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X
R
j
=
¼
R
S
ð
6a
Þ
and
X
N
j
R
j
=
X
N
j
¼
NR
=
N
:
ð
6b
Þ
Noting that parameters N
j
and S are independent, it follows that, even
when individual-predator PPMR is larger than species-averaged PPMR at
the species level, we cannot immediately determine which is higher or lower
at the community level. Meanwhile, if species with a smaller abundance
(i.e. with larger average body mass) have larger PPMR (i.e. N
j
1
N
j
and
R
j
1
R
j
;
Barnes et al., 2010
; also see Section
IV B
), using the Chebyshev's
sum inequality we obtain
N R
:
ð
Þ
NR
7a
This inequality is transformed to
R
=
N
NR
:
ð
7b
Þ
This result means that species averaging overestimates PPMR compared to
individual-predator PPMR at the community level. This result contrasts
with the suggestion of
Woodward and Warren (2007)
, yet supports our
observations that species-averaged PPMR was larger than individual-pred-
ator PPMR (
Figure 2
). The overestimation of PPMR values caused by the
species averaging is explained by the fusion of abundant individuals with
small values of PPMR for a single data point. Therefore, the individual-
predator PPMR may be lower than the species-averaged PPMR at the
community level. For example, this may arise when sample counts of
species with relatively lower species-averaged PPMR are much greater
than the counts of species with high species-averaged PPMRs.
Our present analysis is focused towards comparing species-averaged and
individual-predator PPMRs. However, a similar explanation may be
extended and applied to the comparison between other scales. For example,
the prey count of each predator individual should also affect whether
individual-predator or individual-link PPMR of a community is greater,
as well as the total predator species number and individual count of each
species. By combining this information with the results of
Woodward and
Warren (2007)
, we conclude that different definitions of PPMR may lead to
different values that are higher or lower based on the detail of data
elements that are used, such as body mass and sample counts of prey and
predator individuals and species. Therefore, the argument that the species
averaging leads to a low PPMR (
Woodward and Warren, 2007
; also see
Yvon-Durocher et al., 2011b
) is not a general pattern, with the number of
sample counts being crucial.
