9780521861724htl 1..2

distribution displayed multifractal properties (over all 10 q10;r^2 >0.99),

with widerf( )-spectra than those for the species-size distributions (Fig.8.6b).

The values of the entropy dimension, (1)¼D 1 , differed from 1 (Table8.3),

which implies that particle-size irregularities are unevenly distributed and con-

centrate in several subintervals, which was particularly evident for the SB and

MY (Figs.8.4&8.6b). The various scaling domains of interstitial particles are

similar to those from different soil PSDs (Posadaset al., 2001).

To test the similarity of the multifractal patterns between organisms and

particles the Re ́nyi exponents,q, of the BSD were plotted against theqof the

PSD for each stream (Fig.8.7). All stream ecosystems were characterized by two

linear sections of theqrelationships, where the larger section defined trends

forq<0 and the shorter, steeper section of the curve defined trends forq>0.

The well-defined difference between the two curve sections strengthens the

finding of multifractal patterns governing both BSD and PSDs in streams.

Particularly striking is the similarity between BSD and PSDs scaling exponents

in the first-order LL as shown by the closeness of the curve to a slope of unity

(Fig.8.7). Moreover, the relationship between the Re ́nyi exponents of the BSD

Multifractal spectrum,

f^ (α

)

Lipschitz-Hölder exponent, α

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

0.0

0.2

0.4

0.6

0.8

1.0

(a) (b)

SB

LL

MY

Figure 8.6Multifractal spectra for the (a) species- and (b) particle-size distributions in the streams

SB, LL and MY.

Rényi exponent of particle-size distribution, τq

–20 –15 –10 –5 0 5

Rényi exponent of

body-size distribution,

τq

–20

–15

–10

–5

0

5

10

SB

LL

MY Figure 8.7The Re ́nyi exponents,(q), of

the body-sizeversus(q) of particle-size

distribution in the streams SB, LL and MY.

The dotted line gives a slope of unity.

158 P.E. SCHMID AND J. M. SCHMID-ARAYA