and network of interactions, one can observe from
all to a few of the species surviving. Second, this
means that each network has a limited range of
parameter values under which all species coexist.
Thus, by studying a specific parameterization, for
instance, one could wrongly conclude that a random network has a greater effect on community
persistence than that of an observed network, or
vice versa (10–12). This sensitivity to parameter
values clearly illustrates that the conclusions
that arise from studies that use arbitrary values
in intrinsic growth rates are not about the effects of network architecture on species coexistence, but about which network architecture
maximizes species persistence for that specific
parameterization.
Traditional studies focusing on either local
stability or numerical simulations can lead to
apparently contradictory results. Therefore, we
need a different conceptual framework to unify
results and seek for appropriate generalizations.
Structural stability
Structural stability has been a general mathematical approach with which to study the behavior
of dynamical systems. A system is considered to
be structurally stable if any smooth change in the
model itself or in the value of its parameters does
not change its dynamical behavior (such as the
existence of equilibrium points, limit cycles, or
deterministic chaos) (22–25). In the context of
ecology, an interesting behavior is the stable
coexistence of species—the existence of an equilibrium point that is feasible and dynamically
stable. For instance, in our previous two-species
competition system there is a restricted area in
the parameter space of intrinsic growth rates
that leads to a globally stable and feasible solution as long as r < 1 (Fig. 3, white area). The
higher the competition strength r, the larger the
size of this restricted area (Fig. 3) (19, 26). Therefore, a relevant question here is not only whether
or not the system is structurally stable, but how
large is the domain in the parameter space leading to the stable coexistence of species.
To address the above question, we recast the
mathematical definition of structural stability to
that in which a system is more structurally stable, the greater the area of parameter values
leading to both a dynamically stable and feasible
equilibrium (27–29). This means that a highly
structurally stable ecological system is more likely to be stable and feasible by handling a wider
range of conditions before the first species becomes extinct. Previous studies have used this
approach in low-dimensional ecological systems
(3, 19). Yet because of its complexity, almost no
study has fully developed this rigorous analysis
for a system with an arbitrary number of species.
An exception has been the use of structural stability to calculate an upper bound to the number
of species that can coexist in a given community
(6, 30).
Here, we introduce this extended concept of
structural stability into community ecology in
order to study the extent to which network
architecture—strength and organization of inter-
specific interactions—modulates the range of con-
ditions compatible with the stable coexistence
of species. As an empirical application of our
framework, we studied the structural stability
of mutualistic systems and applied it on a data
set of 23 quantitative mutualistic networks
(table S1). We surmise that observed network
architectures increase the structural stability
and in turn the likelihood of species coex-
istence as a function of the possible set of con-
ditions in an ecological system. We discuss the
applicability of our framework to other types of
interspecific interactions in complex ecological
systems.
Structural stability of mutualistic systems
Mutualistic networks are formed by the mutually
beneficial interactions between flowering plants
and their pollinators or seed dispersers (31).
These mutualistic networks have been shown to
share a nested architectural pattern (32). This
nested architecture means that typically, the mu-
tualistic interactions of specialist species are pro-
per subsets of the interactions of more generalist
species (32). Although it has been repeatedly
shown that this nested architecture may arise
from a combination of life history and comple-
mentarity constraints among species (32–35), the
effect of this nested architecture on community
persistence continues to be a matter of strong
debate. On the one hand, it has been shown that
a nested architecture can facilitate the mainte-
nance of species coexistence (6), exhibit a flexible
response to environmental disturbances (7, 8, 36),
and maximize total abundance (12). On the other
hand, it has also been suggested that this nested
1253497-2 25 JULY 2014 • VOL 345 ISSUE 6195
sciencemag.org SCIENCE
Fig. 1. Stability and feasibility of a two-species competition system. For the same parameters of
competition strength (which grant the global stability of any feasible equilibrium), (A to C) represent the
two isoclines of the system. Their intersection gives the equilibrium point of the system (3, 19). Scenario
(A) leads to a feasible equilibrium (both species have positive abundances at equilibrium), whereas
in scenarios (B) and (C), the equilibrium is not feasible (one species has a negative abundance at
equilibrium). (D) represents the area of feasibility in the parameter space of intrinsic growth rates, under
the condition of global stability. This means that when the intrinsic growth rates of species are chosen
within the white area, the equilibrium point is globally stable and feasible. In contrast, when the intrinsic
growth rates of species are chosen within the green area, the equilibrium point is not feasible. Points “A,”
“B,” and “C” indicate the parameter values corresponding to (A) to (C), respectively.
