(r = 0.2, 0.4, 0.6), we always find that structural
stability is positively associated with nestedness
and mutualistic strength (45). Similarly, structur-
al stability is always associated with the mutual-
istic trade-off by a quadratic function, leading
quite often to an optimal value for maximizing
structural stability (45). These findings reveal
that under the given characterization of inter-
specific competition, there is a general pattern of
network architecture that increases the struc-
tural stability of mutualistic systems.
Yet, one question remains to be answered: Is
the network architecture that we observe in
nature close to the maximum feasibility domain
of parameter space under which species coexist?
To answer this question, we compared the ob-
served network architecture with theoretical
predictions. To extract the observed network
architecture, we computed the observed nested-
ness from the observed binary interaction matrices
(table S1) following (42). The observed mutual-
istic trade-off d is estimated from the observed
number of visits of pollinators or fruits con-
sumed by seed-dispersers to flowering plants
(41, 50, 51). The full details on how to compute
the observed trade-off is provided in (52). Be-
cause there is no empirical data on the relation-
ship between competition and mutualistic strength
that could allow us to extract the observed mu-
tualistic strength g0, our results on nestedness
and mutualistic trade-off are calculated across
different levels of mean mutualistic strength.
As shown in Fig. 5, B to D, the observed
network (blue solid lines) of the mutualistic sys-
tem located in the grassland asclepiads of South
Africa actually appears to have an architecture
close to the one that maximizes the feasibility
domain under which species coexist (reddish/
darker region). To formally quantify the degree
to which each observed network architecture is
maximizing the set of conditions under which
species coexist, we compared the net effect of the
observed network architecture on structural sta-
bility against the maximum possible net effect.
The maximum net effect is calculated in three
steps.
First, as outlined in the previous section, we
computed the partial fitted values of the effect
of alternative network architectures on species
persistence (48). Second, we extracted the range
of nestedness allowed by the network given the
number of species and interactions in the sys-
tem (45). Third, we computed the maximum net
effect of network architecture on structural
stability by finding the difference between the
maximum and minimum partial fitted values
within the allowed range of nestedness and
mutualistic trade-off between d ∈ [0, ..., 1.5]. All
the observed mutualistic trade-offs have values
between d ∈ [0, ..., 1.5]. Last, the net effect of the
observed network architecture on structural
stability corresponds to the difference between
the partial fitted values for the observed archi-
tecture and the minimum partial fitted values
extracted in the third step described above.
Looking across different levels of mean mutu-
alistic strength, in the majority of cases (18 out of
23, P = 0.004, binomial test) the observed net-
work architectures induce more than half the
value of the maximum net effect on structural
stability (Fig. 6, red solid line). These findings
reveal that observed network architectures tend
to maximize the range of parameter space—
structural stability—for species coexistence.
Structural stability of systems with
other interaction types
In this section, we explain how our structural
stability framework can be applied to other types
of interspecific interactions in complex ecological systems. We first explain how structural stability can be applied to competitive interactions.
We proceed by discussing how this competitive
approach can be used to study trophic interactions in food webs.
SCIENCE
sciencemag.org 25 JULY 2014 • VOL 345 ISSUE 6195 1253497-7
Fig. 6. Net effect of network architecture on structural stability. For each of the 23 observed
networks (table S1), we show how close the observed feasibility domain (partial fitted residuals) is as a
function of the network architecture to the theoretical maximal feasibility domain. The network architecture is given by the combination of nestedness and mutualistic trade-off (x axis) across different
values of mean mutualistic strength (y axis). The solid red and dashed black lines correspond to the
maximum net effect and observed net effect, respectively. In 18 out of 23 networks (indicated by
asterisks), the observed architecture exhibits more than half the value of the maximum net effect (gray
regions). The net effect of each network architecture is system-dependent and cannot be used to compare
across networks.
