9780521861724htl 1..2

between the log-partition function and log area remains roughly fulfilled

(Fig.8.9;r^2 >0.95; P<0.001). Excluding slight methodological differences,

these results resemble those obtained for the species-abundance distribution

of a tropical forest in Panama by Borda-de-A ́guaet al.(2002), whereDqwas not

defined forq>1. This more general class of multifractal distributions, for which

not allq-moments exist, also indicates certain scale-related limitations in the

application of Re ́nyi’s generalized dimension for species-abundance distribu-

tions. Nevertheless, if a single fractal dimension does characterize the SAR then

values ofD 0 ,D 1 andD 2 would be all equal. This equality would only be attained if

the densities are equally represented among species. The capacity dimension,

D 0 , is equivalent to the value for the single fractal dimension and to the expo-

nentz of the SAR, which differs between the three streams (Table 8.5).

Comparatively low values of the capacity dimensions indicate that irregularities

in the abundance distribution are more concentrated in samples from smaller

areas, emphasizing small-scale patchiness in streams. An entropy dimension,

D 1 , with a value close to 1 signifies an evenly spread distribution of irregularities

in species-abundance patterns across different scales. However,D 1 values varied

among the three stream communities, ranging from 0.11 in the MY, 0.13 in the

SB, to 0.24 in the LL (Table8.5). Therefore, higher values ofD 1 in the LL implied a

greater evenness in the species-abundance distribution contrasting with the

other stream communities. Similarly, the low values of the correlation dimen-

sion (D 2 ) between communities illustrated uncorrelated irregularities of species-

abundance patterns within the same spatial scale range.

Following Borda-de-A ́guaet al.(2002), results of the generalized dimension

strengthen the positive relationship between the density and spatial range of

species. Considering the extremes of the distribution (e.g.q!10;q!þ10),

that respectively correspond to the rarer but larger and the abundant but

Table 8.5Summary of parameters obtained by multifractal analysis of species-abundance

distributions across different areas in the streams Seebach (SB), Llwch (LL) and Mynach

(MY).Dqare Re ́nyi’s generalized dimensions given with1SE;D 0 is the capacity dimension,

which equals to the slope of the species-area relationship;D 1 is the entropy dimension;D 2 is

the correlation dimension;r^2 are the coefficient of variation of the relationship between log

partition function and log area for eachDq.

Stream Species abundance across areas

D 0 r^2 D 1 r^2 D 2 r^2

SB 0.2330.025 0.959 0.1260.001 0.833 0.1050.013 0.868

LL 0.4480.031 0.973 0.2410.003 0.901 0.1160.021 0.894

MY 0.2000.024 0.968 0.1090.002 0.921 0.0800.009 0.925

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