Determinants of Density–Body Size Scaling Within Food Webs and Tools for Their Detection - Advances in Ecological Research - page 12

population is increased by a factor 1/(1 d), being d the probability of

consumption. When no consumption is experienced, the energetic cost of
one individual is equal to its metabolic demand, and as d increases this cost is
incremented. We considered an arbitrary maximum cost equivalent to 100
individuals for which d takes a value of 0.99.

a 1 M a exp E D CDF m ðÞ R T 1 ð d Þ ð kT Þ ð 4 Þ

=

d can be modelled through different functions; here we used a Hill equation:
d M 5 z ). This is an S-shaped function and the parameter M 50

indicates the value of individual body size at which predation achieves half
of its maximum value (in this case 0.99) and z determines how fast the
transition between zero and 0.99 takes place. Positive values of z determine
an increase in predation with body size; the opposite holds true for negative
values. In addition, when both increasing and decreasing trends in consump-
tion with body size are considered, they can be modelled estimating d as the
sum of two Hill equations with positive and negative values of z.
Equation (4) allows for the exploration of the effect of alternative patterns
of resource availability and predation pressure across body size gradients on
the DMR (see Figure 2 ). When resources and predation are independent of
body size, the expected pattern is a reduction in density with an exponent of

M z /(M z ¼ þ

0.75, meeting the energetic equivalence rule ( Figure 2 A). The incorpora-
tion of mortality can produce a change in the exponent as well as in the
intercept of the DMR ( Figure 2 B). If both predation and resources change
with body size, multimodal distributions could arise, determining changes in
DMR slopes and intercepts and different scalings for different ranges of
body sizes ( Figure 2 C). Notably, when available resources and predation
change smoothly with body size, the expected pattern is a non-linear rela-
tionship without a clear single slope ( Figure 2 D). Consideration of U-shaped
patterns in consumption leads to abrupt changes in scaling exponents
( Figure 2 E). Finally, a preference in consumption for larger size classes
determines steeper slopes when these preferences start operating
( Figure 2 F). As a consequence, variation in a consumer within a population
in the strength of gape limitation or its access to resources is shown here to
have the potential to determine all the main attributes of the DMR.
The proposed analysis makes two contributions to the understanding of
the DMR. First, it provides an explicit consideration of the importance of
gape limitation as a determinant of the DMR affecting the intercept, slope,
and number of modes in the distribution and changes in the DMR at
different ranges of body size. Second, the wide range of patterns expected
for the DMR in a food web context is recognized, as should also be the need
to use methodological approaches that can detect these patterns.