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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

References

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Ecology, R.M. May, ed. 2nd Edition, Sinauer, Sunderland MA (1981).

S2. Ringel, M.S., Hu, H.H., Anderson, G. & Ringel, M.S. The stability

and persistence of mutualisms embedded in community interactions. Theor.

Pop. Biol. 50, 281-297 (1996).

S3. Gross, K. & Cardinale, B.J. The functional consequences of random

versus ordered species extinctions. Ecol. Lett. 8, 409-418 (2005).

S4. V´azquez, D.P., Morris, W.F. & Jordano, P. Interaction frequency as

a surrogate for the total eﬀect of animal mutualists on plants. Ecol. Lett. 8,

1088-1094 (2005).

S5. Goh, B.S. Stability in models of mutualism. Am. Nat. 113, 261-275

(1979).

S6. Fortuna, M.A. & Bascompte, J. Habitat loss and the structure of

plant-animal mutualistic networks. Ecol. Lett., 9, 281-286 (2006).

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