approximated to3/4 within trophic groups, commonly using ordinary least
squares model I regression (OLS). Since body mass scales with metabolic rate to
the 0.75 power, the0.75 exponent betweenN-Mwas taken as evidence that
species abundance is limited by energetic requirements (e.g. Neeet al., 1991). This
‘energetic equivalence rule’ is expected in situations where all species use the
same source of energy from a limited resource. Using model I regressions, this
inverse proportionality with a3/4 exponent was also evident between mean
population density and mean body mass in two geographically separate stream
communities and between different trophic groups of these invertebrate com-
munities (Schmid, Tokeshi & Schmid-Araya, 2000 ). Although the scaling relation-
ship with exponents close to0.75 also holds seasonally for the same streams
(Schmidet al., 2002), wide variations in regression slopes among different taxo-
nomic assemblages makes it unlikely that these assemblages are governed by a
single energetic rule (Schmidet al., 2000).
Density – body-mass scaling with sample area
The area sampled has a distinct influence on a scaling relationship, as the
species number and their densities tend to increase as a power function of the
area (see Multifractal species–area relationships, below). Consequently, N-M relation-
ships must change with area until the area sampled is large enough not to be
dominated by populations with transient individuals. Scale invariance in space
can be expected if the scaling exponents collapse into a single value at a specific
sample area. To test this assumption, theN-Mscaling relationships of the
benthic invertebrate communities of the three streams (SB, LL and MY) were
analyzed from a single sample scale (1 m^2 ) to the scale of a whole river section
(240 m^2 ). By including and amalgamating samples taken at any sampling
position or occasion, it was possible to obtain spatially independent samples
incorporating invertebrate species over different developmental stages, body-
size and density ranges. Preliminary test results showed that the observed trend
in the relationship between the regression slopes of the log density – log body-
mass and area size stabilized after 1000 randomizations. Therefore, 10^4 random
amalgamations of the observed data were conducted (Fig.8.1) as an estimate for
all possible> 1095 sample combinations (e.g.j!/(ji)!i!, wherejis the possible
number of samples of sizei, andi¼2, 3, 4,...,nm^2 ).
The insets of Fig.8.1show the linear regressions of log mean population
density on log mean body-mass for each of the three stream communities over
the total sampled area. In contrast to the communities of the SB and MY, the
community of the LL departed significantly from a 3/4 assumption when con-
sidering OLS model I slopes (t¼9.57;df¼330;P<0.001). However, using the
bisector model II regression approach, which accounts for the error scatter in
the data (Schmidet al., 2000) results in a slopebBIS¼0.727(1 SE: 0.025) not
significantly different from the 3/4 assumption (t¼0.92;df¼330;P¼0.357).
BODY SIZE AND SCALE INVARIANCE 145