For a competition system with an arbitrary number of species, we can assume a standard set of dynamical equations given bydNi dt ¼ Niðai − ∑jbijNj Þ,
where ai > 0 is the intrinsic growth rate, bij > 0
is the competition interaction strength, and Ni
is the abundance of species i. Recall that the
Lyapunov diagonal stability of the interaction
matrix b would imply the global stability of any
feasible equilibrium point. However, in nonsym-metric competition matrices Lyapunov stability
does not always imply Lyapunov diagonal stability (53). This establishes that we should work
with a restricted class of competition matrices
such as the ones derived from the niche space of
(54). Indeed, it has been demonstrated that this
class of competition matrices are Lyapunov diagonally stable and that this stability is independent of the number of species (55). For a
competition system with a symmetric interaction-strength matrix, the structural vector is equal to
its leading eigenvector. For other appropriate
classes of matrices, we can compute the structural vectors in the same way as we did with the
effective competition matrices of our mutualistic
model and numerically simulate the feasibility
domain of the competition system. In general,
following this approach we can verify that the
lower the average interspecific competition, the
higher is the feasibility domain, and in turn,
the higher is the structural stability of the competition system.
In the case of predator-prey interactions in
food webs, so far there is no analytical work
demonstrating the conditions for a Lyapunov–
diagonally stable system and how this is linked
to its Lyapunov stability. Moreover, the computation of the structural vector of an antagonistic
system is not a straightforward task. However,
we may have a first insight about how the network architecture of antagonistic systems modulates their structural stability by transforming a
two-trophic–level food web into a competition
system among predators. Using this transformation, we are able to verify that the higher the
compartmentalization of a food web, then the
higher is its structural stability. There is no universal rule to study the structural stability of
complex ecological systems. Each type of interaction poses their own challenges as a function
of their specific population dynamics.
Discussion
We have investigated the extent to which differ-
ent network architectures of mutualistic systems
can provide a wider range of conditions under
which species coexist. This research question is
completely different from the question of which
network architectures are aligned to a fixed set of
conditions. Previous numerical analyses based on
arbitrary parameterizations were indirectly ask-
ing the latter, and previous studies based on local
stability were not rigorously verifying the actual
coexistence of species. Of course, if there is a
good empirical or scientific reason to use a spe-
cific parameterization, then we should take ad-
vantage of this. However, because the set of
conditions present in a community can be constant-
ly changing because of stochasticity, adaptive
mechanisms, or global environmental change,
we believe that understanding which network
architectures can increase the structural stability
of a community becomes a relevant question.
Indeed, this is a question much more aligned
with the challenge of assessing the consequences
of global environmental change—by definition,
directional and large—than with the alternative
framework of linear stability, which focuses on
the responses of a steady state to infinitesimally
small perturbations.
We advocate structural stability as an integra-
tive approach to provide a general assessment of
the implications of network architecture across
ecological systems. Our findings show that many
of the observed mutualistic network architec-
tures tend to maximize the domain of parameter
space under which species coexist. This means
that in mutualistic systems, having both a nested
network architecture and a small mutualistic
trade-off is one of the most favorable structures
for community persistence. Our predictions could
be tested experimentally by exploring whether
communities with an observed network archi-
tecture that maximizes structural stability stand
higher values of perturbation. Similarly, our re-
sults open up new questions, such as what the
reported associations between network architec-
ture and structural stability tell us about the
evolutionary processes and pressures occurring
in ecological systems.
Although the framework of structural stability
has not been as dominant in theoretical ecology
as has the concept of local stability, it has a long
tradition in other fields of research (29). For
example, structural stability has been key in evo-
lutionary developmental biology to articulate the
view of evolution as the modification of a con-
served developmental program (27, 28). Thus,
some morphological structures are much more
common than others because they are compatible
with a wider range of developmental conditions.
This provided a more mechanistic understand-
ing of the generation of form and shape through
evolution (56) than that provided by a historical,
functionalist view. We believe ecology can also
benefit from this structuralist view. The analo-
gous question here would assess whether the
invariance of network architecture across diverse
environmental and biotic conditions is due to
the fact that such a network structure is the one
increasing the likelihood of species coexistence
in an ever-changing world.
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