Y Kamenir - Quasi-stochastic phytoplankton in the sea of the galilee - страница 1

Страницы:
1  2 

МОРСЬКИЙ ЕКОЛОГІЧНИЙ ЖУРНАЛ

УДК 581.526.325:57.016.2

Y. Kamenir, Ph. D, Research Fellow

Faculty of Life Sciences, Bar-Ilan University, Ramat-Gan 52900, Israel[1] TOHU-VA-VOHU OF THE QUASI-STOCHASTIC PHYTOPLANKTON IN THE SEA OF THE GALILEE

The stability analysis of natural communities and ecosystems requires quantitative means capable of evaluating the structural variability of the aquatic community or assemblage taken as a whole. Size spectrum (SS) provides such assessment and is most often applied to the size distribution of particles or organisms of the given community. An­other old, but rarely used, special case of SS describes the size-frequency distribution of taxonomic units encom­passed by an assemblage, and is called here 'taxonomic size spectrum' (TSS). The phytoplankton assemblage of the Sea of the Galilee (Lake Kinneret), Israel, was used to compare several years (out of the 30-year monitoring period) characterized with the most pronounced abnormalities in many biotic and abiotic parameters. Two types of TSS (TTSS and FTSS) revealed some stable patterns of phytoplankton structure. Simple statistical methods (linear regres­sion) produced quantitative similarity estimates of such patterns. Unpredictability in the annual frequency scores was much higher for particular taxonomic units than for the assemblage size classes. We interpret this distinction as the self-regulation success of the natural aquatic community. A possible explanation can be found in our Ideal Minimal Ecosystem model and in the ideas of G. Cuvier and V.I. Vernadskiy, which provided its theoretical foundation.

Key words: phytoplankton, community structure, taxonomic size spectra, composite size spectra, environmental stress, Lake Kinneret

The stability analysis of natural communi­ties and ecosystems calls for quantitative means capable of evaluating the structural variability of the aquatic community or assemblage taken as a whole. Size spectrum (SS) is one of the most ef­fective and common tools recently available to ecologists trying to characterize whole communi­ties. Using body size, mass, or volume of organ­isms, a whole community can be described by a simple formal plot based on a quantitative group­ing parameter [11].

The 'Sheldon' biomass size spectrum (BSS) and normalized biomass size spectrum (NBS) are special cases of SS, where both size and biomass of the organisms in the community are expressed irrespective of their taxonomy.

Another old, but rarely used, special case of SS describes the size-frequency distribution of taxo-nomic units encompassed by an assemblage [2, 4, 5]. In this case, the individual taxa (rather than particles, cells, colonies, or organisms) are treated as units for distribution into size classes. While the term 'species size-frequency distribution' is most often used for such descriptors of community (as­semblage or a higher taxon) structure, we refer to such size distribution of taxonomic units as taxo-nomic size spectrum (TSS). While BSS, and es­pecially NBS, have gained ever-growing attention in recent years, relatively few studies have consid­ered TSS.

Nevertheless, one can note such studies

© .Ю. Каменир, 2005

5

carried out by very prominent specialists, like Hemmingsen [4], Hutchinson, MacArthur [5], Smith et al. [12]. Such species size-distribution patterns observed for many terrestrial [2, 12], pe­lagic, and benthic assemblages [2, 3] seem to be rather stable, in spite of pronounced changes in community composition. Therefore, TSS, a simple and flexible depiction of community taxonomic structure, seems to be a powerful means needed for quantitative measurements and stability analy­ses. Long, frequent, and ever-growing SS applica­tion experience has already evidenced that natural aquatic communities tend to have inherent pat­terns in biomass size spectrum [2, 11, 14]. The identification and description of such 'typical' pat­terns of biomass size distributions in natural communities can be interpreted as evidence of the existence of some stable aspects of the ecosystem, especially its biota structure. These patterns have stimulated the search for the mechanisms generat­ing them [3, 6, 14] and for the quantitative means needed to analyze the pattern non-ideality and its changes under environmental impacts [14]. At the same time, community structure, examined to its taxonomic details, is often highly variable. Skepti­cism regarding the search for stability patterns in natural phytoplankton assemblages is very strong [10], because, at the species level, plankton dy­namics seen in the field, laboratory, and models, are often difficult to predict. Nevertheless, while the seemingly unpredictable behavior is viewed in detail, some aggregated indices such as total algal biomass can be characterized as quite regular [10].

The most popular recent SS types, specifi­cally BSS and NBS, are based on the counting and sizing of particles (cells, colonies, organisms, or­gans, and sub-organism structures). At the same time, size-frequency distributions of species and other taxonomic units have been studied for at least 70 years. They have also evidenced the existence of ubiquitous and very rigid patterns [2, 4, 5, 12]. These patterns also seem to be capable of surviving very strong environmental impacts. Nevertheless, TSS studies have not been frequent up to now.

In this study, we applied the TSS ap­proach to the phytoplankton of the Sea of the Gali­lee (known lately as Lake Kinneret), Israel, which has been famous for at least 2000 years. A four-year period (1996 - 1999), noted for pronounced deviations from the stable patterns established during the earlier 25 years of lake monitoring, was chosen for analysis. This unstable or 'extreme' period was characterized by declining water lev­els, abnormal blooms, variations in phytoplankton taxonomic composition, and irregular phytoplank-ton dynamics (see below). While phytobenthos plays a very insignificant role in both the biomass and primary production of this ecosystem, the phytoplankton data set for 1996 - 1999 can be valuable to analyze a whole autotrophic assem­blage functioning under extreme conditions.

The primary objective of this study was a comparison between SS patterns of Kinneret phytoplankton during a period of extreme instabil­ity. The working hypothesis was that repeating patterns of annual dynamics could evidence some success in community-structure survival.

Materials and Methods. Site descrip­tion. The Sea of the Galilee, Israel, situated ca. 210 m below mean sea level at 32°45'N, 35°30'E, is a warm, monomictic lake with a surface area of 170 km2, a maximum depth of 44 m, and a mean depth of 26 m. The lake has served as the main regional source of fresh water and fisheries since ancient times. Today, it supplies almost half of Israel's drinking water. Therefore, a continuous monitoring of the lake ecosystem, and especially its plankton, has been carried out since 1969. Until the mid-1990s, its phytoplankton was character­ized by distinct stability and was described as one of the best-attested examples of year-to-year simi­larity in seasonal abundance, distribution, and composition [15].

The most salient features of this seasonal-ity were a high biomass winter-spring bloom of the large (cell volume of 60,000 - 130,000 цш3) thecate dinoflagellate Peridinium gatunense, fol­lowed by the summer assemblage of numerous

nanoplanktonic species. Man-induced alterations, including the artificial lowering of the lake's water level, modifications in the catchment area, and active regulation of fish stock structure, followed by a drought that lasted for several years, have led to pronounced changes in phytoplankton taxo-nomic composition and annual bloom patterns (Fig. 1) since the mid-1990s [15].

Phytoplankton data acquisition and processing. As part of the routine monitoring pro­gram, phytoplankton samples were collected at 2-week intervals from 9- 11 discrete depths throughout the water column from a fixed pelagic station at the deepest part of the lake.

Fig. 1. The phytoplankton annual dynamics in Lake Kinneret (known as the Sea of the Galilee). The total assemblage depth-integrated biomass per unit area (gww m -2). A comparison of four annual cycles for a period of pronounced variability of numerous biotic and abiotic parameters. Bxx corresponds to the year number (xx) of 1996 - 1999. Data from [8]. Рис. 1. Годовая динамика фитопланктона в оз. Кин-нерет (известном как Галилейское море). Суммар­ная биомасса (г сырого веса) на м2. Сопоставление четырех годовых циклов периода максимальной нестабильности ряда биотических и абиотических параметров. Вхх соответствует году (хх) 1996 -1999. По [8].

Lugol-preserved samples were brought to the lab and prepared for microscopic counting us­ing the inverted microscope technique.

Phytoplankton were identified and counted according to species, and for the most im­portant species with variable cell size (like Perid-

Морський екологічний журнал, № 4, Т. IV. 2005

inium gatunense), also according to size categories (e.g., small, medium, and large). All species with individual cells having a diameter greater than 2 pm were identified and counted. From the smaller cell range (picoplankton), only common colony-forming cyanobacteria were counted. The sample processing is described in detail by [15]. Some 4000-9000 records of phytoplankton taxonomic units have been added to the Kinneret Laboratory database.

The model. The model we are using, the Ideal Minimal Ecosystem (IMES; Fig. 2), is a hi­erarchical structure of cyclic fluxes and processes implemented through huge numbers of 'flow-through' (i.e., undergoing birth or division, devel­opment, growth, posterior production, death, and resource-regeneration) elements.

Fig. 2. The Ideal Minimal Ecosystem (IMES) model [6]. The hierarchy of the 'flow-through' element recy­cling flow (Ic). The intrinsic losses (Iout) are balanced with the compensatory inflow (Iin). Рис. 2. Идеальная Минимальная Экосистема (ИМЭС) модель [6]. Иерархия потоков рециклинга проточных элементов (Ic). Неизбежные потери (Iout) возмещаются компенсирующим притоком (Iin).

Their stores, inertia and nonlinear charac­teristics produce the buffering properties of the medium and equilibrium of the main parameters. Viewed as a whole, this structure (Fig. 2) seems to be far too complicated for analysis of the dynam­ics of its parts, but quite suitable for application of spectral analysis and statistical analysis of its steady state patterns [6]. The resemblance of this

structure to structures studied by hydro- and aero­dynamics seems to reflect very profound and im­portant properties of the natural ecosystem taken as a whole. This model strongly resembles the 'tourbillon vital of G. Cuvier and the 'living gas' concept of V.I. Vernadskiy, which are discussed in more detail in [7]. IMES is described as a dissipa-tive structure created by and existing only due to the constant flux of energy dissipated as heat. The most effective methods for dealing with such ob­jects are those based on rather sophisticated mathematics, mainly statistics and spectral analy­sis. Statistical parameters of the multitude of ele­ments should be used to estimate the object struc­ture variability [6, 7].

Operational taxonomic unit and size class allocation. The mean cell volume (V, um3) of each species, or of a size category for species counted under several size categories, was calcu­lated from linear microscope measurements and the closest geometrical shape [8, 15]. This cell volume was the parameter used for allocating a taxon to a size class. Since our individual taxa were not strictly species but in some cases also size categories within a species, we refer to each as an 'operational taxonomic unit', or OTUj [13]. Doubling the cell volume, i.e., standard incre­ments of the cell size logarithm created size classes. The smallest size class was up to 0.5 pm3, followed by 0.5 - 1 pm3, etc., to the largest size-class of 131,072 - 262,144 pm3. The LoVxx nota­tion is used throughout this paper for size classes, where xx is the logarithm of the class right bound­ary [e.g., LoV2 means log10V (pm3) = 2, i.e., cell volume from 50 to 100 pm3].

Composite size spectrum (CSS). Let us suppose that the monitoring results produce a set (a database or a matrix) of M records of the form: [x, y, z, t, OTUj, Nj] (1),

where t is the registration time, x, y, z are the spa­tial coordinates of a specific sample, OTUj is the species (here, the OTUj code) characterized by mean cell volume Vj or body mass Wj, and Nj is the OTUj abundance registered on the base of this 8 sample.

Such a data set can be transformed to a

form:

[OTUj, Wx, Wy, Wz, Wt, Nj, Wj] (2),

where Wx, Wy, Wz, and Wt are weight coeffi­cients accounting for the time-space volume repre­sented by this sample, e.g., the space grid scales and the time interval between the samples.

Then it can be compressed to a form: [OTUj, FRj, W4, Njbn, Wjbw] (3),

where W4 is the four-dimensional weight coeffi­cient:

W4 = Wt W3, W3 = Wx Wy Wz (4),

FRj is the frequency rate which represents the num­ber of samples including OTUj during a described time interval, like a year, season, month, etc.

A one-year interval was used in this study. Certainly, Nj can vary from sample to sample, and should be replaced by the average (for the group of FRj records) estimate. The same is true for W4 (W3, Wt) weights if they vary from sample to sample. Here, for simplicity purposes, we assume

that W4=const=1.

Such a data set can be further transformed to a form of size spectrum (SS) if the OTUj re­cords are distributed into size classes:

[FRi, W4, Njbn, Wjbw] (5),

where FRi = I FRj (6), integrates the recorded counts of all OTUj-s fal­ling into size class i; [i = 1 to n], where n is the number of size classes representing each spec­trum; n = 20 in this study.

We shall call this structure (formula 5) 'composite size spectrum' (CSS).

If bn = bw = 1, each size class amounts to the cumulative biomass of particles (organisms) of size class i, and the CSS can be transformed to a regular BSS or 'Sheldon' spectrum [11]. If bn =1 and bw = 0, it represents the cumulative organism (cell) abundance producing an NBS [14]. The CSS can also describe the size distribution of metabolic fluxes (like respiration) and the 'metabolic surface' distribution, when bn =1 and bw = 0.75 - 0.80 [7].

While the above variants of size spectra are widely studied, we shall apply here two vari­ants with bn = bw = 0. Then Njbn = 1, Wjbw =1, and expressions (5), (6) will represent a [1, n] matrix, i.e., a numerical vector of length n, with elements: [FRi] [i=1 to n] (7).

"Switching-off" the OTUj frequency-weights [i.e., substituting FRj = const = 1] turns form (7) into:

[Ki] [I = 1 to n] (8),

where Ki is the number of OTUj-s falling into size class i.

While form (8) represents the distribution of the cumulative number of taxonomic units (e.g., species or OTUj-s) between size classes, it corre­sponds to the above-mentioned traditional size-frequency species distribution [2, 4, 5]. We shall call them taxonomic size spectra while using OTUj, which can represent not only species but also subspecies, higher taxons, or even just groups of particles that are interesting for the investigator and suitable for automated distinguishing from other such groups [13]. Then forms (8) and (7) can be called 'traditional' and 'frequency-weighted' TSS (TTSS, FTSS, respectively).

Both forms of TSS were built here using the 'Histogram' procedure of the SPSS statistical program.

Annual taxonomic size spectra. One TTSS and one FTSS were built for each annual period. The total number of OTUj-s recorded dur­ing one year (range: 91 - 111) was distributed into size classes to create an annual TTSS. A fixed cell volume was used for each OTUj [2, 12]. Hence, a TTSS was created as the frequency distribution (histogram) of the cumulative number of OTUj-s recorded during one year into size classes, using the histogram procedure of the statistical package

SPSS.

FTSS was created using the same histo­gram allocation procedure, but the OTUj registra­tion records were the counted elements distributed between the size classes. TSSyy notation was ap­plied, where yy was the year, for example TSS82 was the TSS for the year 1982.

Морський екологічний журнал, № 4, Т. IV. 2005

Estimation of similarity between TSS curves. Each histogram, i.e., a column of num­bers, can be interpreted as a numerical vector, and hierarchical cluster analysis can be applied to evaluate the similarity between histograms, to find and compare the most closely related histogram shapes [1]. The Pearson correlation coefficient (r) was used here as an effective similarity estimate

[13].

Linear regression and correlation analysis were also used to estimate the Pearson correlation (r) between the TSS numerical vectors. The SPSS program, version 12.0, was used for all statistical procedures.

Results.

Taxonomic size spectra. The TSS-s for the most contrasting pair of years, i.e., 1996 and 1998 (Fig. 1), are compared in graphic form (Fig.

3).

Two types of TSS (specifically, TTSS and FTSS) are shown (Fig. 3 a). All four curves re­semble each other by having two major peaks near the center of the curve, at LoV2.1 and at LoV3.0, with lower but still elevated frequencies in the outer-edge regions. These secondary peaks appear at both the high and low ends of the spectrum. Distinct minima occur at LoV0, LoV1.2, and LoV3.3, representing size classes with only a few and rarely occurring OTUj-s.

Especially important, for the aims of this study, distinctions are evidenced via a comparison of the TTSS-FTSS pairs. The most pronounced differences are seen within the extreme right re­gion, which corresponds to the Kinneret dominant, bloom-forming species P. gatunense (see above).

This difference is explained by the OTUj frequency-weight coefficients (Fig. 3 b), which are ignored (more precisely, are set to the const=1.0 value) in the case of TTSS. These frequency weights (FRj), summed up according to formula (6) within each size class i, produce size class i frequency rate, FRi. Size class averaged OTUj weights FRj96 and FRj98 are shown by two sepa­rate curves in Fig. 3 b.

Y. Kamenir

—*—

FTSS96

—•—

FTSS98

Л

TTSS96

o—

TTSS98

1500 -, 1250 -

Fig. 3. Taxonomic size spec­tra, i.e., frequency distribu­tions of taxonomic units into size classes. Comparison of one of the most contrasting pairs of years (1996 and 1998) in Lake Kinneret's 30-year   monitoring period.

TTSS and FTSS are the 'tra­ditional' and 'frequency-weighted' Taxonomic Size Spectrum (TSS), respec­tively. Size classes of dou­bling cell volume V were created in [8] and are pre­sented on a logarithmic scale (log V). FRj is the frequency rate per OTUj of size class i 'teams'. OTUj is the opera­tional taxonomic unit (see Methods).

Рис. 3. Таксономические размерные спектры (ТРС) т.е. частотные распределе­ния таксономических еди­ниц по размерным клас­сам. Сопоставление одной из наиболее контрастных пар лет (1996 и 1998) из 30-летнего периода мони­торинга оз. Киннерет. ТТРС и ЧТРС (TTSS и FTSS) - традиционный и частотно-взвешенный так­сономические размерные спектры (ТРС) соответст­венно. Размерные классы (удвоение объема клетки) созданы в [8] и представ­лены на логарифмической шкале (log V). FRj - часто­та регистрации таксономической единицы OTUj из размерного класса i. OTUj - operational taxonomic unit (см. Методы).

The lowest - zero - FRj values correspond to OTUj-s, which were never recorded during the specific period (in this case, one year). The high­est frequency rates give emphasis to OTUj-s most often met both in time and space. Those values are seen in the region of small (LoV < 1.5) and in the region of the largest algae, LoV4-LoV6, encom­passing the bloom-dominant P. gatunense. This region is very different for years 1996 and 1998 10 (Fig. 3 b), which leads to a pronounced difference between the four TSS (i.e. TTSS and FTSS for 1996 and 1998; Fig. 3 a). As those dissimilarities reflect the difference in the number of the days and depth horizons of registration of the largest algae, which normally produce the main part of the average annual phytoplankton biomass, those dissimilarities help us better understand the differ­ence between the assemblage structures producing

the annual biomass dynamics of 1996 and 1998

(Fig. 1).

Quantification of similarity between curves. Conceptually, an 'ideal' situation of sta­bility can be imagined as the persistence of the same phytoplankton structure and seasonality from year to year. Thus, for any pair of years compared, each size class of one year will have the same number of OTUj-s as the same size class in the second year, and the pair of TTSS curves will completely overlap. The same is true for the FTSS pairs, where each FRi should be kept the same, resulting in survival of the FRi set. If the data for such ideal pairs of years are plotted as an

XY-scatter diagram, with the number of OTUj-s per size class of the first year of the pair plotted on axis X against the same size class for the second year on axis Y, the data points should fall along a straight line, describing the regression, and the determination coefficient (r2) for this line should reach the 'ideal' value of 1.0. The proximity to this ideal situation can be measured by the regression determination coefficient r2, where r is the Pearson correlation coefficient. While the OTUj-set com­parison shows a very chaotic scatter (Fig. 4a) and determination coefficients from 0.22-0.57, the size class (i.e., FRi) set comparison evidences a rather small dispersal of points for each pair of years, with r2 values from 0.72­0.92, which is much closer to the ideal situation of 1.0.

a

erf

LL

300 250 □ 200 150 100 50

0

y97 = 0.86x + 32.05; R 2 =0.574 y98 = 0.55x + 39.66;R2 = 0.220 y 99= 0.54x + 50.05;R2 = 0.298

50    1 00   1 50   200 250 FRj96

b

9

CD

or

F

1500

1250 -

1000

750

500 250 0

y97 = 1.11x + 69.55;R2 = 0.922 y 98= 0.85x + 12 7.88; R 2 = 0.718 y99 = 1.11x + 99.62;R2 = 0.722

0      250    500    750   1 000

FRi96

~0 FTSS97 FTSS98 Д FTSS99

- - .Linear

(FTSS98) Linear

(FTSS99)

- — Linear

(FTSS97)

Fig. 4. A geometric model as a means of comparison be­tween pairs of data sets which use X and Y axes: a, b - the annual frequency rate scores (FRj) of Kinneret phytoplank-ton individual taxonomic units (OTUj), and of cumula­tive size class i scores (FRi), respectively. The years 1997 - 1999 (axis Y) vs. 1996 (axis X). All regressions are sig­nificant (p<0.05); n= 167 in a, and n=20 and b, respectively. Рис. 4. Геометрическая мо­дель как средство сопостав­ления пар наборов данных, использующих оси X и Y соответственно: a, b - годо­вые частоты регистрации (FRj) индивидуальных так­сономических единиц (OTUj) и кумулятивные частоты размерных классов i (FRi) соответственно. Го­ды 1997 - 1999 (ось Y) в сравнении с 1996 (ось X). Все регрессии статистиче­ски значимы (p<0.05); n=167 для a, and n=20 для b, соответственно.

Discussion. TSS and stability of the community structure. By plotting the size distri­bution of all taxonomic units creating the phyto-plankton assemblage each year and comparing pairs of such spectra, we were able to evidence some type of similarity and estimate quantitative differences between cumulative annual scores dur­ing a period of instability. The frequency rate com­parisons (Fig. 4a and b) helped us evidence the ef­fect of some community-level force that links the 'separate and independent' units (OTUj) into size class subsets sharing almost the same 'team quote' each year.

Linear regression was used to compare the histogram pairs, bearing a resemblance to the lin­ear regression application for the NBS shape (slope) and non-ideality (r2) estimation [14]. Here we use a pair of numerical vectors, each of them representing an SS. Each pair of numerical vectors represents two years, i.e., the annually-accumulated values, and a special kind of weight coefficient is used here - the species (more pre­cisely - OTUj) number per size class in TTSS, and OTUj success level (FRj) in FTSS.

The two types of TSS evidence that the pronounced changes in the annual dynamics of the Kinneret phytoplankton biomass (Fig. 1) and taxo-nomic composition (Fig. 4 a) were paralleled with some changes in the fine structure of the phyto-plankton taxonomic size spectra (Fig. 3), at the same time as the general pattern of the phyto-plankton assemblage TSS turned out to survive even during the most pronounced environmental anomalies registered in some three decades of lake monitoring. Whereas the species characteristics (time-space distribution, abundances, and bio-masses) were highly dynamic, the species list and even species number changed every year; a specific form of similarity in the phytoplankton structure was discernible via taxonomic size spectra (Fig. 3).

While the numerical vector length was sufficiently different for the FRj and FRi compari­sons (Fig. 4 a, b; n=167 and n=20, respectively), we produced also n=20 vectors of taxonomic units, via a random distribution of n=167 lines used for Fig. 4a, into n=20 line subsets. Then the correla­tion coefficients (r) were calculated for all possible pairs of such numerical vectors (Fig. 5a), with the histogram of r-estimates (Fig. 5b). The average value and SD were calculated for this population of r-values, using Fisher's Z-transformed estimates. Afterwards, the confidence interval was trans­formed to the r-coefficient estimates via the Fisher's backward transformation. The average estimate r=0.291, with the 95% confidence interval from -0.174 to 0.650 show that the r-values given on Fig. 4b turned to be high, well outside the upper end of this confidence interval (r2 ~ 0.370), which confirms the importance of the above difference between FRj and FRi comparisons.

Typical patterns: BSS and TSS. Two types of taxonomic size spectra (TTSS and FTSS) have revealed some stable structural properties of aquatic communities, known as 'typical patterns' of size spectra. While using such SS, simple statis­tical methods (linear regression, cluster analysis) can provide quantitative similarity estimates [1] helpful in the comparison of structural patterns of integral aquatic communities and assemblages. Using the schemes and mathematical methods de­veloped for the 'ataxonomic' biomass size spectra (BSS, NBS), the taxonomic size spectra can help in the search for and analysis of the natural as­semblage stability phenomena via integration of ataxonomic and taxonomic approaches.

While size spectra mostly consider the size distribution of the organisms' biomass or abundance [11, 14], some authors have already paid attention to the size distribution of taxonomic units [2 - 5]. TSS, like NBS, seems to be very consistent. For example, in several lakes in the USA, patterns of pelagic community taxonomic size structure were very conservative. In spite of substantial dissimilarities in numerous limnologi-cal properties, including the morphometry, trophic status, food web structure, and even species diver­sity, the patterns of species size distribution were very conservative [3].

r[Z]

1,0

0,8 0,6 0,4 0,2

0,0

-0,2 -0,4 -]

-0,6

♦ ♦

♦ ♦

♦ ♦ ♦

♦ ♦

NN

 

N_Req

200 n

A

150

 

100

 

yd

 

 

 

-0,50

-0,25

0,00

0,25

0,50

0,75

1,00

Fig. 5. The cor­relation coeffi­cient (r) value distribution de­pending on the vector pair number (a) and as a histogram (b), respectively

Страницы:
1  2 


Похожие статьи

Y Kamenir - Quasi-stochastic phytoplankton in the sea of the galilee