The parameters of this mutualistic system correspond to the values describing intrinsic growth
rates (ai), intra-guild competition (bij), the benefit received via mutualistic interactions (gij), and
the saturating constant of the beneficial effect of
mutualism (h), commonly known as the handling time. Because our main focus is on mutualistic interactions, we keep as simple as possible
the competitive interactions for the sake of analytical tractability. In the absence of empirical
information about interspecific competition,
we use a mean field approximation for the
competition parameters (6), where we set
bðPÞ ii ¼ bðAÞ ii ¼ 1 and bðPÞ ij ¼ bðAÞ ij ¼ r < 1 (i ≠ j).
Following (39), the mutualistic benefit can be
further disentangled by gij ¼ ðg0yijÞ=ðkd i Þ, where
yij = 1 if species i and j interact and zero otherwise, ki is the number of interactions of species
i, g0 represents the level of mutualistic strength,
and d corresponds to the mutualistic trade-off.
The mutualistic strength is the per capita effect
of a certain species on the per capita growth rate
of their mutualistic partners. The mutualistic
trade-off modulates the extent to which a species
that interacts with few other species does it strongly,
whereas a species that interacts with many partners does it weakly. This trade-off has been justified on empirical grounds (40, 41). The degree
to which interspecific interactions yij are organized into a nested way can be quantified by the
value of nestedness N introduced in (42).
We are interested in quantifying the extent to
which network architecture (the combination of
mutualistic strength, mutualistic trade-off, and
nestedness) modulates the set of conditions compatible with the stable coexistence of all species—
the structural stability. In the next sections, we
explain how this problem can be split into two
parts. First, we explain how the stability conditions can be disentangled from the feasibility
conditions as it has already been shown for the
two-species competition system. Specifically, we
show that below a critical level of mutualistic
strength (g0 < gr 0), any feasible equilibrium point
is granted to be globally stable. Second, we explain how network architecture modulates the
domain in the parameter space of intrinsic growth
rates, leading to a feasible equilibrium under the
constraints of being globally stable (given by the
level of mutualistic strength).
We investigated the conditions in our dynamical system that any feasible equilibrium point needs to satisfy to be globally stable. To derive these conditions,
we started by studying the linear Lotka-Volterra approximation (h = 0) of the dynamical model (Eq. 2).
In this linear approximation, the model reads
¼ diag PA
;; ; ;
where the matrix B is a two-by-two block matrix
embedding all the interaction strengths.
Conveniently, the global stability of a feasible
equilibrium point in this linear Lotka-Volterra
model has already been studied (14–17, 43). Particularly relevant to this work is that an interaction matrix that is Lyapunov–diagonally stable
grants the global stability of any potential feasible equilibrium (14–18).
Although it is mathematically difficult to verify
the condition for Lyapunov diagonal stability, it
is known that for some classes of matrices,
Lyapunov stability and Lyapunov diagonal stability are equivalent conditions (44). Symmetric
matrices and Z-matrices (matrices whose off-diagonal elements are nonpositive) belong to
those classes of equivalent matrices. Our interaction strength matrix B is either symmetric when
the mutualistic trade-off is zero (d = 0) or is a
Z-matrix when the interspecific competition
is zero (r = 0). This means that as long as the
real parts of all eigenvalues of B are positive
(18), any feasible equilibrium point is globally
stable. For instance, in the case of r < 1 and g0 =
0, the interaction matrix B is symmetric and
Lyapunov–diagonally stable because its eigenvalues are 1 – r, (SA – 1)r + 1, and (SP – 1)r + 1.
For r > 0 and d > 0, there are no analytical
results yet demonstrating that Lyapunov diagonal stability is equivalent to Lyapunov stability.
However, after intensive numerical simulations
we conjecture that the two main consequences of
Lyapunov diagonal stability hold (45). Specifically, we state the following conjectures: Conjecture 1: If B is Lyapunov-stable, then B is D-stable.
Conjecture 2: If B is Lyapunov-stable, then any
feasible equilibrium is globally stable.
We found that for any given mutualistic trade-off and interspecific competition, the higher the
level of mutualistic strength, the smaller the maximum real part of the eigenvalues of B (45). This
means that there is a critical value of mutualistic
strength (g0r) so that above this level, the matrix B
is not any more Lyapunov-stable. To compute gr 0,
we need only to find the critical value of g0 at
which the real part of one of the eigenvalues of
the interaction-strength matrix reaches zero (45).
This implies that at least below this critical value
1253497-4 25 JULY 2014 • VOL 345 ISSUE 6195
Fig. 3. Structural stability in a two-species competition system. The figure shows how the range of
intrinsic growth rates leading to the stable coexistence of the two species (white region) changes as a
function of the competition strength. (A to D) Decreasing interspecific competition increases the area of
feasibility, and in turn, the structural stability of the system. Here, b11 = b22 = 1, and b12 = b21 = r. Our goal is
extending this analysis to realistic networks of species interactions.