Geoscience Reference
In-Depth Information
with a predator individual. Assume that predator individuals of size class m
i
(m
i
1
m
i
) have the individual-predator PPMR r
i
(i.e. their average prey
mass is m
i
/r
i
) and that their proportion in the population is p
i
. We also
consider that each predator includes n
i
prey individuals in the gut, with the
biologically reasonable assumption that larger predator individuals consume
more prey (i.e. n
i
1
n
i
). Then, the average individual-predator PPMR of
the predator species is given as
X
p
i
r
i
¼
r
;
ð
2a
Þ
where the bar represents the averaging among the predator size classes based
on the proportion p
i
. Meanwhile, the species-averaged PPMR is calculated
by dividing the average predator body mass with the average prey body mass:
,
P
p
i
n
i
m
i
=
X
p
i
m
i
r
i
nm
=
r
P
p
i
n
i
¼
m
:
ð
2b
Þ
n
ab
According to the Chebyshev's sum inequality (i.e.
ab if a
i
1
a
i
and
b
i
1
b
i
), it holds that
n
m
nm
:
ð
3
Þ
If the individual-predator PPMR is identical among all individuals (i.e.
r
i
1
¼
r
i
¼
r) as is conventional, it follows that
n
m
r
nm
=
r
:
ð
4
Þ
This inequality illustrates that the individual-predator PPMR (Eq.
(2a)
)is
higher than the species-averaged PPMR (Eq.
(2b)
) for any particular
species. If the individual-predator PPMR varies within the predator species,
and larger individuals have smaller values (i.e. 1/r
i
1
1/r
i
), using the
Chebyshev's sum inequality we obtain
nm1
=
r
nm
=
r
ð
5
Þ
bec
ause the arithmetic mean is always larger than the harmonic mean (i.e.
1
r).Together with inequalities
(3) and (5)
, inequality
(4)
always
holds, illustrating that the individual-predator PPMR is higher than the
species-averaged PPMR. This seems consistent with the suggestion of
Woodward and Warren (2007)
, whereby species averaging underestimates
PPMR. However, this is not the case. In fact, the species-averaging effect
may not lower PPMR values in the presence of small modifications that are
added to the assumptions of the above equations. For example, when
larger individuals have larger PPMR values (i.e. r
i
1
=
r
1
=
r
i
;
Barnes et al.,
2010
; also see Section
IV.B
), or when there is no regularity in the
