often proportional to their body size (Schmid & Schmid-Araya, 1997 , 2002),
particles were analyzed within the size range of organisms at 5mm increments
using the Galai CIS100L laser analyzer within situvideo analysis (Ankersmid,
2002 ). Because the organic fraction (FPOM) constituted the majority of particles
(>95%) in that size range, samples were softly stirred and sonicated in the liquid
flow controller of the CIS for 10 s to break up possible flocs (aggregates) of
particles before each measurement in the CIS100L.Power-law and scaling relationships
Power-law scaling characterizes many patterns observed in ecology, ranging
from individuals to the level of ecosystems (Margalef, 1996 ; Calder, 1996 ;
Brown & West, 2000 ). Many fundamental allometric characteristics of organ-
isms scale with body size as power laws (Peters, 1983 ), with exponents often
being simple multiples of 1/4 (Brownet al., 2002; Brown, Allen & Gillooly, this
volume). These power laws often hold for scales ranging over several orders of
magnitude, sufficiently large to justify a unifying theoretical framework. Power
laws are scale-invariant (Schmid, 2000 ; Marquetet al., 2005), where a change in
scale of the independent variablel preserves the functional form of the original
relationship. This can be demonstrated with the function:gðlÞ¼alb ( 8 : 1 )
by considering a scale transformation inl so thatl!l,gðLÞ¼ðaLbÞlb ( 8 : 2 )Therefore, even with a change in scale, the functional form of the original
relationship with the scaling exponentbremains unchanged. Only the propor-
tionality constant changes fromatoaballowing for changes of resolution by
altering the value of. A scale-invariant (¼scale-free) pattern denotes that a
characteristic property will look the same no matter which scale is chosen (sensu
Gisiger, 2001 ). Therefore, scale-free patterns arise where the same principles or
processes are at work no matter what is the scale of analysis (Milne, 1998 ).
Scaling relationships often emerge from non-experimental approaches empha-
sizing the existence of statistical patterns in the structure of assemblages or
communities that seemingly reflect the operation of universal principles
(Brownet al., 2004). A potentially universal rule for the scaling of energy meta-
bolism across species could be based on the assumption that an optimal design
of fractal-like structure and function operates across all species (West, Brown &
Enquist, 1999 ; Brownet al., 2004). However, this view of a 3/4 scaling rule, has
also been questioned by several authors (e.g. Dodds, Rothman & Weitz, 2001 ).
One intriguing scaling relationship found in nature is the population density –
body-mass relationship, with body mass (M) explaining often more than 50%
of the variation in species density (N). The slope of the relationship often144 P.E. SCHMID AND J. M. SCHMID-ARAYA