# Effects of heterogeneous interaction strengths on food web complexity

**ABSTRACT** Using a bioenergetic model we show that the pattern of foraging preferences greatly determines the complexity of the resulting food webs. By complexity we refer to the degree of richness of food-web architecture, measured in terms of some topological indicators (number of persistent species and links, connectance, link density, number of trophic levels, and frequency of weak links). The poorest food-web architecture is found for a mean-field scenario where all foraging preferences are assumed to be the same. Richer food webs appear when foraging preferences depend on the trophic position of species. Food-web complexity increases with the number of basal species. We also find a strong correlation between the complexity of a trophic module and the complexity of entire food webs with the same pattern of foraging preferences.

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Page 1

Effects of heterogeneous interaction strengths on food web

complexity

Josep L. Garcia-Domingo and Joan Saldan ˜a

J. L. Garcia-Domingo and J. Saldan ˜a (jsaldana@ima.udg.edu), Dept. Informa `tica i Matema `tica Aplicada, Univ. de Girona, ES-17071

Girona, Catalunya, Spain. JLGD also at: Dept. d’Economia, Matema `tica i Informa `tica, Univ. de Vic, ES-08500 Vic, Catalunya, Spain.

Using a bioenergetic model we show that the pattern of foraging preferences greatly determines the complexity of the

resulting food webs. By complexity we refer to the degree of richness of food-web architecture, measured in terms of some

topological indicators (number of persistent species and links, connectance, link density, number of trophic levels, and

frequency of weak links). The poorest food-web architecture is found for a mean-field scenario where all foraging

preferences are assumed to be the same. Richer food webs appear when foraging preferences depend on the trophic

position of species. Food-web complexity increases with the number of basal species. We also find a strong correlation

between the complexity of a trophic module and the complexity of entire food webs with the same pattern of foraging

preferences.

The introduction of interaction strengths (IS) in the

description of ecological networks is a key require-

ment in understanding their dynamics (Yodzis 1981,

Berlow 1999, Dell et al. 2005) and amounts to the so-

called weighted networks, where nodes represent species (or

local populations in metapopulation models) and links

represent pairwise interactions among them. In food webs

such interactions are given by trophic relationships and

their strength, which gives the corresponding link weight,

has been defined in several different ways according to the

approach used for the food-web analysis. In perturbation

analysis, IS refer to the impact of a change in the properties

of a given link on the dynamics of other species; when

modeling food-web dynamics, they refer to a feature of an

individual link that does not take into account the impact

on the rest of the network (Berlow et al. 2004).

Recently, the influence of IS on the stability of food

webs has been discussed for real communities (Berlow

1999, Navarrete and Berlow 2006, Neutel et al. 2002) and

in food-web models with a large number of species

(Haydon 2000, Kokkoris et al. 2002, Jansen and Kokkoris

2003, Emmerson and Raffaelli 2004, Quince et al. 2005).

In Martinez et al. (2006), the authors studied the effect of

the variation of the IS of omnivore links on the stability of

large food webs. In these models analytical results are in

general unexpected (but see Chen and Cohen 2001).

In this paper, our main concern is to explore the

influence of the IS on the food-web complexity of a

dynamic bioenergetic model. By complexity we refer to the

richness of the food-web architecture, measured in terms of

number of species and links, link density, connectance,

number of trophic levels, and frequency of weak links.

Changes in the IS are obtained via changes in the pattern

of foraging preferences. The foraging preference of a

predator for one of its prey is the maximum assimilation

rate of predator species when consuming its prey. In our

model, this feature of the predator-prey relationship is

assumed to depend on the trophic position of species

classified as top (T, species without predators), basal

(B, species without prey) and intermediate (I, species with

prey and predators). This yields four possible values

(namely, T-B, T-I, I-I, and I-B) that constitute what we

call the pattern of foraging preferences.

When modeling trophic interactions, IS are supposed to

depend on factors like metabolic efficiencies, handling

times, foraging efficiencies, or frequencies of encounters,

for example. In all these cases, per capita IS have been

traditionally taken constant in time. However, in recent

works, a more realistic modeling of food web dynamics has

introduced time-varying interactions by considering adap-

tive foragers (Brose et al. 2003, Kondoh 2003, 2006,

Garcia-Domingo and Saldan ˜a 2007, Uchida and Drossel

2007). Foraging adaptation permits changes in the food

diet of a predator so that it can optimize its energy intake by

updating the effort invested in each of its prey. This

adaptation modifies predator-prey IS and, thus, acts on the

topology of the network by eventually removing those links

with zero strength. As a consequence, the complexity of the

food web is molded by the action of the dynamics with

adaptive predators.

Oikos 117: 336?343, 2008

doi: 10.1111/j.2007.0030-1299.16261.x,

# The Authors. Journal compilation # Oikos 2007

Subject Editor: Jordi Bascompte, Accepted 28 September 2007

336

Page 2

In Kondoh (2003) and Brose et al. (2003), the influence

of the foraging adaptation on the connectance-persistence

relation was studied, and these results were argued in

Garcia-Domingo and Saldan ˜a (2007). In all these works, a

mean-field approach was considered since all IS were

independent of the species trophic position. We will show

that this homogeneous scenario leads to food webs with a

very poor structure, far away from real ecosystems. On the

contrary, more complex food webs arise assuming hetero-

geneous patterns of foraging preferences. Following the

discussion in Berlow et al. (2004), our results illustrate the

need of plausible estimations of such biological parameters.

We also analyze the role of the number of basal species

in the emergence of complexity in the food-web structure.

This issue was considered in Kondoh (2006) in terms of the

directed connectance for a particular choice of parameter

values that leads to unrealistic final food-web configurations

regardless the number of basal species. By contrast, we will

see that for other parameter values there is a rise in species

richness and in link density as the number of basal species

increases, although this gain is not directly translated into a

rise in connectance unless a minimum number of basal

species is present.

The question of whether ‘‘the details of population

dynamics in one or few modules change the structure of

whole systems over time’’ (Sabo et al. 2005) is longstanding

in food-web theory. It has been addressed from two

independent viewpoints. One is structural, analyzing the

frequency distribution of some simple trophic structures, or

motifs, appearing in real food webs (Milo et al. 2002,

Bascompte and Melia ´n 2005, Camacho et al. 2007). The

other one is analytical and concerns with the study of the

properties of trophic modules in order to infer properties of

ecosystem dynamics. For instance, Bascompte et al. (2005)

study the effect of introducing IS obtained from data into

the dynamics of tri-trophic food-chain models, and Gross et

al. (2005) analyze the stability of n-trophic food chains. The

influence of varying IS on the stability of trophic modules

and, in particular, the role of weak links has been studied in

McCann et al. (1998), Fussmann and Heber (2002),

Emmerson and Yearsley (2004), and Rooney et al.

(2006). However, in multispecies dynamical models, the

stability of a small trophic module does not guarantee the

stability of the whole food web where it is embedded (May

1974, Yodzis 1989), since an isolated analysis forgets the

interplay of the module with its surrounding in the food

web.

To overcome this difficulty and to bridge the two

mentioned approaches, our method lies in adopting a set of

foraging preferences for the dynamics of large food webs

(initially generated by the niche model) according to the

robustness that this set confers on a fixed trophic module.

This basic module (Fig. 1) is complex enough to comprise

different kinds of trophic relationships (omnivory, intra-

guild predation, apparent competition), and some well-

known motifs appearing in food webs (Camacho et al.

2007) are simpler subgraphs of it.

We claim that the larger is the module complexity, the

larger is food-web complexity. Indeed, a higher complexity

of the persistent module implies an increase in the number

of motifs within the module that persist, and their stability

provokes an enrichment in complexity in food webs

containing them. To illustrate this claim, we have chosen

different sets of foraging preferences that generate different

degrees of persistence of the module, from the loss of the

most of the trophic interactions to the full persistence of

connections. Our complementary approach tries to under-

stand to what degree conditions determining the complexity

and structure of trophic modules scale to account for the

complexity of entire food webs.

Model and methods

We use a bioenergetic model with the assumption of

dynamic foraging efforts (Brose et al. 2003, Kondoh 2006,

Garcia-Domingo and Saldan ˜a 2007). The evolution in time

of the abundance xi?xi(t) of species i (i?1, ... , S) is

governed by

dxi

dt?xi

where Pi(x)?1?aS

for basal species and ri?0 otherwise), liis the death rate, fij

is the maximum assimilation rate of species i when

consuming species j (i.e. the foraging preference of predator

i for prey j), and aijis the foraging effort that predator i

invests in prey j. We use the Holling type II functional

response that guarantees saturation in predation rate as the

number of prey increases.

The dynamics of the foraging efforts aij(t) are governed

by a differential equation that compares the energetic gain

that predator i would obtain in consuming only prey j with

its actual energetic gain, that is

?

ri(1?xi)?li?

X

S

j?1

?fij

Pi(x)aij?

fji

Pj(x)aji

?

xj

?

(1)

k?1aikxk; riis the growth rate (ri?1

daij

dt?

gi

Pi(x)

?

fijxj?

X

k?1

S

fikaikxk

?

aij

(2)

where giis the adaptation rate of species i. This dynamic

approach for foraging efforts was firstly introduced by

Kondoh (2003), where the author also used a linear

nonsaturating functional response and an intrinsic growth

term for every species dynamics. Instead, Brose et al. (2003)

considered a nonlinear saturating functional response and

assigned intrinsic growth term only to basal species. We

follow the latter approach with an abbreviated set of

parameters. The relationship of our fijwith the original

parameters is simply given by fij?mi?yij, with mibeing

the mass-specific metabolic rate of species i and yijbeing the

maximum rate at which species i assimilates species j per

unit metabolic rate of species i.

Since Eq. 2 implies that /aS

follows that the efforts of every predator i satisfy the linear

constraint

j?1daij(t)=dt?0 for all t, it

X

j?1

for all t whenever the initial foraging efforts aij(0) satisfy

Eq. 3.

Foraging efforts aij(t) give the time-varying structure of

the food web. If aij(T)?aji(T)?0 for some T?0 (and

hence for all t]T), then the network at time T does not

have a link between i and j (and nevermore), while aij(t)?0

S

aij(t)?1

(3)

337

Page 3

indicates that i is a predator of j at time t, and thus the

network has a (directed) link from j to i at this time.

Consequently, the topology of the network is molded by

the action of dynamics. In this sense, from Eq. 2 it is clear

that effort dynamics will depend not only on foraging

preferences but also on species abundances. In this setting,

we consider aij(t) as a measure of the per capita interaction

strength between predator i and its prey j at time t?0.

We assume that the foraging preferences fijwill depend

only on the trophic positions of both species i and j,

classified as top (T), basal (B) and intermediate (I). Hence,

four possible values fTB, fIB, fTI, and fIIwill be assigned to

every fijdepending on the trophic classification of species i

and species j. The parameters liare also dependent on this

classification, with values lT, lB, and lI, respectively.

The parameter sets we have considered in the simula-

tions are given in Table 1. For each of them, we observe the

action of the dynamics on the trophic module shown at the

top of Fig. 1 after a suitable time T?1000. The resulting

structure after dynamics is shown at the bottom of Fig. 1.

This module is rich enough to embrace all kind of trophic

interactions (like omnivory, intraguild predation or appar-

ent competition). Those species preying on more than one

species (namely, species T1and I1) are capable of adaptive

predation and, initially, they invest the same effort in all

their prey (i.e. aT1j(0)?1/3, j?I1,I2,B1, and aI1j(0)?1/2,

j?I2,B1). The initial values for the species densities xi(0),

i?1, 2, 3, 4, are randomly taken between 0 and 1, and

have no influence on the final structure of the module. Note

that in all cases the final configurations contain the same

species than the initial one, i.e. the system is species

persistent. By contrast, the set of persistent links (by

definition, those with a foraging effort aij(T)?10?10)

will depend on the parameter choice.

The first choice is given by the parameter set A which

corresponds to the set of values used in Brose et al. (2003).

According to these authors, such values are based on

empirically supported estimates and may correspond to a

community dominated by invertebrates. The rest of

parameter sets are obtained from the set A by introducing

some heterogeneity in the fijaccording to the B/I/T-trophic

classification, and are always within the same order of

magnitude than the original values.

In particular, the parameter set B leads to the same final

pattern of interactions among species than the set A but

with different preferences for intermediate and basal species.

The parameter set D has the same foraging preferences of

the top species than parameter set B, but higher foraging

preferences of intermediate species. Finally, in comparison

with the set B, the parameter set C has lower foraging

preferences of top species, and higher preferences of

intermediate species. Clearly, many other parameter com-

binations are possible, but these are enough to show

noticeable differences among final food-web configurations.

B1

I1

T1

I2

fII

fIB

fIB

fTI

fTI

fTB

A and B

CD

Fig. 1. (TOP) Trophic module of four species (one top T1, two intermediate I1and I2, and one basal B1) with omnivory (fTB) and

intraguild predation (fII), and (DOWN) trophic structure after dynamics for the parameter sets A, B, C and D.

338

Page 4

For each parameter set, we then generate 100 food webs

under the niche model (Williams and Martinez 2000), with

an initial number of species S0?20 and initial connectance

C0?0.4. This high value of C0allows us to generate initial

food-web configurations containing a large number of

trophic links in order to see the effects of the dynamics.

After the action of dynamics, most of the final food-web

connectances are lower and in agreement with field

observations (Dunne et al. 2002).

Every generated adjacency matrix gives the initial

configuration of the trophic relationships (that is, those

efforts aij(0)?0). For each matrix, we assign a growth rate

equal to 1 (ri?1) for basal species and 0 otherwise.

Moreover, in order to guarantee that constrain Eq. 3 is

initially fulfilled, we fix an initial interaction strength aij?

1/(initial number of prey of species i). Finally, we classify

each species in the food web as top, intermediate or basal

and assign the values of parameters liand fijaccording to the

set chosen from Table 1. As before, the initial values of the

species densities xi(0) are randomly taken between 0 and 1.

We have considered in our simulations that the 75% of

predators are adaptive, since not all predator species have

switching behavior. As seen in precedent works with

adaptive predators (Brose et al. 2003, Kondoh 2003,

Garcia-Domingo and Saldan ˜a 2007), this threshold assures

that foraging adaption has a significant role in the

configuration of the evolving food web. Following Garcia-

Domingo and Saldan ˜a (2007), we assign adaptation (gi?1)

to the 75% of predators having more prey, keeping as non

adaptive (gi?0) the remaining 25%. This choice implies

that adaptation and population dynamics take place at the

same time scale, and thus that adaptation is due to

behavioral changes rather than to evolutionary ones. The

fact that foraging adaptation has a major effect on the

complexity-stability relationship when efforts and popula-

tion dynamics have comparable time scales has been

pointed out by several authors (Dell et al. 2005).

We then let the system given by Eq. 1 and 2 to evolve for

a large period of time (until T?1000). Afterwards, we

remove species and links for which xi(T) or aij(T) fall below

10?10, respectively. If specie i is removed, so are all links

emanating from/arraving to i (all aijand aji). Finally, we

collect the complexity indicators of the food web.

For every choice A, B, C, D of parameter values in Table

1, we repeat the experiment with 100 generated matrices or

replicas, avoiding matrices that contain disconnected non-

basal species. For the numerical integration, we use the

Runge-Kutta 7.8 method.

Results

We use different measures to capture the complexity of the

final food-web configuration: number of persistent species

S, number of persistent links L, link density L/S, and

directed connectance C?L/S2. The food-web composition

in terms of top (T), intermediate (I), and basal (B) species is

computed. The number of trophic levels of the food web is

TL?1?maxiQBd(i, B), where d(i, B) is the distance from i

to the set B of basal species, that is, the shortest path in the

network from i to any basal species.

A link between species i and j is called weak (at time t) if

aij(t)B10?1. Thus, in the model the feature of being a

weak link can vary in time. This definition seems to be a

priori independent of the foraging preferences fij but,

indeed, the dynamics of the foraging efforts aij(t) depends

on them (Eq. 2). The variable WLbin reports the number of

food webs having weak links over a total of 100 replicas. For

these webs, we have collected the percentage of weak links

WL. A cycle is a directed closed path. The variable Cycbin

gives the number of food webs with presence of cycles over

a total of 100 replicas. The results for all these indicators are

summarized in Table 2.

In the simulations, we have obtained 46.9%, 38.7%,

10.6%, 3.1% and 0.7% of initial food webs with 1, 2, 3, 4

Table 1. The values of growth rates (ri), adaptation rates (gi), foraging

preferences (fij), and death rates (li) used in the simulations. The

action of dynamics on the trophic module for each parameter set is

drawn in Fig. 1.

ABCD

ri

1 (basal)

0 (non-basal)

gi

1 (adaptive)

0 (non-adaptive)

fTB?1, fTI?5

fIB?3, fII?6

lB?0.15

lT?lI?0.5

fij

3fTB?2, fTI?6

fIB?2, fII?5

fTB?2, fTI?6

fIB?3, fII?9

li

0.5

Table 2. Indicators describing food-web complexity after dynamics: number of persistent species S, number of persistent links L,

connectance C, link density L/S, number of trophic levels TL, food-web composition in terms of the fraction of basal B, intermediate I and top

T species, percentage of food webs with presence of weak links WLbin and percentage of weak links in a food web WL, percentage of food

webs with presence of cycles Cycbin. Each result shows mean values over 100 food webs with 1, 2 or 3 basal species, with standard

deviations in parentheses. WL are mean values over food webs with presence of weak links.

ABCD

S

L

C

L/S

TL

B, I, T (%)

WLbin (%)

WL (%)

Cycbin (%)

7.97 (4.16)

7.83 (5.63)

0.14 (0.06)

0.89 (0.25)

2.00 (0.00)

24, 0, 76

9

12.69 (5.79)

0

11.76 (3.90)

15.53 (9.27)

0.12 (0.07)

1.24 (0.45)

2.57 (0.50)

15, 12, 73

35

29.57 (11.92)

2

11.90 (4.00)

16.47 (10.81)

0.12 (0.04)

1.29 (0.54)

2.58 (0.52)

14, 12, 74

33

28.62 (11.44)

2

14.24 (3.64)

27.43 (17.80)

0.13 (0.05)

1.80 (0.89)

2.90 (0.44)

12, 26, 62

61

29.29 (14.53)

20

339

Page 5

or 5 basal species respectively. Therefore, 96.2% of them

have a number of basal species less or equal to three. So, we

have neglected those food webs with four and five basal

species because of their lack of statistical significance. This

low appearance of food webs with more than three basal

species is due to the high initial connectance we have

considered to generate the initial configurations (for more

details see Kondoh 2006). Moreover, such a restriction on

the number of basal species reduces the variability of the

final connectance in those cases where the food-web

configurations exhibit a significant degree of complexity

(cases C and D).

From A to D, there is a large increase in complexity. The

number of persistent species is doubled while the number of

persistent links is multiplied by four. The link density also

experiences an important gain. The trophic levels go from

collapsed webs with TL?2 (Fig. 2 and Discussion) up to

values close to real food webs, and weak links and species in

cycles rise in frequency. The connectance is damped in B

and C due to a quadratic dependency on persistent species S

in front of the linear dependency on persistent links L, but

attains greater values for D. The food web composition

changes from an unrealistic configuration without inter-

mediate species to a more realistic one, but always keeping a

surplus in top predators abundance.

Interestingly, although parameter sets A and B amount

to the same results in the module test, the complexity

indicators for case B greatly improve the ones of A. The

species persistence is 20% upper and the number of links is

doubled. In addition, a more interesting food-web structure

arises in case B: intermediate species are present and the

trophic levels have more realistic values. Moreover, weak

links appear in a significant fraction of webs. Only the

connectance is diminished as a consequence of the inverse

quadratic dependence on the number of persistent species.

In contrast with this improvement, case C does not

represent a remarkable gain in complexity with respect to

case B. Finally, case D gives rise to greater values for all

indicators. The average species persistence is 71%, and the

mean link density is approximately 2. The mean number of

trophic levels increases to 2.9 which is closer to the observed

values (Williams et al. 2002).

Figure 3 shows the relationship between persistent

species and final connectance in model food webs corre-

sponding to all the parameter sets. In case A, there exists a

good power-law correlation of the form C?aSbwith b?

?0.65 and a coefficient of determination R2?0.86. This

result is in agreement with previous observations in Garcia-

Domingo and Saldan ˜a (2007). However, this fitted regres-

sion between species persistence and final connectance is

lost as we go from case A to case D. Remarkably, this loss of

correlation goes hand by hand with an increase in complex-

ity of the final food webs.

The presence of weak links is quite poor under the

parameter sets A, B and C. Moreover, their distribution in

the food web is far from the pattern of few strong

interactions embedded in a majority of weak interactions

reported for real food webs (McCann et al. 1998). This

changes in case D, where 61% of webs have a mean fraction

of 29.29% of weak links (Table 1).

It is also remarkable the increase in the variability of the

final food-web connectance in cases C and D, specially for

large values of S. This fact is due to the role of basal species

in the final food-web configurations. While, in cases A and

B, having more basal species is not translated into a higher

degree of food-web complexity (Discussion), this is no

longer true in cases C and D. In these cases, the increase in

the number of final food-web configurations goes in parallel

with the number of basal species which leads to scattered

plot showing no correlation between C and S. Such a degree

of dispersion is even greater if one includes food webs with

four or even five basal species.

A way to illustrate this claim is to compute the same

food-web descriptors by segmenting the results with respect

to the number of basal species appearing in. In Fig. 4 one

can clearly see that both species richness S and link density

L/S increase monotonously with the number of basal

species. This growth in complexity is noticeable when

varying from one to two basal species, although connec-

tance C diminishes because of its quadratic dependence on

the number of persistent species. Only when the number of

basal species goes from two to three there exists an increase

of food-web connectance due to the lower increment of

species persistence with respect to that of the number of

links occurring in this case.

Discussion

The influence of variation in IS has been widely studied in

few-species trophic modules revealing new insights on food-

web stability. The observed behavior is then inferred to

apply to complex food webs. By contrast, large food-web

models are not analytically tractable and are concerned with

regularities in the structure of food webs (see the discussion

of detail vs resolution in Sabo et al. 2005). The role of IS in

conferring whole system stability in complex food-web

B1

B1

B2

B2

A

B

Fig. 2. The extreme cases of food webs with S?7, B?2 and

TL?2, without intermediate species: (A) Fully connected and (B)

star-shaped configuration.

340

Page 6

models has been studied only in recent years (Neutel et al.

2002, Kondoh 2003, Emmerson and Raffaelli 2004).

In this paper we have used the adaptive foraging

mechanism introduced in Kondoh (2003) for the time

evolution of a food web. Extensions of such a mechanism

which allow for nonlinear constraints for both foraging

efforts and predator avoidance have been recently consid-

ered (Uchida and Drossel 2007). In any case, the advantage

of using models with foraging adaptation is that, once the

initial food-web structure is given, there is no a priori

constraints on species trophic interactions. This modeling is

based on the interplay of local optimization processes of

energy intake that leads to the appearance of properties at

food-web scale.

One of the questions we wanted to address is the relation

between module stability and food-web stability. It is worth

noting that the smaller trophic structures with a basal

species included in the basic module considered in this

study are also stable under some of our sets of parameter

values. This gain of stability of smaller trophic motifs

enhances complexity at the food-web level. Nevertheless,

variability in species persistence and in complexity at this

level arises due to the interplay with the trophic structures

surrounding a given module of the food web (Fig. 3).

Fig. 3. Plots for the persistence-connectance relationship of food web data obtained with each set A, B, C and D of parameter values,

corresponding to 100 food webs with 1, 2 or 3 basal species. Boxes show mean value9standard error.

Fig. 4. Changes in species persistence (S), connectance (C), and link density (L/S) in terms of number of basal species (Bas)

for each parameter set corresponding to food webs with 1, 2 or 3 basal species. Points are mean values, and boxes show mean value9SE.

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Taken together, only general trends between both kinds of

stability are expected.

In this respect, results evidence the existence of a positive

correlation between module complexity and food-web

complexity. The same is true if we focus on species

persistence, for example, since the percentage of surviving

species goes up from 40% to 71% when we move from the

parameter set A to parameter set D. But what is important

to note is that this increase in species survival goes in

parallel with a sprouting of food web complexity.

The parameter set A has been used in previous works by

Brose et al. (2003) and Kondoh (2006) in the study of the

influence of adaptive predators in the stability-complexity

relationship. A mathematical analysis shows that, with this

mean-field choice, simple structures like tri-trophic chains

are not persistent since the top predator goes extinct. In our

module test, the top predator survives thanks to the

omnivore link (Fig. 1), but the final module structure is

simply given by a star-shaped configuration. This is

reflected in the number of trophic levels which is constantly

equal to 2 for all food webs (TL?2). This number of

trophic levels and the absence of intermediate species mean

that only basal species and species feeding on them have

finally survived.

In this setting of parameter values, we can analyze two

extreme configurations. On one hand, if every non-basal

species feeds only on one basal species, the food web

configuration is the union of disconnected stars with center

a basal species (Fig. 2B). In this case, denoting by Bas the

number of basal species, the connectance is C?(S?Bas)/

S2and the link density is L/S?(S-Bas)/S. On the other

hand, if every non-basal species feeds on all basal species

(Fig. 2A), then C?Bas(S?Bas)/S2and L/S?Bas(S-Bas)/

S. From Fig. 4, we see that connectance and link density in

case A when Bas?2 (mean S?10.54) and Bas?3 (mean

S?12.71) are closer to the extreme case where the food-

web architecture is given by the union of local star-shaped

configurations. This is the reason why having more basal

species does not contribute to an increase of the food-web

complexity. Such a simple architecture resulting under this

parameter values and for webs with only one basal species

was already observed in Garcia-Domingo and Saldan ˜a

(2007). In summary, for the set A of parameter values,

foraging adaptation renders food webs with unrealistic

architectures.

On the other hand, although we find a trend between

module complexity and food-web complexity, the correla-

tion between them is not one-to-one. For instance,

parameter sets A and B give the same module but render

quite different indicators of food-web complexity. The

reason is that parameter set A leads to a strongly homo-

geneous scenario where all species compete for resources

with the same efficiency. This causes a great degree of

rigidity in the food web which amounts to the extinction of

those species not feeding on basal species. By contrast,

parameter set B offers a higher flexibility to the food web

which allows for a greater species persistence and more

complex structure. This increase in complexity clearly

illustrate our claim that interactions among the trophic

structures in the food web can lead to new food-web

features that are not included in the configuration of an

isolated module. In turn, parameter set C shares with B

such a food-web flexibility and the final food-web con-

nectances are practically the same for both sets due to the

similar values of species persistence and link density. In

both cases, the mean number of trophic levels is the same

(TL?2.58) even though the trophic module in case B has

TL?2, and, in case C, it has TL?3. The highest value of

the number of trophic levels is obtained under the

parameter set D (TL?2.9) which is almost 50% higher

than that of the corresponding trophic module.

In all cases, the practical non-existence of cycles in the

final food-web configuration can be seen as a consequence

of the energy intake optimization underlying the foraging

adaptation. This fact has been pointed out when thinking

the food web as a transportation network in Allesina and

Bodini (2005) where the adjacency matrices of food webs

are seen close to those of directed acyclic graphs. This

phenomenon has been also observed in Barthe ´lemy and

Flammini (2006) where optimal traffic networks are proven

to have tree-like structures.

The importance of weak links in conferring food-web

stability is reasserted. Moreover, what is relevant here is that

in models with foraging adaptation, weak interactions

appear in a very natural way as a consequence of local

optimization processes.

In terms of the number of basal species, the results show

that, in order to attain a high level of complexity, a

significant number of basal species is needed. In this sense,

it is remarkable how the rise in food-web complexity

appears, firstly, as an increase of species persistence when

passing from one to two basal species and, secondly, as an

increase in the total number of links when passing from two

to three basal species. This is reason for the non-

monotonous behavior of connectance with the number of

basal species observed in Fig. 4.

In summary, our study supports the idea that the

patterning of IS is essential to maintaining community

persistence (Yodzis 1981, Dell et al. 2005). We have seen

how different choices of foraging preferences amounts to

different interaction strengths which, in turn, lead to very

different final food-web architectures, supporting the

principle of higher flexibility for better adaptability. With

this respect, two main points can be stressed. First, mean

field scenarios where all trophic preferences are the same

regardless of trophic position give rise to poor food webs,

whereas heterogeneous patterns of foraging preferences lead

to a sprouting of food-web complexity. Second, the pattern

of foraging preferences that accounts for complexity scales

from trophic modules to entire food webs.

Acknowledgements ? This work has been supported by project FP6-

2003-NEST-PATH-1 named ‘‘Unifying Networks for Science

and Society’’ of the Sixth European Framework Programme (J.L.

Garcia-Domingo and J. Saldan ˜a) and by grant MTM2005-07660-

C02-02 of the Spanish government (J. Saldan ˜a).

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