architecture can minimize local stability (9), have
a negative effect on community persistence (10),
and have a low resilience to perturbations (12).
Not surprisingly, the majority of these studies
have been based on either local stability or numerical simulations with arbitrary parameterizations [but, see (6)].
Model of mutualism
To study the structural stability and explain the
apparently contradictory results found in studies
of mutualistic networks, we first need to introduce
an appropriate model describing the dynamics
between and within plants and animals. We use
the same set of differential equations as in (6).
We chose these dynamics because they are sim-
ple enough to provide analytical insights and yet
complex enough to incorporate key elements—
such as saturating, functional responses (37, 38)
and interspecific competition within a guild (6)—
recently adduced as necessary ingredients for a
reasonable theoretical exploration of mutualistic
interactions. Specifically, the dynamical model
has the following form
dPi
dt
¼ Pi aðPÞ i − ∑jbðPÞ ij Pj þ ∑jgðPÞ ij Aj 1 þ h∑jgðPÞ ij Aj !
dAi
dt
¼ Ai aðAÞ i − ∑jbðAÞ ij Aj þ ∑jgðAÞ ij Pj 1 þ h∑jgðAÞ ij Pj!
8
>
>
>
>
>
<
>
>
>
>
>
:
ð2Þ
where the variables Pi and Ai denote the abundance of plant and animal species i, respectively.
SCIENCE
sciencemag.org 25 JULY 2014 • VOL 345 ISSUE 6195 1253497-3
Fig. 2. Numerical analysis of species persistence as a function of model
parameterization. This figure shows the simulated dynamics of species
abundance and the fraction of surviving species (positive abundance at the
end of the simulation) using the mutualistic model of (6). Simulations are
performed by using an empirical network located in Hickling, Norfolk, UK
(table S1), a randomized version of this network using the probabilistic model
of (32), and the network without mutualism (only competition). Each row
corresponds to a different set of growth rate values. It is always possible to
choose the intrinsic growth rates so that all species are persistent in each of
the three scenarios, and at the same time, the community persistence
defined as the fraction of surviving species is lower in the alternative
scenarios.
