less useful form, most prominently through friction. In aquatic ecology, examples of dissipative

systems are numerous, if not the rule. For instance, in the well-known turbulent energy cascade, the

kinetic energy generated at a large scale by processes such as wind and tide are transferred without

dissipation through the inertial subrange (that is, a hierarchy of eddies of decreasing size) to the

viscous Kolmogorov scale where it is dissipated into heat. On the other hand, the solar energy rst

converted into chemical energy is subsequently transferred from one trophic level to the other one

with considerable losses, responsible for the low energetic efciency of both benthic and pelagic

food chains.

Sensitive dependence on initial conditions is not only observed in complex systems but also in

the simplest logistic equation model in population biology (May 1976). This equation describes

the size of a self-reproducing population, P, at time t + 1 as a nonlinear function of the popula-

tion at time t:

t+1

= P

(a − bP

) (6.1)

Considering the normalized population size x

= bN

/a, Equation (6.1) rewrites as:

t+1

= ax

(1 − x

) (6.2)

or equivalently as:

t+1

= ax

− ax

(6.3)

* Energy able to perform work.

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202 Fractals and Multifractals in Ecology and Aquatic Science

where the driving parameter a is positive, thus ax

represents a linear growth.* When x

is small

(that is, x

→

0), the nonlinear term ax

can be neglected, thus x

t+1

≈ ax

, and the growth of the

population is indeed linear. It then comes that when a > 1, the population should increase.

However, because of the recursive nature of Equation (6.3), ax

becomes signicant as x

increases; rst an inection point occurs and then a steady state (Figure 6.1A) when the two

terms oppose one another because of their opposite signs. In contrast, when a < 1, the popula-

tion will always decrease and eventually becomes extinct regardless of the size of the initial

population (Figure 6.1B). More specically, when a > 1 the value of parameter a determines

whether a population stabilizes at a constant size, oscillates between a limited sequence of

sizes, or behaves erratically in an unpredictable pattern (Figure 6.2).

The dichotomy between those different states can be established using bifurcation diagrams

(Figure 6.3). A bifurcation diagram is a plot of the normalized population size x

as a function of

the values of the driving parameter a. This is illustrated in Figure 6.3 starting with an initial value

of x

(here, x

= 0.05) and the driving parameter a in the range 2.95 < a < 4.00. For low values of

a (that is, a < 3), x

(as t goes to innity) eventually converges to a single number that represents

the population of the species (Figure 6.3A). Now, when a = 3, x

no longer converges but oscillates

* The growth will be steady if a is constant, but we will see that a nonconstant a may lead to counterintuitive results.

0.8

0.6

0.4

0.2

020406080 100

x(t)

0.05

0.04

0.03

0.02

0.01

Figure 6.1 The logistic equation, x

t+1

= ax

(1 − x

), shown for values of ranging from (A) a = 0.50, 0.95, 1.00,

1.50, 2.00, and 3.90 from bottom to top, and (B) from a = 0.50, 0.95, and 1.00 from bottom to top. The initial

value x

was set at x

= 0.05.

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Fractal-Related Concepts: Some Clariﬁcations 203

between two values (Figure 6.3B). This characteristic change in behavior is called a bifurcation.

Turn up the driving parameter even further and x

oscillates between four values (Figure 6.3C),

and further goes through bifurcations of period 8, then 16, and then chaos (Figure 6.3E). When the

value of the driving parameter a equals 3.57, P

neither converges nor oscillates; its value becomes

completely random (Figure 6.3D). For values of r larger than 3.57, the behavior is largely chaotic

(Figure 6.3E,G). However, in the range 3.00 < a < 4.00, there is a particular value of a where the

sequence again oscillates with period of three (a = 3.828427) (Figure 6.3F). The bifurcations then

begin again with period 6, 12, 24, and then back to chaos (Figure 6.3G). Within the chaotic regime

are evident bands or windows containing few points that can result in attractors (Rasband 1990; see

Section 6.1.2). Ultimately, for a ≥ 4, values of x

> 1 can be generated, and therefore x

t+1

< 0; that is,

the equation becomes unphysical (unbiological) beyond this point.

As stressed above, a bifurcation diagram contains regions with singular equilibrium populations

for low values of a, bifurcating into an oscillating population as the parameter a increases and in

turn deteriorating into a chaotic pattern as a reaches a critical value (Strogatz 1994). Within the

chaotic region, however, smaller areas of stable periodicity are discernible (associated with certain

minute ranges of the value of a), and these stable areas appear over and over on every possible scale

of examination. Note that the repetition of this pattern across different scales (Figure 6.4) presents

the intrinsic properties of self-similarity reviewed above, and has thus widely been identied as a

fractal (Turcotte 1992; Strogatz 1994).

1.0

0.5

x(t)

0.0

1.0

0.5

x(t)

0.0

1.0

0.5

x(t)

0.0

1.0

0.5

x(t)

0.0

1.0

0.5

x(t)

0.0

1.0

0.5

x(t)

0.0

Figure 6.2 100 realizations of the logistic equation, x

t+1

= ax

(1 − x

), for values of a ranging from a =

2.000000, 3.261205, and 3.511687 from top to bottom (left), and from a = 3.57494, 3.683735, and 4.000000

from top to bottom (right). The initial value x

was set at x

= 0.05.

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