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Asymmetric Coevolutionary Networks
Facilitate Biodiversity Maintenance
Jordi Bascompte,
1
*
Pedro Jordano,
1
Jens M. Olesen
2
The mutualistic interactions between plants and their pollinators or seed dispersers have
played a major role in the maintenance of Earth’s biodiversity. To investigate how coevolutionary
interactions are shaped within species-rich communities, we characterized the architecture of an
array of quantitative, mutualistic networks spanning a broad geographic range. These coevolutionary
networks are highly asymmetric, so that if a plant species depends strongly on an animal species,
the animal depends weakly on the plant. By using a simple dynamical model, we showed that
asymmetries inherent in coevolutionary networks may enhance long-term coexistence and facilitate
biodiversity maintenance.
I
t is widely acknowledged that mutualistic
interactions have molded biodiversity (1, 2).
In the past decade, much has been learned
about how communities shape coevolutionary
interactions across time and space (3). Howe v-
er, although most studies on coevolution focus
on pairs or small groups of species, recent work
has highlighted the need to understand how
broader networks of species coevolve (4–7).
Such knowledge is critical to understanding the
persistence and coevolution of highly diverse
plant-animal assemblages.
Recent research on the architecture of plant-
animal mutualistic networks has been based
mostly on qualitative data, assuming that all
realized interactions are equally important (Fig.
1A) (5–7). This has precluded a deeper assess-
ment of network structure (8) and strongly
limited our understanding of its dynamic impli-
cations. To understand how mutualistic networks
are organized and how such an organization
affects spec ies coexiste nce, we co mpiled from
published studies and our own work 19 plant-
pollinator and 7 plant-frugivore quantitative
networks (Fig. 1 and Database S1). These net-
works range from arctic to tropical ecosystems
and illustrate diverse ecological and biogeo-
graphical settings. Each network displays infor-
mation on the mutual dependence or strength
between each plant and anim al species, mainly
measured as the relative frequency of visits (9).
Thus, our networks describe ecological inter-
actions, and evolutionary inferences should be
made with caution. However, frequency of vis-
its has been shown to be a surrogate for per
capita reproductive performance (10). Our re-
sults could be more directly r elated to coev o-
lution when the reproductive success of one
species depends directly on visitation frequen-
cy. This seems to be the case when there is a
high variation of dependences among species
(10). Unlike previous studies on food webs
(11–16), for each plant-animal species pair , we
have now two estimates of mutual dependence
(defined in two adjacency matrices P and A):
the dependence d
P
ij
of plant species i on animal
species j (i.e., the fractio n of al l an imal visits
coming from this particular animal species) and
the dependence d
A
ji
of animal species j on plant
species i (i.e., the fraction of all visits by this
animal species going to this particular plant
species) (Fig. 1, B and C). Therefore, one can
calculate an index of asymmetry for each
pairwise interaction (17), depicting the relative
dissimilarity between the two mutual depen-
dences (Fig. 1, B and C).
Regardless of the type of mutualism, the
frequency distribution of dependences is right-
skewed,mostlywithweakdependencesanda
few strong ones (Fig. 2). This is in agreement
with previous work on ecological networks
(9, 11–16). This heterogeneous distribution is
highly significant and cannot be predicted on
the basis of an independent association between
plants and animals. On the contrary, the dis-
tribution of animal visits is highly dependent on
plant species (P G 0.00001, G-test in all nine
communities in which the test can be per-
formed). To illustrate the effect of such weak
dependences on community coexistence, we
used a mutualistic model (18–21). For the
simplest case, there is a positive community
steady state (community coexistence) if the
following inequality holds (21)
ab G
ST
mn
where a and b are the average per capita effects
of the animals on the plants, and of the plants
on the animals, respectively. Hereafter, such
per capita effects are estimated by the mutual
dependence values (21). S and T are the aver-
age intraspecific competition coefficients of
plants and animals, and n and m are the number
of plant and animal species, respectively.
As community size increases, the product of
mutual dependences has to become smaller for
the community to coexist (fig. S1). Two situ-
ations fulfill this requirement: (i) either both
dependences are weak; or (ii) if one dependence
is strong, the accompanying dependence is very
weak (so the product remains small). The
dominance of weak dependences (Fig. 2)
contributes to situation i. To assess the likeli-
hood of scenario ii, we next look at the asym-
metry of mutual dependences.
For each pair of plant species i and animal
species j, we calculated the observed asym-
metry of mutual dependences using (17). The
frequency distribution of asymmetry values is
also very skewed, with the bulk of pairwise
interactions being highly asymmetric (Fig. 3).
The question now is whether dependence pairs
are more asymmetric than expected by chance.
To answer this question, we calculated a null
frequency distribution of asymmetry values to
compare with the observed one by means of a
c
2
test. We achieved this by fixing the observed
dependence d
P
ij
of plant species i on animal
species j and randomly choosing d
A
ji
without
replacement from the set of all dependen ces of
the animals on the plants in this p articular com-
munity. This procedure was repeated 10,000
times; the null asymmetry frequency distribu-
tion is the average of these replicates.
For pollination, only seven out of 19 com-
munities (36.8%) showed a frequency distri-
bution of asymmetry values that deviates
significantly from the null frequency distribution
(46.1% when considering only networks with at
least 100 pairs). For seed dispersal, only one out
of seven communities (14.3%) showed a fre-
quency distribution of asymmetry values that
deviates significantly from the null frequency
distribution (20.0% when considering only net-
works with at least 100 pairs). These results
show that in the bulk of the cases, the frequency
distribution of asymmetry values originates
exclusively from the skewed distribution of
dependences. That is, most communities show
mutual dependences that are asymmetric, but no
more asymmetric than what we would expect by
chance, given the distribution of dependence
values.
Because strong interactions h ave the potential
to destabilize ecological networks (16, 18, 22–24),
we repeated the above calculations considering
only dependence pairs in which at least one
value is larger than or equal to 0.5 (other thresh-
old values do not significantly affect our
results). The fraction of large pollination net-
works (at least 100 pairs) with a frequency
distribution of asymmetry significantly depart-
ing from expectation increased to 87.5% (seven
out of eight communities). Similarly, for seed
dispersal, the three largest communities (n Q 20
pairs) also have frequency distributions of
asymmetry values significantly departing from
random (100%). Overall, these results suggest
that there are constraints in the combination of
strong mutual dependence values. Next, from the
significant comparisons, we explored which in-
tervals of asymmetry contribute to significance.
Asymmetry values range from 0 to 1 (Fig. 3).
Within this range, some values may be over-
1
Integrative Ecology Group, Estacio´n Biolo´gica de Don
˜
ana,
Consejo Superior de Investigaciones Cientı
´ficas,
Apartado
1056, E-41080 Sevilla, Spain.
2
Department of Ecology and
Genetics, University of Aarhus, Ny Munkegade, Building
540, DK-8000 Aarhus, Denmark.
*To whom correspondence should be addressed. E-mail:
bascompte@ebd.csic.es
www.sciencemag.org SCIENCE VOL 312 21 APRIL 2006
431
REPORTS
represented and some underrepresented, relative
to random expectation (again comparing the
null frequency distribution with the observed
frequency distribution by using a c
2
proba-
bility distribution). We found that the first half
of the range (low to moderate asymmetry) is
significantly underrepresented (P 0 3.81
10
j6
for pollination and P 0 0.0156 for seed
dispersal; binomial test). This underrepresenta-
tion of low asymmetry values implies that a
strong dependence value for one of the partners
in the mutualistic interaction tends to be
accompanied by a weak dependence value of
the other partner. That is, two strong inter-
actions tend to be avoided in a pair, which
agrees with the analytic prediction (scenario ii).
Our above analysis of mutual dependences,
however, is based on isolated analysis of pair-
wise interactions and thus provides only limited
information on the complexity of the whole
mutualistic network (25). For example, how does
the pattern of skewed dependences and strong
asymmetries scale up to account for properties at
the community level? A more meaningful
measure of network complexity is provided by
the concept of species strength (25). The strength
of an animal species, for example, is defined as
the sum of dependences of the plants relying on
this animal. It is a measure of the importance of
this animal from the perspective of the plant set
(Fig. 1, D and E). This measure is a quantitative
extension of the species degree, which is the
number of interactions per species in qualitative
networks (5). Previous work showed that mu-
tualistic networks are highly heterogeneous
(i.e., the bulk of species have a few interactions,
but a few species have many more interactions
than expected by chance) (5). Next, we con-
sidered how this result stands when quantitative
information is considered.
In all but one case, there is a significant
positive relationship between species strength
and species degree (Fig. 4). To explore devia-
tions from linearity, we performed a quadratic
regression and teste d for the significance of the
quadratic term. The quadratic term is signifi-
cant in 35 out of the 52 cases (for each com-
munity, we looked at both plants and animals
independently). This fraction increases to 24
out of 30 cases when considering only com-
munities with at least 30 species. That is,
species strength increases faster than species
degree (Fig. 4), a pattern previously found for
the worldwide airport network, but not for the
scientific c ollabor ation network (25). The
strength of highly connected species is even
higher than expected based on their degree,
because specialists tend to interact exclusively
with the most generalized species (6, 7)andso
depend completely on them. Thus, specialists
contribute disproportionately to increase the over-
all strength of the generalists they depend on.
Overall, previous results based on qualitative
networks (i.e., their high heterogeneity in the
numberoflinksperspecies)(5)areconfirmed
by our analysis of quantitative networks. Second,
previous work (i.e., asymmetry at the species
level) (6, 7) provides a mechanistic explanation
for some of the new results presented here as the
higher-than-expected strength of generalist spe-
cies. However, our results go a step further, be-
causeweshowherethatasymmetryisalsoa
property at the link level based on species-specific
mutual dependences.
Our results suggest that the architecture of
quantitative mutualistic networks is character-
ized by the low number of strong dependences,
their asymmetry, and the high heterogeneity in
species strength, all of which may promote com-
munity coexistence. Community coexistence, in
turn, may favor the long-term persistence of re-
ciprocal selective forces r equired for the coevo-
lution of these species-rich assemblages (2, 3).
By considering mutualistic networks as coevol-
ved structures rather than as diffuse multi-
specific interactions, we can better understand
how these networks develop (3). There are two
forces that, acting in combination, may lead to
networks with the reported architecture: coevolu-
Fig. 1. A network approach to plant-
animal mutualisms. (A) Example of a
community of plants and their seed dis-
persers in Cazorla, SE Spain (see Database
S1 for references and data sets). Green
circles represen t plant species and red
squares repr esen t animal species. A plant
and an animal interact if there is a
qualitative link between them. (B and C)
Each of the above plant-animal interac-
tions is described by two weighted links
(arrows) depicting the relative depen-
dence of the plant on the animal (green
arrow) and the animal on the plant (red
arrow). The asymmetry of the pairwise in-
teraction is proportional to the difference
between the thickness of both arrows.
Here we show a symmetric (B) and an
asymmetric (C) example. (D and E)A
species degree is the number of inter-
actions it has with the other set. Species
strength is the quantitat ive extension of
species degree, and can be defined as the
sum of dependences of the animals on
the plant (D) and the plants on the ani-
mal (E). Although the degree is four in
both(D)and(E),thestrengthoftheanimal(E)ishigherthanthatoftheplant(D).
A
D
E
B
C
0
0.2
0.4
0.6
0
0.2
0.4
0.6
Probability
0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.2 0.4 0.6 0.8 1
Dependence
0.2 0.4 0.6 0.8 1
A
B
C
F
I
H
E
D
G
Fig. 2. Frequency distributions of dependence values within a mutualistic community. Green solid
histograms (A to F) represent dependences of plants on pollinators, and red dashed histograms (G to
I) represent dependences of seed dispersers on plants. See Database S1 for references and data sets.
21 APRIL 2006 VOL 312 SCIENCE www.sciencemag.org
432
REPORTS
tionary complementarity and coevolutionary con-
vergence (3). Pairwise interactions build up on
complementary traits of the plants and the ani-
mals (e.g., corolla and pollinator tongue lengths),
whereas the convergence of traits allows other
species to attach to the network as this evolves
(e.g., convergence in fruit traits among plants
dispersed by birds rather than mammals) (3).
These forces differ from those shaping antago-
nistic interactions such as coevolutionary alte r -
nation (i.e., selection favoring herbivores attacking
less defended plants) (2, 3). Thus, one could pre-
dict differences in the architecture of mutualis-
tic and antagonistic networks. Other types of
biological interactions also show high asym-
metry values. For example, a large fraction of
competitive interactions are asymmetric, espe-
cially in the marine intertidal (26, 27). Our re-
sults highlight the importance of asymmetric
interactions in mutualistic networks. Asym-
metry seems to be the key to both their diver-
sity and coexistence. Whether asymmetry
extends to other types of complex networks
remains to be seen.
References and Notes
1. P. R. Ehrlich, P. H. Raven, Evolution 18, 586 (1964).
2. J. N. Thompson, The Coevolutionary Process (Univ. of
Chicago Press, Chicago, IL, 1994).
3. J. N. Thompson, The Geographic Mosaic of Coevolution
(Univ. of Chicago Press, Chicago, IL, 2005).
4. J. Memmott, Ecol. Lett. 2, 276 (1999).
5. P. Jordano, J. Bascompte, J. M. Olesen, Ecol. Lett. 6,69
(2003).
6. J. Bascompte, P. Jordano, C. J. Melia´n, J. M. Olesen,
Proc. Natl. Acad. Sci. U.S.A. 100, 9383 (2003).
7. D. P. Va´zquez, M. A. Aizen, Ecology 85, 1251 (2004).
8. L.-F. Bersier, C. Banasek-Richter, M.-F. Cattin, Ecology 83,
2394 (2002).
9. P. Jordano, Am. Nat. 129, 657 (1987).
10. D. P. Va´zquez, W. F. Morris, P. Jordano, Ecol. Lett. 8, 1088
(2005).
11. R. E. Ulanowicz, W. F. Wolff, Math. Biosci. 103, 45 (1991).
12. R. T. Paine, Nature 355, 73 (1992).
13. W. F. Fagan, L. E. Hurd, Ecology 75, 2022 (1994).
14. D. Raffaelli, S. Hall, in Food Webs, Integration of Patterns
and Dynamics, G. Polis, K. Winemiller, Eds. (Chapman &
Hall, New York, 1995), pp. 185–191.
15. J. T. Wootton, Ecol. Monogr. 67, 45 (1997).
16. J. Bascompte, C. J. Melia´ n, E. Sala, Proc. Natl. Acad. Sci.
U.S.A. 102, 5443 (2005).
17. The asymmetry of a pairwise mutualistic interaction is
estimated as follows: AS(i, j) 0kd
P
ij
j d
A
ji
k/max(d
P
ij
, d
A
ji
),
where d
P
ij
and d
A
ji
are the relative dependences of plant
species i on animal species j and of animal species j on
plant species i, respectively; max(d
P
ij
, d
A
ji
) refers to the
maximum value between d
P
ij
and d
A
ji
. Related measures of
asymmetry are highly correlated to this equation, so
results are insensitive to the particular asymmetry
measure used.
18. R. M. May, Stability and Complexity in Model Ecosystems
(Princeton Univ. Press, Princeton, NJ, 1973).
19. R. M. May, in Theoretical Ecology, R. M. May, Ed. (Sinauer,
Sunderland, MA, ed. 2, 1981), pp. 78–104.
20. M. S. Ringel, H. H. Hu, G. Anderson, M. S. Ringel, Theor.
Pop. Biol. 50, 281 (1996).
21. Materials and methods are available as supporting
material on Science Online.
22. K. McCann, A. Hastings, G. R. Huxel, Nature 395, 794
(1998).
23. G. D. Kokkoris, A. Y. Troumbis, J. H. Lawton, Ecol. Lett. 2,
70 (1999).
24. A. Neutel, J. A. P. Heesterbeek, P. C. Ruiter, Science 296,
1120 (2002).
25. A. Barrat, M. Barthe´lemy, R. Pastor-Satorras, A. Vespignani,
Proc. Natl. Acad. Sci. U.S.A. 101, 3747 (2004).
26. R. T. Paine, J. Anim. Ecol. 49, 667 (1980).
27. T. W. Schoener, Am. Nat. 122, 240 (1983).
28. We thank P. Amarasekare, J. E. Cohen, W. Fagan, M. A.
Fortuna, P. Guimara
˜
es, T. Lewinsohn, N. Martinez, R. M.
May, C. J. Melia´n, R. T. Paine, A. G. Sa´ ez, G. Sugihara,
J. N. Thompson, and A. Valido for comments on a previous
draft. J. E. Cohen and M. A Fortuna provided technical
assistance. Funding was provided by the Spanish Ministry
of Science and Technology (grants to J.B. and P.J.), the
Danish Natural Sciences Research Council (to J.M.O.), and
the European Heads of Research Councils and the
European Science Foundation through an EURYI award
(to J.B.).
Supporting Online Material
www.sciencemag.org/cgi/content/full/312/5772/431/DC1
Materials and Methods
Fig. S1
Database S1
References
5 December 2005; accepted 27 February 2006
10.1126/science.1123412
0 5 10 15 20
0
2
4
0 10203040
0
5
10
0 5 10 15 20 25
0
4
8
0 50 100
0
40
80
Species strength
010203040
0
10
20
30
010203040
0
20
40
0 5 10 15
0
2
4
6
0 5 10 15 20
Species degree
0
4
8
0 5 10 15 20 25
0
10
20
A
B
C
D
E
F
G
H
I
Fig. 4. Relationship between the number of interactions per species (degree) and its quantitative
extension, species strength. (A to C) Pollinator species in plant-pollinator communities. (D to F)
Plant species in plant-pollinator communities. (G and H) Animal species in plant seed–disperser
communities. (I) Plants in a plant seed–disperser community. A quadratic regression is represented
when the quadratic term is significant; otherwise a linear regression is plotted (G). As noted, in all
cases but (G), species strength increases faster than species degree. See Database S1 for references
and data sets.
0
0.2
0.4
0.6
0
0.2
0.4
0.6
Probability
0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.2 0.4 0.6 0.8 1
Asymmetry
0.2 0.4 0.6 0.8 1
AB
C
F
I
H
E
D
G
Fig. 3. Frequency distributions of asymmetry values of mutual dependences within a mutualistic
community. (A to F) Plant-pollinator communities. (G to I) Plant seed–disperser communities. See
Database S1 for references and data sets.
www.sciencemag.org SCIENCE VOL 312 21 APRIL 2006
433
REPORTS
www.sciencemag.org/cgi/content/full/312/5772/431/DC1
Supporting Online Material for
Asymmetric Coevolutionary Networks Facilitate Biodiversity
Maintenance
Jordi Bascompte,* Pedro Jordano, Jens M. Olesen
*To whom correspondence should be addressed. E-mail: bascompte@ebd.csic.es
Published 21 April 2006, Science 312, 431 (2006)
DOI: 10.1126/science.1123412
This PDF file includes:
Materials and Methods
Fig. S1
References
Other Supporting Online Material for this manuscript includes the following:
(available at www.sciencemag.org/cgi/content/full/312/5772/431/DC1)
Database S1 as zipped archive
SUPPORTING ONLINE MATERIAL
The Database
A compressed Excel file is sent separately with the quantitative database
(Data-BA.zip). It contains a description of each community, its reference, a
list of plant and animal species, and the dependence and asymmetry values
for each pairwise interaction.
Materials and Methods
The following model, an extension of the two-species mutualistic model
by Robert May and others (S1, S2), describes the dynamics of a set of n
plant species and m animal species interacting mutualistically:
dP
i
dt
= r
i
P
i
− S
i
P
2
i
+
m
X
j=1
α
ij
P
i
A
j
, (1)
dA
j
dt
= q
j
A
j
− T
j
A
2
j
+
n
X
i=1
β
ji
P
i
A
j
, (2)
where P
i
and A
j
represent the abundances of plant i and animal j, respec-
tively; r
i
and q
j
are the growth rates of plant i and animal j, respectively;
S
i
and T
j
are the intraspecific competition coefficients of plant i and animal
j, respectively; α
ij
is the per-capita effect of animal j on plant i; β
ji
is the
per-capita effect of plant i on animal j; n is the number of plant species, and
m is the number of animal species.
1
Note that the above model can be generalized to describe the dynamics
of other 2-mode networks describing, for example, the interactions between
hosts and their parasitoids or plants and their herbivores.
For the sake of analytical simplification, let us assume that all plant
species are equivalent (r
i
= r, S
i
= S, α
ij
= α), and all animal species are
equivalent (q
j
= q, T
j
= T , β
ji
= β). Although a strong simplification, this
strategy is commonly used in ecology to obtain analytic, general conclusions
(see, e.g. ref. S3). In the steady state (P
i
= P
∗
∀i, A
j
= A
∗
∀j), the
previous system can be rewritten as.
dP
i
dt
= 0 = rP
∗
− SP
∗2
+ mαP
∗
A
∗
, (3)
dA
j
dt
= 0 = qA
∗
− T A
∗2
+ nβP
∗
A
∗
. (4)
There are four different solutions of the above system: (0, 0), (r/S, 0),
(0, q/T ), and a non-trivial coexistence solution given by:
P
∗
= −
rT + mαq
nβmα − T S
, (5)
A
∗
= −
qS + nβr
nβmα − T S
. (6)
The above non-trivial steady state will be positive if and only if:
αβ <
ST
mn
, (7)
2
provided that all parameters are positive. Positive growth rates can be as-
sumed for facultative mutualisms.
Note that for the case of one plant and one animal, equation (7) becomes
also the condition for the stability of the feasible steady state (S2). Although
equation (7) necessarily rests on the simplifying assumption of identical pa-
rameter values for each set, the results are robust with respect to departures
from this symmetric case as shown by numerical simulations (fig. S1). The
term αβ can thus be generalized as the average product of per-capita effects
across the plant-animal pairs.
While our empirical measure of dependence d
P
ij
is a static index represent-
ing a relative frequency of visits, the parameter α
ij
in the model represents
a dynamic measure. Our approach, thus, assumes that per-capita effects
can be estimated by dependences. A recent paper (S4) provides strong sup-
port for this assumption, as the frequency of interactions has been shown to
be highly correlated with the total reproductive effect in plant populations.
Also, one can theoretically show that the correlation between total per-capita
reproductive effect and interaction frequency will be higher the greater the
variation of dependences among species (S4). This is fulfilled by the high
heterogeneity in dependence values reported in here.
As with any theoretical exercise, our model makes strong assumptions to
be able to provide simple, straightforward predictions. Our model assumes
a fully connected, randomly interacting network, while we now know that
plant-animal mutualistic networks are highly structured (i.e., non-random),
3
and have a much lower density of links (see however inset in fig. S1). Mod-
els of mutualisms are also intrinsically destabilizing, so model (1-2) is only
bounded if inequality (7) holds (left side of isocline in fig. S1). Previous work
has looked at stabilizing factors such as temporal or spatial variability or third
species such as predators or competitors (S2). However, despite the simpli-
fications of the model, it can be shown that close to equilibrium, it behaves
qualitatively similarly to related models incorporating more realism, so con-
clusions derived from this model about coexistence are not significantly dif-
ferent from those derived from more realistic models (S2, S5). Also, one has
to look at this model not as a realistic representation of mutualistic networks,
but as an exercise of the type other things being equal, larger communities
have to contain weak, asymmetric mutual dependences to coexist. An impor-
tant follow up of this paper will be to explore more realistic dynamic models
accounting for the structure of real mutualistic networks. A recent paper
(S6) has used such a model and compared its predictions with predictions
from a random network of interactions. Although persistent quantitative
differences were found, the overall results were qualitatively similar.
4
0
25 50 75
100
Community Size
0
0.1
0.2
0.3
0.4
0.5
Product of Mutual Dependences
Non-Coexistence
Coexistence
0
25 50 75
100
Community Size
0.0
0.5
1.0
1.5
2.0
Product of Mutual Dependences
Non-Coexistence
Coexistence
Figure S1. Robustness of the analytical result (community coexistence
criterion in main text) as we relax the assumption of symmetry in parame-
ter values across species. The critical average product of mutual per-capita
effects (dependences hereafter) separating the domain of coexistence of the
feasible community steady state is plotted as a function of community size
(animal species × plant species). Solid red line represents the isocline given
by expression (7). Parameters are: S = 1, T = 2. Dots (interpolated by bro-
ken lines) correspond to numerical simulations of system (1-2). We assume
communities with the same number of plants and animals and the following
parameter values: r
i
and q
j
are randomly sampled from a uniform distribution
with means 1 and 0.65, respectively, and variance 0.2 and 0.1, respectively;
S
i
and T
j
are sampled from a uniform distribution with means 1 and 2 (as
the analytic case) and variance 0, 10, 20, and 30%, respectively shown by dif-
ferent colors. To tune the average product of mutual dependences, all plants
and animals in system (1-2) have the same dependence value; the square of
such a value is the product of mutual dependences. Inset represents a similar
analysis considering a connectivity equal to 0.2 instead of a fully connected
matrix, and sampling each dependence value from a uniform distribution with
the same mean as before and variance of 20%. Rest of parameters as before.
5
We represent the average and SD of 10 replicates. Solid line is the power
regression of the means.
6
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7
