Topological Properties of FoodWebs: From Real Data toCommunity Assembly Models José M. MontoyaRicard V. Solé

SFI WORKING PAPER: 2001-11-069

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SANTA FE INSTITUTE

Topological properties of food webs: from real

data to community assembly models

Jose M. Montoya1,2 and Ricard V. Sole1,3

(1) Complex Systems Research Group, FENUniversitat Politecnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain

(2) Department of EcologyUniversity of Alcala, 28871 Alcala de Henares, Madrid, Spain

(3) Santa Fe Institute, 1399 Hyde Park Road, New Mexico 87501, USA

Type of article: Report.

Running title: Food web topology and assembly dynamics.

Keywords: food webs, scale-variant properties, degree distributions, assemblymodels, functional responses, community fragility, weak interaction strengths.

Number of words in the abstract: 148

Number of words in the report (abstract and figure legends included ): 4731

Biosketch

Jose M. Montoya works on theoretical ecology and macroevolution. Currently heis focusing on food web theory, specifically on how observed patterns emerge fromcommunity assembly models, and on how food web structure influences communityresistance to different kind of human-induced perturbations.

Mailing address:Jose M. Montoya, Complex System Research Group. Dept. Physics FEN-UPC,Campus Nord B4. 08034 Barcelona-Spain.FAX: +34-93-401-7100e-mail: montoya@complex.upc.es

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Abstract

We explore patterns of trophic connections between species in the largest andhighest-quality empirical food webs to date, introducing a new topological prop-erty called the degree distribution, defined as the frequency of species SL with Llinks. Non-trivial differences are shown in degree distributions between species-richand species-poor communities, which might have important consequences for ecosys-tems’ responses to disturbances. Species richness-connectance (SC) and number oflinks-species richness (LS) relationships observed provide no support for the theoryof LS scaling law or constant connectance for food webs. We further explore theseobservations by means of simulated food webs resulting from random multitrophicassembly models. SC and LS relationships of real food webs are reproduced, butdegree distributions are not properly predicted. The best agreement between em-pirical and simulated webs occurs when weak interaction strengths between speciesgovern food web dynamics.

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1 Introduction

Patterns in community structure do exist in nature. Studies of food web properties

have provided some of the clues for the understanding of ecosystem organization

and their relationship with different types of ecological stability (Pimm 1991; War-

ren 1994). Many of these properties have been shown to be artifacts due to the

incompleteness and biases of the data in which they are based (Polis 1991; Cohen et

al. 1993; Winemiller et al. 2001). Nonetheless, over the last decade, there has been

a great effort in documenting high-quality food web data, rejecting some previously

observed regularities and confirming others (Williams & Martinez 2000). Most of

these studies have been centered in the scale-invariant nature of food web patterns,

that is, whether observed patterns are roughly constant among webs with widely

varying size (in terms of number of species S), in order to determine if species-rich

and species-poor ecosystems are organized in a similar way.

Basic average properties of the entire food webs, like S, the number of actual

links L or direct connectance C (i.e., L divided by the maximum possible number

of links S2) have received considerable attention over the last decade (Pimm et

al. 1991; Martinez 1992; Havens 1992; Warren 1994; Murtaugh & Kollath 1997).

But none of these studies has dealt with the distribution of trophic links between

the species present in the community (the so-called degree distribution). In this

article we show that surprisingly, larger food webs exhibit skewed degree distribution

that strongly departs from what would be expected from a random wiring. In this

context, the importance of the topological patterns displayed by rich communities

has been recently shown to play a key role in community fragility (Sole & Montoya

2001). The deletion of the most-connected species (measuring both inward and

outward links) that are typical from food webs with skewed degree distributions

might trigger the coextinction of many others species from the community via direct

or indirect effects (Pimm 1980; Pimm 1991; Sole & Montoya 2001).

One central question in food web theory is how structural and static patterns

emerge from population dynamics of the interacting species. We have explored the

3

similarities and differences in food web patterns between real and simulated webs,

the latest constructed through multitrophic assembly models considering different

types of functional responses. We have found that basic average properties are

almost the same in real and constructed webs. This occurs in particular when

low values of per capita interaction strength between species are selected in the

model, thus predicting that most interactions in complex food webs might by weak,

confirming some previous theoretical and empirical findings (Paine 1992; Raffaelli

& Hall 1996; McCann et al. 1998). However, fine-grained properties as the degree

distributions of links can not be explained through random assembly dynamics,

providing an important clue for the future of food web modelization.

2 Food webs analyzed

We have studied the largest and highest-quality empirical food webs available in

the literature that were originally documented to study food web properties, al-

though they are still far from perfect, as their authors emphasize (Table 1). In

this collection there are webs belonging to different habitat types. Three are from

freshwater habitats: Skipwith pond, Little Rock lake and Bridge Brook lake; one is

from a marine habitat: the Benguela ecosystem; two belong to marine freshwater

interfaces: Chesapeake bay and Ythan estuary; the other five are from different

terrestrial habitats: Coachella valley from a desert, St. Martin from a caribbean

island, El Verde from a tropical rain forest, an UK grassland parasitoid community,

and Silwood park from the species related with the Scotish Broom Cytisus scoparius.

All of them are community food webs, with the exception of the Silwood Park web,

which is a source food web (Cohen et al. 1990; Memmot et al. 2000).

The number of species fluctuates from 30 (Coachella valley) to 182 (Little Rock

lake), reflecting a wide range of species richness. Some of these food webs differ in

their taxonomic resolution. Trophic species (i.e., groups of real taxonomic species

sharing a fraction of prey and predators) are present in most of the food webs. For

instance, the Coachella valley, Little Rock lake and El Verde webs present a high level

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of aggregation, whereas in the Silwood park or UK grassland webs all the species

are real taxonomic species. This should produce some bias in our results if well-

taxonomically resolved food webs would belong mainly to species-poor or species-rich

communities. But that is not the case. Differences in the level of aggregation are

present in some rich communities as well as in some poor communities. Furthermore,

connectance has been shown to be very robust under trophic aggregation (Martinez

1991, 1993). Another aspect that must be considered when comparing food web

data is the sampling effort (Cohen et al. 1993). The 12 food webs analyzed here

are far from being poor sampled. Sampling effort has been reported to have little

effect on the topological properties analyzed here. Connectance has also appeared

to be robust under different sampling efforts for some of the food webs present in

this study: for the Little Rock lake (Martinez 1991, 1994) and the UK grassland

(Martinez et al. 1999) webs. Thus, both a wider range of S and a higher degree of

taxonomic resolution is present for the food webs explored here than for previous

data collections (Sugihara et al. 1989; Cohen et al. 1990; Schoenly et al. 1991;

Martinez 1992, 1994).

Degree distributions of links among species has been also shown to be robust

for different levels of trophic aggregation and sampling effort. The two versions

of the Ythan estuary food web share a similar degree distribution despite their

differences in taxonomic detail: the second version was expanded by adding 42

metazoan parasite species that in the former version were lumped in a simple trophic

species (Huxham et al. 1996; Montoya & Sole 2001). We have measured the total

number of links (inward and outward) for each species in each food web. In figure

1 we represent some degree distributions along the SC curve corresponding to some

of the webs analyzed here. A clear trend can be observed as S increases: the degree

distributions become more skewed and far from what would be predicted from a

random distribution of links per species, which is typically Poissonian (Bollobas

1985). The SC relationship fits very well with a power-law and, in this manner, C

changes with the size of the webs.

5

Name S L C = L/S2 ReferenceSkipwith pond 37 351 0.26 Warren (1989)Little Rock lake 182 2371 0.07 Martinez (1991)

Bridge Brook lake 75 555 0.1 Havens (1992)Benguela ecosystem 29 187 0.22 Yodzis (1998)

Chesapeake bay 36 475 0.3 Baird & Ulanowicz (1989)Ythan estuary (1) 93 407 0.05 Hall & Raffaelli (1991)Ythan estuary (2) 134 583 0.03 Huxham et al. (1996)Coachella valley 30 241 0.27 Polis (1991)St. Martin island 44 217 0.11 Goldwasser & Roughgarden (1993)

El Verde 156 1428 0.06 Reagan & Waide (1996)UK grassland 87 128 0.02 Dawah et al. (1995)Silwood park 154 366 0.02 Memmot et al. (2000)

Table I: Summary of the average properties of the food webs analyzed. S refers to

the number of taxonomic species or species aggregations into trophic species. L are

the number of binary links contained in each web and C is the direct connectance

(Warren 1994). Two different versions of the Ythan estuary web are included,

showing that degree distributions of links between species are robust under different

sampling effort (see text for details).

3 Multitrophic assembly models

Food web patterns can be seen as the result of population interactions through time.

Can topological properties observed in empirical food webs be reproduced introduc-

ing community dynamics? One possible way to explore this crucial aspect of food

web theory is to perform community assembly experiments in silico. Most of the lit-

erature on assembly dynamics of multispecies ecosystems has dealt with competitive

communities (Case 1990; Morton et al. 1996) or randomly wired ecosystems without

trophic structure (May 1973; Pimm 1991 and references therein). Assembly models

of multispecies ecosystems with trophic structure have been fairly less developed,

starting from the early work by Pimm and collaborators (Pimm & Lawton 1978;

6

10−2 10−1 100

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Silwood El Verde Little Rock

Ythan (1)

Skipwith

Chesapeake

BenguelaSt. Martin

−0.55

Figure 1: SC relationship for the empirical food webs analyzed (central figure) anddegree distribution for some of them (figures at the peryphery). S scales with Cfollowing a power-law function SC−γ, being γ = 0.55 (power-law fit with least-squares regression on log-transformed data: r2 = 0.67, p < 0.002). Actual degreedistributions (vertical bars, where SL is the number of species with L links) areshown together with expected distributions from a random wiring (continuous lines),predicting Poissonian behaviour with mean equal to linkage density (L/S). Notethat as S increases, actual distributions depart from a random food web, showinglong tails.

7

Pimm 1979; Lockwood et al. 1997; see also Law & Blackford 1992). Here we follow

this latest approach, using an extended version of Pimm’s multitrophic assembly

model, introducing different types of functional responses in predator-prey dynam-

ics: linear (Holling type I), where per capita consumptiom rate of prey per predator

increases linearly with prey abundance, and prey-dependent where it increases in a

decelerating (Holling type II) or in a sigmoidal (Holling type III) fashion up to some

maximal attack rate (Holling 1965). Consider a set of Lotka-Volterra equations of

the form:

dxi

dt= xi

βi +

S∑j=1

αijΦ(xj)

(1)

where xi(i = 1, ..., S) indicate the population size for each species, βi are positive for

basal species and negative for non-basal species, and the matrix αij define the per

capita interaction strength of species i on species j, i and j belonging to different

trophic levels (for Holling types II and III, αij is the maximal attack rate, and hence,

a measure of interaction strength). The functional response of species i is given by:

Φ(xj) = xj, for linear functional response

Φ(xj) = xj

D+xj, for Holling type II

Φ(xj) =x2

j

D2+x2j, for Holling type III

where D is the half-saturation density of prey consumed by predator, and it deter-

mines the shape of the increase of both functional responses.

The values of βi are chosen randomly from an uniform distribution between 0

and 1; D is fixed to a constant value for all the species and αij are chosen from a

given distribution ρ(α) (i. e.,∫

ρ(α)dα = 1). Here we take the simplest one, an

uniform distribution ρ(α) ≡ 1/αm, where αm is the maximum allowed per capita

interaction strength. Variations in this maximum value result in different model

outcomes. The signs of the interactions are chosen accordingly by considering the

nature of the interacting species. Here three sets of species of equal size are used:

8

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Figure 2: SC relationship for simulated food webs (central figure) and degree distri-bution for some of them (figures at the peryphery). In this case, linear functionalresponse has been selected, the exponent for the power-law is γ = 0.5 (power-lawfit with least-squares regression on log-transformed data: r2 = 0.84, p < 0.0001).Actual degree distributions (vertical bars) and expected random ones are as infigure 1. The trend observed in empirical food webs is not present here. Hereαm = 0.2, ti=100, and C, S and degree distributions are the average values overthe latest 15000 iterations. Fits for other functional responses: Holling II: γ = 0.54(r2 = 0.61, p < 0.001); Holling III: γ = 0.6 (r2 = 0.72, p < 0.001)

9

Sb = {x1, ..., xs}, Si = {xs+1, ..., x2s} and St = {x2s+1, ..., x3s}, where S = 3s and

the subindexes b, i and t stand for basal, intermediate and top species, respectively.

Omnivory is allowed in our system, so that top species can feed both on intermediate

and basal species.

All these model features are predefined in a matrix that represents the regional

species pool. Degree distributions of links per species in species pools are constructed

randomly, and therefore, they are Poissonian. Each ti time iterations one species is

randomly chosen from the species pool to enter into the community at a population

density of 0.001 (as previous studies, e.g. Lockwood et al. 1997). One species is

considered extinct if its density is under that value.

We have performed different simulations for each of the functional responses, as

well as for diverse combinations between them. Calculations of S, C, L and degree

distributions are mean values over 15000 time steps. In doing so, model’s results

are more comparable to empirical data, which are collected through several years,

reflecting various community snapshots put together (Cohen et al 1993). In figure 2

the SC relationship is showed for simulated food webs with linear functional response

and αm = 0.2, as well as the degree distributions for some of them.

Several studies on theoretical and empirical measures of interaction strengths

advocate concentraiting in per capita interaction strengths rather than in other

measures (Laska & Wooton 1998; Berlow et al. 1999). Specifically, Lotka-Volterra

coefficients αij have been shown as one of the best approximations, because (1)

they do not necessarily require an assumption of equilibrium conditions as other

measures, so they are potentially applicable to a extensive set of situations and

(2) they are less sensitive to variation in species densities than another measures

that require almost fixed population values (see Laska & Wooton 1998 for a recent

review).

10

10-2

10-1

100

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101

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S

100

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103

S

100

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L

A B

Figure 3: SC (A) and LS (B) relationships for empirical food webs (black circles) andsimulated food webs (with linear functional response) with different αm: αm=0.2(empty squares), αm=0.1 (crosses), and αm=1.0 (plus signs). The best fit to empir-ical data is obtained for αm=0.2 (see figures 1 and 2 for regression statistics). Linesin B indicate predictions of the link-species scaling law (LSSL) (continuous line,L = 2S) and the constant connectance hypothesis (CCH) (dashed line, L = 0.14S2).No significant regressions are obtained for both real and simulated food webs overany of the two predictions: Real webs: with LSSL r2 = 0.011, p >> 0.05; with CCHr2 = 0.06, p >> 0.05. Simulated webs: with LSSL r2 = 0.62, p >> 0.05; with CCHr2 = 0.52, p >> 0.05.

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4 Results

Topological properties of the food webs generated are almost the same despite the

functional response selected. They reproduce basic average properties (SC and LS

relationships) of real webs, with the best agreement for low values of maximum

interaction strenght αm. However, they do not discriminate differences in degree

distributions for species-rich and species-poor ecosystems observed in real data. The

basic results can be summarized as follows:

(1) C is a scale-variant property with a very particular relationship with S: a

power-law S ∼ c−ε with an exponent ε ≈ −1/2 (figure 3A). The constant con-

nectance hypothesis (that is, C ≈ 0.14 despite changes in S) showed in some previ-

ous studies (Pimm et al. 1991; Martinez 1992; Havens 1992) does not hold for the

12 best described food webs to date.

(2) L increases with S in a different manner from what is predicted by both the

link-species scaling law (LSSL) and the constant connectance hypothesis (CCH).

Assuming the simplest relationship between L and S in the form of L = αSγ , the

LSSL states that γ must be close to one and that, on average, the number of links

per species in a web is constant and scale invariant at roughly two, and therefore,

L ≈ 2S (continuous line in figure 3B)(Cohen et al. 1990; see also Martinez 1992). By

contrast, the CCH states that the number of links in a web increases approximately

as the square of the number of trophic species, with α < 1, where α is the direct

connectance C of the web, which is assumed constant. Thus, L = CS2 (dashed

line in figure 3B, with C = 0.14) (Martinez 1991, 1992) Other studies rejecting the

link-species scaling law have found values of γ ≈ 1.5 (Sugihara et al. 1989; Schoenly

et al. 1991; Havens 1992; Martinez 1994). Real data as well as the model outcomes

do not follow any of the two hypotheses (see Figure 3 for regression statistics),

suggesting a more complex relationship than previous findings, that does not fit

any typical statistical distribution. The reason is twofold. First, the power-law

relationship between S and C, indicates that C is not constant, and hence CCH is

not supported. Second, the number of links per species is not constant in all the

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webs analyzed and therefore, the LSSL is not reproduced. Understanding the latest

reason needs the next point.

(3) Degree distributions of total links (outward plus inward) between species

vary in real food webs over a continuous transient from species-poor to species-rich

ecosystems. In this transition, communities with high S show skewed distributions

with long tails (e.g. power law fit of the log-transformed data with Ls = S−γ for

Ythan (2): γ = 1.04 ± 0.05 (r2 = 0.83, p < 0.01), and for Silwood γ = 1.13 ± 0.06

(r2 = 0.79, p < 0.01), Montoya & Sole 2001; Camacho, Guimera & Amaral, un-

published), a topology far from an expected random distribution of links, always

Poissonian (Bollobas 1985). By contrast, species-poor communities present a degree

distribution closer to that expected from a random wiring of connections. That

is, when species richness is high, most species from the community have very few

connections (i.e., there are many specialists) and only a few species are highly con-

nected (i.e., generalist preys and predators), whereas when species richness is low,

the number of connections of each species fluctuates around mean linkage density

< k >. Thus, beacuse of the high variability in links per species in species-rich food

webs, the LSSL can not adequately reproduce the relationship between L and S.

This hypothesis was originally based in a set of species-poor webs (Cohen & Briand

1984; Cohen et al. 1990). The introduction of species-rich communities with degree

distributions showing great fluctuations around < k > contradicts this hypothesis.

By contrast, the shape of degree distributions in simulated food webs is always Pois-

sonian for both species-poor and species-rich ecosystems (Figure 2). This indicates

an important caveat in current multispecies assembly models for the understanding

of fine-grained food web patterns and their relationship with community fragility.

(4) Simulated food webs whose patterns are as those observed in empirical food

webs are limited to a certain value of interaction strengths. For high values of αm,

no good fit is obtained. Besides, for values under αm = 0.2, SC and SL relationships

and degree distributions are inadequately reproduced (figure 3A). This suggests that

real communities migth be governed by weak interactions between species through

their development.

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5 Discussion

We have explored some of the basic food web patterns and addressed a new one, the

degree distribution of links per species, trying to understand their origin through

community dynamics. The shape of the network of trophic interactions in ecological

communities has been shown to be highly dependent from the number of species

present. These topological properties might be the result of an assembly process

and can be partially reproduced through simple multitrophic assembly models where

dynamics are dominated by weak interaction strengths between species.

The analysis of the 12 best-described food webs belonging to different habitat

types has shown that C is a scale-variant property and L does not follow neither the

link-species scaling law nor the constant connectance hypothesis. A recent study

concerning six different data sets of consumer taxocenes also reflect this lack of

congruence between real webs and these food web theories (Winemiller et al 2001).

Some previous analyses focused on different compilations have shown just the op-

posite for the SC relationship, suggesting that webs of different size are constrained

toward a roughly constant value of C ∼ 0.1 − 0.15 (Martinez 1992 and references

therein; Warren 1994). We have shown that C varies in real food webs as S in-

creases, from 0.27 to 0.02. For simulated webs the range is almost the same. The

fact that trophic aggregation and sampling effort is not higher for species-rich or for

species-poor communities gives robustness to this observation (see Williams & Mar-

tinez 2000). It is interesting to notice that our models (for values of αm ≈ 0.2) were

not able to generate food webs with C < 0.25, suggesting the existence of at least

two different types of dynamical constraints through the assemblage of ecological

communities: (a) a threshold of C over which communities can not be assembled

and, (b) (under this threshold) a power law decay of C as S increases, with an

exponent close to −1/2. The latest is a rediscovery of the inverse hyperbolic rela-

tionship between S and C obtained from the EcoWeb database (Cohen & Briand

1984; Cohen et al. 1990), but the food web data in which our observation is based

were documented specifically to address food-web patterns, whereas previous data

14

collections were not, providing a poor basis for assesing the SC relationship (Polis

1991; Cohen et. al. 1993). Another food web patterns seem to underlay in SC re-

lationships: the niche model of Williams & Martinez (2000) (with one-dimensional

trophic niche, without temporal dynamics), predicts a dozen properties of some of

the webs analyzed here with only two parameter inputs: S and C.

Why do these patterns in food web connectance happen? Many hypotheses have

been proposed for explaining them (see Warren 1994 for a review). The most de-

veloped and stimulating is the relationship of C, as a measure of complexity, with

different types of ecosystem stability. Actually, there are two different hypotheses,

generating the complexity-stability debate. Several ecological models have suggested

that lower values of connectance involve higher local (May 1973; Pimm 1991 ) and

global (Cohen et al. 1990) stability, that is, the system recovers faster after a dis-

turbance. By contrast, new theoretical results suggest that more-connected systems

may have a greater number of viable reassembly pathways, and hence, they could

recover faster from perturbation (Law & Blackford 1992). But connectance is a

coarse-grained property not enough to determine wheter a species-rich community

is more or less stable than a species-poor one; we need to break up connectance into

parts. In doing so, we have incorporated two basic aspects of ecosystem organization

that might have important consequences for the complexity-stability debate: degree

distributions of links per species and per capita interaction strengths.

Degree distributions of links per species have very important consequences for

community fragility and persistence (Sole & Montoya, 2001). Food webs with skewed

degree distributions (species-rich communities in our analysis) experiment a dico-

tomic behaviour: they display a high homeostasis when species are removed at

random from the community but they are very fragile under removals directed to

generalist or most-connected species (considering total number of links, outward plus

inward, as well as considering only outward links, see Sole & Montoya 2001). Here

fragility is measured as the number of coextinctions triggered and the degree of frag-

mentation of the food web into sub-webs disconnected among them. By contrast,

food webs with Poissonian degree distributions (similar to species-poor communities

15

in our analysis) are equally highly fragile to both types of species removals (random

or directed). Thus, the fragility (a measure of stability) of real communities to

perturbations that eventually lead into the extinction of some species is related to

the shape of degree distributions. The observation that this shape changes from

richer to poorer real and simulated communities might have deep implications for

ecosystem organization and its resistance to perturbations. Perturbations in a low-S

community with low S might have large effects despite the number of connections

that the species mainly affected have. This is due to the fact that in these com-

munities species tend to have a number of connections around mean linkage density

< k >. An increase in S implies a more complex heterogeneous distribution, where

few species are playing a key trophic role in the persistence of the community (the

most-connected ones) whereas a huge number of them (the ones with only a few

connections) do not seem to be so important in terms of community persistence

(at least from a strictly trophic point of view). In this manner, stochastic environ-

mental fluctuations migth have less effect on species-rich communities whereas some

human-induced perturbations, such as habitat loss and fragmentation, which quite

often affect to highly-connected species (Owen-Smith 1987; Wilson 1992) might have

large effects on the system.

The assembly dynamics incorporated in our models could not reproduce the

skewed distributions observed in the richest-species empirical food webs, even in-

troducing explicitly the interference between predators generated through the con-

sumption of shared preys (see Arditi & Ginzburg 1989). Thus, multitrophic assem-

bly models with randomly-constructed species pools, can not explain the statistical

relationships between specialist and generalist species observed in ecosystems with

high S. Future multitrophic assembly models should incorporate more ecological re-

alism, perhaps with the introduction of space. In this way, species with large home

ranges could have more preys to feed on and more predators, increasing their number

of connections. Species where individuals have large home ranges are particularly

vulnerable to habitat loss, fragmentation and degradation, specially to edge effects

(Woodrofe & Gingsberg 1998; Purvis et al. 2000), and in most cases are reported

16

as the generalist species of the ecosystem (Owen-Smith 1987; Wilson 1992).

It has been extensively argued that consumer-resource interaction strengths are

skewed toward weak interactions (Paine 1992; Berlow et al. 1999; McCann 2000).

Some of the webs analyzed here support this observation (UK grassland, Dawah et

al. 1995; Ythan estuary, Raffaelli & Hall 1996; in some sense, St. Martin Island, see

Goldwasser & Roughgarden 1993). This have important consequences for commu-

nity stability and species persistence (May 1973; Laska & Wooton 1998; McCann

et al. 1998; McCann 2000) since: (1) the weak-interaction effect generates negative

covariances between resources (preys) that promotes community-level stability, and

(2) these negative covariances ensure that the species that interact weakly dampen

the destabilizing potential of strong interactions (McCann 2000). The topological

properties of the empirical food webs analyzed here were more accurately reproduced

by using a certain value of maximum per capita interaction strength αm = 0.2. Very

strong interactions are not present, so population fluctuations can be easily damped

in general (negative covariances) as well as the relative strong interactions present.

Hence, weak interactions between species must underly the food web dynamics of

the communities reported here both in species-rich and species-poor systems. There-

fore, we can conclude that, among others, there might be three basic requirements

for maintaining high levels of biodiversity in ecological communities from a trophic

perspective: low mean connectance, skewed degree distributions and the dominance

of weak interactions between species.

Acknowledgments

The authors thank Nikita Davinof, David Alonso and Javier Gamarra for help in

the analysis of some of the food webs and useful discussions on theoretical ecology.

Santa Fe Institute night radio station was a great source of inspiration. This work

has been supported by a grant Comunidad de Madrid FPI-4506/2000 (JMM) and

CICYT PB97-0693 and The Santa Fe Institute (RVS).

17

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