Brose et al. Unified spatial scaling: Supplementary Information
1
Supplementary Information
We have introduced new metacommunity link-area and link-species relationships1 that
are based on the assumption that consumer as well as resource species richness follows
power-law species-area relationships:
S  cA z ,
(1)
where S is species richness, A area and c and z are constants1. Based on these
assumptions, we argued that the probability pi that a species chosen at random from the
metacommunity pool will be present in a random local community patch of size Ai,
follows
pi 
Si
 ai ,
S0
(2)
where a=2-z. Note that pi is a central tendency among species that doesn’t necessarily
describe the distribution of each individual species. Specifically,
pi  pi (k ) k ,
where pi(k) is the probability for species k to be in a patch of area Ai, and the
(3)
k
indicates an average over the species labeled by k.
The mean number of links, Li, in a local patch of area Ai can be calculated from
the spatial distribution of species by
S0
Li    pi (k | r ) pi (r ) ,
k 1 r k 
(4)
Brose et al. Unified spatial scaling: Supplementary Information
2
where pi(k|r) is the probability that a consumer species k is present in an Ai given that a
resource species r is present there and

rk 
is a sum over all species which are eaten by
species k, and pi(r) is defined as in (3). As assumed in empirical studies, (4) assumes that
species trophically linked somewhere in the metacommunity will be linked any local
community where both species are present. By defining the co-occurrence factor
i (k , r )  pi (k | r ) / pi (k ) as the ratio of the occupancy of species k in Ai where species r is
present, to its occupancy in random Ai, (4) becomes
S0


Li    pi (k )    i (k , r ) pi (r )  S 0 pi (k )  L0,k   i (k , r )
k 1 
lk 

where
r k
r k
 pi (r )
r k
,
(5)
k
indicates an average over species r which are eaten by species k, and
k
indicates an average over species k. The second step in (5) assumes that  i (k , r ) and
pi (r ) are not significantly correlated with one another.
If the quantities within
k
are not significantly correlated with one another, (5)
simplifies to
Li  L0 i a 2i ,
where  i   i (k , r )
(6)
rk
is the average co-occurrence factor across consumer-resource
k
combinations, and (2) has been applied. Expression (6) can be written explicitly as a linkarea relationship, and using (1) this can be transformed to a link-species relationship:
L  K ( A) A2 z  ( K / c 2 ) ( A)S 2
(7)
Brose et al. Unified spatial scaling: Supplementary Information
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where K  L0 / A02 z , and the subscripts i have been dropped because these expressions
apply with any x in Ai=A0/xi, and hence apply to any A. Note that (7) is the same as
model (6) in our main text1, and that θ(Ai) is the average co-occurrence factor at scale i,
θi. If the spatial distribution of all consumer species is independent of the distribution of
their resource species, i.e. if pi k | r   pi k  for all combinations of consumers k and
resources r, and all scales i, then θ(A) = 1. If the relationship between consumer and
resource distribution is scale independent, then θ(A) = constant. Otherwise, θ(A) depends
on the spatial extent of the community, i.e. it is scale-dependent. In the empirical data
sets used to test (7) and the two trophic group model below, θ(A) was found to have scale
-dependence1. In particular, we successfully fitted power-law θ(A) to the data, indicating
that the tracking efficiency is higher at smaller spatial scales. Although this form for θ(A)
would yield power-law link-area and link-species relationships in (7), the generality of
this result remains to be clarified.
It is possible that the independence assumptions going into (5) and (6) do not hold,
and hence (7) would be more complicated. For example, consumers may rely more on
widespread resources than on spatially rare resources (  i (k , r ) may increase with pi (r ) ),
and species who consume a wider variety of resource species may on average rely less on
a given resource ( i (k , r )
rk k
may decrease with L0,k). However, here we take these
simplifying assumptions and find that the resulting predictions work well on the data sets
we examine1. A similar theoretical development could be carried out for more
complicated circumstances.
Brose et al. Unified spatial scaling: Supplementary Information
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Equations (4) and the subsequent equations (5-7) can be applied to macroecological
scales by assigning to a given metacommunity the scale Ai that will hold its Si species
according to the power-law species-area relationship. At these scales species may be
spatially isolated from one another by occurring in different local communities and hence
not all potential links between species will be realized. Equation (4) accounts for this
approximately by assigning links only between species that are trophically linked in at
least one local community in the overall metacommunity (for each species k, it sums only
over species r linked to species k somewhere in the overall metacommunity, i.e. it uses

rk 
), but it overestimates the number of links because this local community link may
occur outside of the smaller metacommunity Ai. However, in contrast to the “community
model” (Equations (2) and (3) in our main text1), which assumes general co-occurrence
of species at all scales and ignores metacommunity structure, the unified model in
equation (7), and its more general form in equation (8) below, use the structure of the
metacommunity web and the degree of co-occurrence of species at local scales to predict
the scaling of local community food web structure, and at least approximately account for
the degree of co-occurrence of species at local scales in its predictions at macroecological
scales. Hence the unified model uses the metacommunity web structure and the degree
of co-occurrence of species to predict the scaling of food web structure from local
communities to metacommunities.
Up until now we have assumed invariance of the species-area exponent z and
constant c with trophic rank. If one accounts for differences in the species-area exponent
between two overlapping groups of species: all species that are consumed by another
Brose et al. Unified spatial scaling: Supplementary Information
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species (resources), and all species that consume another species (consumers), the
metacommunity link-area and link-species relationships follow:
L  Kθ ( A) A zk  zr 
K
K
θ ( A) S k S r 
θ ( A) S u ,
ck cr
ck cr
(8)
where k and r label properties of the consumers and resources, respectively, St, zt, and ct,
are the species richness, species-area exponent, and species-area constant for trophic


level t, K  L0 A0z,kk A0z,rr , and u  ln S k S r  ln S  . A0,k and A0,r are the area sizes large
enough to hold all of the consumer and resource species in the metacommunity
respectively (according to their respective species-area relationships). Note that (8) is the
same as model equation (7) in our main text1.
The link-species exponent u in (8) can be expressed approximately in terms of the
species-area exponents. Technically, if we can separate the species in the
metacommunity into T species groups that each satisfy power-law species-area
relationships, we expect the total species richness in local communities to depend on a
sum of power-laws:
T
S  d  A ct A zt ,
(9)
t 1
where d(A) is a scale dependent fraction taking into account that the species groups of
trophic levels t may overlap. However, if we assume that the total species richness is also
well approximated by a power-law species-area relationship S  cA z , we can transform
(8) into a scale-independent link-species relationship:
Brose et al. Unified spatial scaling: Supplementary Information
L
K ( A) u
S
cu
6
(10)
where u  ( zk  zr ) / z . If this is a good approximation for u at all scales, and if  ( A) is
independent of A, we get a power-law link-species relationship.
We tested the fit of our link-area and link-species model (8) in two empirical data
sets1. Extending this approach, the link-species relationship (10) gives excellent fits to the
data sets of the Adirondack lakes (r2=0.865) and the Santa Clara Valley streams
(r2=0.923). For both of these empirical data sets, all three species groups (consumers,
resources, and all species) followed significant power-law species-area relationships1.
However, as mentioned above, if non-overlapping groups of species each separately
exhibit power-law species-area relationships, total species richness will not follow a
power-law dependence on area. For the empirical data sets we examined, there was a
large degree of overlap between consumer and resource species groups, and as a
consequence the species-area exponent varied only slightly between these groups. We
prescribe caution in using this species-area exponent method for predicting u as the
degree of overlap in trophic groups decreases and the species area exponent shows
greater variation across trophic groups
Up until now, we have distinguished two trophic levels in (8) and (10). In the case
where z and c instead are allowed to vary across a larger number of trophic levels, which
again may be overlapping, the assumptions which lead to (8) instead lead to
T T
T T
K 
L    K tt ' tt ' ( A) A zt  zt '    tt '  tt ' ( A) S t S t ' .
t  2 t '1
t  2 t '1  ct ct ' 
(11)
In (11), St, zt, and ct, are the species richness, species-area exponent, and species-area
constant for trophic level t, and T is the number of trophic levels in the metacommunity.
Brose et al. Unified spatial scaling: Supplementary Information
K tt '  L0,tt '
A
zt
0 ,t
7

A0z,tt' ' , where L0,tt’ is the number of links in between consumers of trophic
level t and resources of level t’ in the metacommunity, and A0,t is the patch size which
would contain all of the species in the metacommunity from trophic level t.
 tt ' ( Ai )   i (k , l )
lt ',k
is the average of the co-occurrence factor across all consumerkt
resource combinations of the trophic levels t and t’. (11) is based on the assumption that
each trophic level t separately follows a power-law species-area relationship. Under the
assumption that total species richness also follows an approximate power-law (note the
caution we prescribed above), the link-species relationship in (11) can be transformed
into a sum of powers:
T
t
L  
t  2 t '1
K tt ' tt ' utt '
S ,
c utt '
(12)
where utt '  ( z t  z t ' ) / z . Testing the fit of (12) in the two empirical data sets1 did not
lead to improved fits in comparison to (8).
Additional references
1 Brose, U., Ostling, A., Harrison, K. A and Martinez, N.D., Unified scaling of species
and their trophic interactions. Nature.