g0 < g0r, any feasible equilibrium is granted to be
locally and globally stable according to conjectures 1 and 2, respectively. We can also grant the
global stability of matrix B by the condition of
being positive definite, which is even stronger
than Lyapunov diagonal stability (14). However,
this condition imposes stronger constraints on
the critical value of mutualistic strength than does
Lyapunov stability (39).
Last, we studied the stability conditions for the
nonlinear Lotka-Volterra system (Eq. 2). Although
the theory has been developed for the linear
Lotka-Volterra system, it can be extended to the
nonlinear dynamical system. To grant the stability of any feasible equilibrium (Pi > 0 and Ai > 0
for all i) in the nonlinear system, we need to
show that the above stability conditions hold on
the following two-by-two block matrix (14, 43)
Bnl : ¼
bðPÞ ij − gðPÞ ij 1þh∑kgðPÞ ik Ak
−
gðAÞ ij
1 þ h∑kgðAÞ ik Pk bðAÞ ij
2
6
6
6
6
4
3
7
7
7
7
5
ð4Þ
Bnl differs from B only in the off-diagonal block
with a decreased mutualistic strength. This im-
plies that the critical value of mutualistic strength
for the nonlinear Lotka-Volterra system is larger
than or equal to the critical value for the linear
system (45). Therefore, the critical value g0r derived
from the linear Lotka-Volterra system (from the
matrix B) is already a sufficient condition to grant
the global stability of any feasible equilibrium in
the nonlinear case. However, this does not imply
that above this critical value of mutualistic
strength, a feasible equilibrium is unstable. In
fact, when the mutualistic-interaction terms are
saturated (h > 0), it is possible to have feasible
and locally stable equilibria for any level of mu-
tualistic strength (39, 45).
Feasibility condition
We highlight that for any interaction strength
matrix B, whether it is stable or not, it is always
possible to find a set of intrinsic growth rates
so that the system is feasible (Fig. 2). To find
this set of values, we need only to choose a
feasible equilibrium point so that the abundance of all species is greater than zero (A* i > 0
and P* j > 0) and find the vector of intrinsic
growth rates so that the right side of Eq. 2 is
equal to zero: aðPÞ i ¼ ∑jbðPÞ ij P* j − ∑jgðPÞ ij A* j
1 þ h∑jgðPÞ ij A j
andaðAÞ i ¼ ∑jbðAÞ ij A j − ∑jgðAÞ ij P j
1 þ h∑jgðAÞ ij P* j . This reconfirms that the stability and feasibility conditions
are different and that they need to be rigorously
verified when studying the stable coexistence of
species (3, 16, 17, 19). This also highlights that the
relevant question is not whether we can find a
feasible equilibrium point, but how large is the
domain of intrinsic growth rates leading to a
feasible and stable equilibrium point. We call
this domain the feasibility domain.
Because the parameter space of intrinsic growth
rates is substantially large (RS, where S is the
total number of species), an exhaustive numerical search of the feasibility domain is impossible.
However, we can analytically estimate the center
of this domain with what we call the structural
vector of intrinsic growth rates. For example, in
the two-species competition system of Fig. 4A
the structural vector is the vector (in red), which
is in the center of the domain leading to fea-
sibility of the equilibrium point (white region).
Any vector of intrinsic growth rates collinear to
the structural vector guarantees the feasibility of
the equilibrium point—that is, guarantees spe-
cies coexistence. Because the structural vector is
the center of the feasibility domain, then it is also
the vector that can tolerate the strongest devia-
tion before leaving the feasibility domain—that
is, before having at least one species going extinct.
In mutualistic systems, we need to find one
structural vector for animals and another for plants.
These structural vectors are the set of intrinsic
growth rates that allow the strongest perturba-
tions before leaving the feasibility domain. To
find these structural vectors, we had to transform
the interaction-strength matrix B to an effective
competition framework (45). This results in an
effective competition matrix for plants and a dif-
ferent one for animals (6), in which these matrices
represent respectively the apparent competition
among plants and among animals, once taking into
account the indirect effect via their mutualistic part-
ners. With a nonzero mutualistic trade-off (d > 0),
the effective competition matrices are nonsymmet-
ric, and in order to find the structural vectors, we
have to use the singular decomposition approach—
a generalization of the eigenvalue decomposition.
This results in a left and a right structural vector
for plants and for animals in the effective compe-
tition framework. Last, we need to move back from
the effective competition framework in order to
obtain a left and right vector for plants (aðPÞ L and
aðPÞ R ) and animals (aðAÞ L and aðAÞ R ) in the observed
mutualistic framework. The full derivation is
provided in the supplementary materials (45).
Once we locate the center of the feasibility
domain with the structural vectors, we can ap-
proximate the boundaries of this domain by
quantifying the amount of variation from the
structural vectors allowed by the system before
having any of the species going extinct—that is,
before losing the feasibility of the system. To
quantify this amount, we introduced propor-
tional random perturbations to the structural
vectors, numerically generated the new equilib-
rium points (21), and measured the angle or the
deviation between the structural vectors and the
perturbed vectors (a graphical example is provided
in Fig. 4A). The deviation from the structural vec-
tors is quantified, for the plants, by hP (a(P)) = [1 −
cos(y(P) L)cos(y(P) R)]/[cos(y(P) L)cos(y(P) R)], wherey(P) L and
y(P) R are, respectively, the angles between a(P)
and a(P) L and between a(P) and a(P) R. The pa-
rameter a(P) is any perturbed vector of intrinsic
growth rates of plants. The deviation from the
structural vector of animals is computed similarly.
As shown in Fig. 4B, the greater the deviation
of the perturbed intrinsic growth rates from the
structural vectors, the lower is the fraction of
surviving species. This confirms that there is a
restricted domain of intrinsic growth rates cen-
tered on the structural vectors compatible with
the stable coexistence of species. The greater the
tolerated deviation from the structural vectors
SCIENCE
sciencemag.org 25 JULY 2014 • VOL 345 ISSUE 6195 1253497-5
Fig. 4. Deviation from the structural vector and community persistence. (A) The structural vector
of intrinsic growth rates (in red) for the two-species competition system of Fig. 1. The structural vector is
the vector in the center of the domain leading to the feasibility of the equilibrium point (white region) and
thus can tolerate the largest deviation before any of the species go extinct. The deviation between the
structural vector and any other vector (blue) is quantified by the angle between them. (B) The effect of
the deviation from the structural vector on intrinsic growth rates on community persistence defined as
the fraction of model-generated surviving species. The example corresponds to an observed network
located in North Carolina, USA (table S1), with a mutualistic trade-off d = 0.5 and a maximum level of
mutualistic strength g0 = 0.2402. Blue symbols represent the community persistence, and the surface
represents the fit of a logistic regression (R2 = 0.88).
