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 foodweb 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 foodweb architecture is found for a meanfield 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. Foodweb 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. GarciaDomingo and Joan Saldan ˜a
J. L. GarciaDomingo and J. Saldan ˜a (jsaldana@ima.udg.edu), Dept. Informa `tica i Matema `tica Aplicada, Univ. de Girona, ES17071
Girona, Catalunya, Spain. JLGD also at: Dept. d’Economia, Matema `tica i Informa `tica, Univ. de Vic, ES08500 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 foodweb 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 foodweb architecture is found for a meanfield 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. Foodweb 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 foodweb 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 foodweb 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 foodweb 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 foodweb complexity of a
dynamic bioenergetic model. By complexity we refer to the
richness of the foodweb 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 predatorprey 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, TB, TI, II, and IB) 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 timevarying interactions by considering adap
tive foragers (Brose et al. 2003, Kondoh 2003, 2006,
GarciaDomingo 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 predatorprey 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.2007.00301299.16261.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 connectancepersistence
relation was studied, and these results were argued in
GarciaDomingo and Saldan ˜a (2007). In all these works, a
meanfield 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 foodweb 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 foodweb 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 foodweb 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 tritrophic foodchain models, and Gross et
al. (2005) analyze the stability of ntrophic 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 foodweb 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,
GarciaDomingo 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 massspecific 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 timevarying 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/Ttrophic
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 foodweb 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
foodweb 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 foodweb
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,
GarciaDomingo 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
complexitystability 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
RungeKutta 7.8 method.
Results
We use different measures to capture the complexity of the
final foodweb configuration: number of persistent species
S, number of persistent links L, link density L/S, and
directed connectance C?L/S2. The foodweb 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 (nonbasal)
gi
1 (adaptive)
0 (nonadaptive)
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 foodweb complexity after dynamics: number of persistent species S, number of persistent links L,
connectance C, link density L/S, number of trophic levels TL, foodweb 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 foodweb
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 foodweb 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 powerlaw 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 foodweb 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 foodweb configurations. While, in cases A and
B, having more basal species is not translated into a higher
degree of foodweb complexity (Discussion), this is no
longer true in cases C and D. In these cases, the increase in
the number of final foodweb 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
foodweb 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 foodweb 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
fewspecies trophic modules revealing new insights on food
web stability. The observed behavior is then inferred to
apply to complex food webs. By contrast, large foodweb
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 foodweb
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)
starshaped 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 foodweb 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
foodweb scale.
One of the questions we wanted to address is the relation
between module stability and foodweb 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 foodweb 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 persistenceconnectance 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 foodweb
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 stabilitycomplexity
relationship. A mathematical analysis shows that, with this
meanfield choice, simple structures like tritrophic 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 starshaped 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 nonbasal
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?(SBas)/S. On the other
hand, if every nonbasal species feeds on all basal species
(Fig. 2A), then C?Bas(S?Bas)/S2and L/S?Bas(SBas)/
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 starshaped
configurations. This is the reason why having more basal
species does not contribute to an increase of the foodweb
complexity. Such a simple architecture resulting under this
parameter values and for webs with only one basal species
was already observed in GarciaDomingo 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 foodweb complexity, the correla
tion between them is not onetoone. For instance,
parameter sets A and B give the same module but render
quite different indicators of foodweb 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 foodweb
features that are not included in the configuration of an
isolated module. In turn, parameter set C shares with B
such a foodweb flexibility and the final foodweb 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 nonexistence of cycles in the
final foodweb 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 treelike structures.
The importance of weak links in conferring foodweb
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 foodweb 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 foodweb 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 foodweb 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
2003NESTPATH1 named ‘‘Unifying Networks for Science
and Society’’ of the Sixth European Framework Programme (J.L.
GarciaDomingo and J. Saldan ˜a) and by grant MTM200507660
C0202 of the Spanish government (J. Saldan ˜a).
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