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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ABSTRACT: Ecological networks are often composed of different subcommunities (often referred to as modules). Identifying such modules has the potential to develop a better understanding of the assembly of ecological communities and to investigate functional overlap or specialization.The most informative form of networks are quantitative or weighted networks. Here, we introduce an algorithm to identify modules in quantitative bipartite (or two-mode) networks. It is based on the hierarchical random graphs concept of Clauset et al. (2008 Nature 453: 98–101) and is extended to include quantitative information and adapted to work with bipartite graphs. We define the algorithm, which we call QuanBiMo, sketch its performance on simulated data and illustrate its potential usefulness with a case study.Modules are detected with a higher accuracy in simulated quantitative networks than in their binary counterparts. Even at high levels of noise, QuanBiMo still classifies 70% of links correctly as within- or between-modules. Recursively applying the algorithm results in additional information of within-module organization of the network.The algorithm introduced here must be seen as a considerable improvement over the current standard of algorithms for binary networks. Due to its higher sensitivity, it is likely to lead to be useful for detecting modules in the typically noisy data of ecological networks.Methods in Ecology and Evolution 12/2013; · 5.32 Impact Factor
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ABSTRACT: The fundamental building blocks of any ecosystem, the food webs, which are assemblages of species through various interconnections, provide a central concept in ecology. The study of a food web allows abstractions of the complexity and interconnectedness of natural communities that transcend the specific details of the underlying systems. For example, Fig. 1 shows a typical food web, where the species are connected through their feeding relationships. The top predator, Heliaster (starfish) feeds on many gastropods like Hexaplex, Morula, Cantharus, etc., some of whom predate on each other . Interactions between species in a food web can be of many types, such as predation, competition, mutualism, commensalism, and ammensalism (see Section 1.1, Fig. 2).12/2008: pages 57-72;
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ABSTRACT: Many analyses of ecological networks in recent years have introduced new indices to describe network properties. As a consequence, tens of indices are available to address similar questions, differing in specific detail, sensitivity in detecting the property in question, and robustness with respect to network size and sampling intensity. Furthermore, some indices merely reflect the number of species participating in a network, but not their interrelationship, requiring a null model approach. Here we introduce a new, free software calculating a large spectrum of network indices, visualizing bipartite networks and generating null models. We use this tool to explore the sensitivity of 26 network indices to network dimensions, sampling intensity and singleton observations. Based on observed data, we investigate the interrelationship of these indices, and show that they are highly correlated, and heavily influenced by network dimensions and connectance. Finally, we re-evaluate five common hypotheses about network properties, comparing 19 pollination networks with three differently complex null models: 1. The number of links per species ("degree") follow (truncated) power law distributions. 2. Generalist pollinators interact with specialist plants, and vice versa (dependence asymmetry). 3. Ecological networks are nested. 4. Pollinators display complementarity, owing to specialization within the network. 5. Plant-pollinator networks are more robust to extinction than random networks. Our results indicate that while some hypotheses hold up against our null models, others are to a large extent understandable on the basis of network size, rather than ecological interrelationships. In particular, null model pattern of dependence asymmetry and robustness to extinction are opposite to what current network paradigms suggest. Our analysis, and the tools we provide, enables ecologists to readily contrast their findings with null model expectations for many different questions, thus separating statistical inevitability from ecological process.The Open Ecology Journal 02/2009; 2(1).
Effects of heterogeneous interaction strengths on food web
Josep L. Garcia-Domingo and Joan Saldan ˜a
J. L. Garcia-Domingo and J. Saldan ˜a (email@example.com), 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
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
Oikos 117: 336?343, 2008
# The Authors. Journal compilation # Oikos 2007
Subject Editor: Jordi Bascompte, Accepted 28 September 2007
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
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
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
k?1aikxk; riis the growth rate (ri?1
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
j?1daij(t)=dt?0 for all t, it
for all t whenever the initial foraging efforts aij(0) satisfy
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
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.
A and B
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.
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.
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.
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.
B, I, T (%)
24, 0, 76
15, 12, 73
14, 12, 74
12, 26, 62
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.
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
Fig. 2. The extreme cases of food webs with S?7, B?2 and
TL?2, without intermediate species: (A) Fully connected and (B)
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
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.
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
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
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
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.
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2003-NEST-PATH-1 named ‘‘Unifying Networks for Science
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