within which all species coexist, the greater the
feasibility domain, and in turn the greater the
structural stability of the system.
Network architecture and
structural stability
To investigate the extent to which network
architecture modulates the structural stability of
mutualistic systems, we explored the combination of alternative network architectures (
combinations of nestedness, mutualistic strength, and
mutualistic trade-off) and their corresponding
feasibility domains.
To explore these combinations, for each observed mutualistic network (table S1) we obtained 250 different model-generated nested
architectures by using an exhaustive resampling
model (46) that preserves the number of species
and the expected number of interactions (45).
Theoretically, nestedness ranges from 0 to 1 (42).
However, if one imposes architectural constraints
such as preserving the number of species and
interactions, the effective range of nestedness
that the network can exhibit may be smaller (45).
Additionally, each individual model-generated
nested architecture is combined with different
levels of mutualistic trade-off d and mutualistic
strength g0. For the mutualistic trade-off, we explored values d ∈ [0, ..., 1.5] with steps of 0.05 that
allow us to explore sublinear, linear, and super-linear trade-offs. The case d = 0 is equivalent to
the soft mean field approximation studied in (6).
For each combination of network of interactions
and mutualistic trade-off, there is a specific critical value gr 0 in the level of mutualism strength
g0 up to which any feasible equilibrium is globally stable. This critical value gr 0 is dependent on
the mutualistic trade-off and nestedness. However,
the mean mutualistic strength g ¼ ⟨gij⟩ shows no
pattern as a function of mutualistic trade-off and
nestedness (45). Therefore, we explored values of
g0 ∈ ½0; :::;gr 0; with steps of 0.05 and calculated
the new generated mean mutualistic strengths.
This produced a total of 250 × 589 different network architectures (nestedness, mutualistic trade-off, and mean mutualistic strength) for each
observed mutualistic network.
We quantified how the structural stability (
feasibility domain) is modulated by these alternative
network architectures in the following way. First,
we computed the structural vectors of intrinsic
growth rates that grant the existence of a feasible
equilibrium of each alternative network architecture. Second, we introduced proportional random perturbations to the structural vectors of
intrinsic growth rates and measured the angle or
deviation (h(A), h(P)) between the structural vectors
and the perturbed vectors. Third, we simulated
species abundance using the mutualistic model
of (6) and the perturbed growth rates as intrinsic
growth rate parameter values (21). These deviations lead to parameter domains from all to a few
species surviving (Fig. 4).
Last, we quantified the extent to which network architecture modulates structural stability
by looking at the association of community persistence with network architecture parameters,
once taking into account the effect of intrinsic
growth rates. Specifically, we studied this association using the partial fitted values from a
binomial regression (47) of the fraction of surviving species on nestedness (N), mean mutualistic strength (g), and mutualistic trade-off (d),
while controlling for the deviations from the structural vectors of intrinsic growth rates (h(A), h(P)).
The full description of this binomial regression and
the calculation of partial fitted values are provided
in (48). These partial fitted values are the contribution of network architecture to the logit of the
probability of species persistence, and in turn,
these values are positively proportional to the size
of the feasibility domain.
Results
We analyzed each observed mutualistic network
independently because network architecture is
constrained to the properties of each mutualistic
system (11). For a given pollination system lo-
cated in the KwaZulu-Natal region of South
Africa, the extent to which its network archi-
tecture modulates structural stability is shown in
Fig. 5. Specifically, the partial fitted values are
plotted as a function of network architecture. As
shown in Fig. 5A, not all architectural combina-
tions have the same structural stability. In par-
ticular, the architectures that maximize structural
stability (reddish/darker regions) correspond to
the following properties: (i) a maximal level of
nestedness, (ii) a small (sublinear) mutualistic
trade-off, and (iii) a high level of mutualistic
strength within the constraint of any feasible
solution being globally stable (49).
A similar pattern is present in all 23 observed
mutualistic networks (45). For instance, using
three different levels of interspecific competition
1253497-6 25 JULY 2014 • VOL 345 ISSUE 6195
sciencemag.org SCIENCE
Fig. 5. Structural stability in complex mutualistic systems. For an observed mutualistic system with 9
plants, 56 animals, and 103 mutualistic interactions located in the grassland asclepiads in South Africa
(table S1) (58), (A) corresponds to the effect—colored by partial fitted residuals—of the combination of
different architectural values (nestedness, mean mutualistic strength, and mutualistic trade-off) on the
domain of structural stability. The reddish/darker the color, the larger the parameter space that is
compatible with the stable coexistence of all species, and in turn the larger the domain of structural
stability. (B), (C), and (D) correspond to different slices of (A). Slice (B) corresponds to a mean mutualistic
strength of 0.21, slice (C) corresponds to the observed mutualistic trade-off, and slice (D) corresponds to
the observed nestedness. Solid lines correspond to the observed values of nestedness and mutualistic
trade-offs.
