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capita effects of one species on another (gij P and gij A) (41).
Second, the networks provide information on the number of
species one species interacts with (its degree or generalization
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level of a species has been found to be proportional to its
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number of visits by the product of the degree of plants and
animals can be assumed to be proportional to the interaction
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explicit dependence between interaction strength and
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(g0)/[(kiP)d]—we obtain (qij)/(kiPkjA) º (g0)/[(kiP)d] and
(qij)/(kiAkjP) º (g0)/[(kiA)d]. In order to estimate the value of
d, we can just take the logarithm on both sides of the previous
equations for the data excluding the zeroes. Then, d is simply
given by the slope of the following linear regressions:
log qij kP i kA j
¼ aP − dlog kP i
and log qij kA i kP j
¼ aA − dlog kA i
where aP and aA are the intercepts for plants and animals,
respectively. These two regressions are performed
simultaneously by lumping together the data set. The intercept
for the effect of animals on plants (aP) may not be the same as
the intercept for the effect of the plants on the animals (aA).
53. The competition matrix given by b ¼
3 3:1 1
is Lyapunov-stable but not D-stable.
Because Lyapunov diagonal stability implies D-stability, this
matrix is also not Lyapunov–diagonally stable.
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We thank U. Bastolla, L.-F. Bersier, V. Dakos, A. Ferrera, M. A. Fortuna,
L. J. Gilarranz, S. Kéfi, J. Lever, B. Luque, A. M. Neutel, A. Pascual-García,
D. B. Stouffer, and J. Tylianakis for insightful discussions. We thank
S. Baigent for pointing out the counter-example in (53). Funding was
provided by the European Research Council through an Advanced
Grant (J.B.) and FP7-REGPOT-2010-1 program under project 264125
EcoGenes (R.P.R.). The data are publicly available at
Materials and Methods
Figs. S1 to S14
17 March 2014; accepted 3 June 2014
sciencemag.org 25 JULY 2014 • VOL 345 ISSUE 6195 1253497-9