Introduction

A major challenge in ecology is to understand how ecological networks are assembled. Network assembly ultimately reflects how interactions between individuals of different species scale up to organize ecological communities [1], [2]. The study of food webs, which are networks of trophic interactions among species, has benefited from the proposal of probabilistic, topological models that are able to reproduce the structure of trophic interactions by incorporating simple ecological processes (reviewed by Stouffer [3]). These models offer a way to build realistic food webs using a few parameters such as the number of interacting species and the number of interactions that can be estimated in the field [4], [5], [6], [7]. By connecting the structure of real food webs with candidate underlying processes, such models provide a basis for investigating the implications of food web organization for ecological dynamics [8], species persistence [9], [10], and ecosystem services [11]. Moreover, differences in how closely each model fits the structure of empirical food webs provide insight into the fundamental rules organizing trophic interactions in ecological systems [7], [12], [13].

The majority of studies on how such models reproduce ecological networks have focused on food webs, but there is an increasing body of theory that relies on probabilistic models to understand the structure of networks formed by other kinds of ecological interactions such as mutualisms [2], [14], [15], [16]. The theoretical background for devising specific models for mutualistic networks stems from the fact that antagonisms and mutualisms differ in their fundamental ecological and evolutionary implications [17], [18]. Additionally, mutualistic networks share some marked structural regularities that differ from antagonistic networks such as food webs [18], [19], [20]. For instance, mutualistic networks are best described as two-mode networks in which there are two sets of nodes (e.g., animals and plants) and there are no interactions among species within the same set [21]; in contrast, food webs are organized into several loosely defined trophic levels [12]. Moreover, mutualistic networks tend to be highly nested, that is, a given species interacts with a subset of the partners of species that have more interactions whereas antagonistic networks have lower degrees of nestedness [18], [19](but see [22]). An additional feature of mutualistic networks is that they exhibit right-skewed distributions of the number of interactions per species [21], whereas in food webs, this skewness is, in general, less pronounced [23].

The well-established differences between food webs and mutualistic networks (e.g., [18], [24]) have been counterbalanced by increasing evidence that ecological networks share some basic similarities. For instance, modularity, which was previously predominantly related to antagonistic networks [25], [26], was reported in a large set of mutualistic networks [27]. Along the same lines, although nestedness is often higher in mutualistic than in antagonistic two-mode networks [18], a recent study [22] showed that food webs are actually composed of interconnected, nested, two-mode sub-webs.

Another way in which mutualistic networks and food webs converge is that most mutualistic relationships are, in fact, rooted in consumer-resource relationships [28], [29]. For example, pollination is a type of mutualism that often involves animals foraging for resources provided by flowering plants [30]. Similarly, the frugivores that disperse seeds away from parental trees are usually foraging on fruit pulp [31]. Therefore, even though food webs and mutualistic networks differ in some key aspects of their structure, we should expect that ecological processes related to resource use partially shape these interactions in similar ways. In fact, all of the models proposed for food webs are inspired by processes shaping the consumer-resource interactions in a given locality. These consumer-resource interaction rules are quite general and may also apply to other types of interactions. In this sense, we hypothesize that food web models are able to reproduce the structure of mutualistic networks. To test this hypothesis we adapted food web models to reproduce two-mode networks and compared their performance, and that of a model designed for mutualisms, in reproducing real mutualistic networks. We first calculated summary statistics that described the structural properties of real food webs and used a likelihood-based model selection approach [32] in which we computed the likelihood of obtaining the observed values under a set of candidate network models. Finally, we explored whether simple topological features of mutualistic networks explain the performance of network models.

Methods

The models

To test the performance of food web models in reproducing the structure of mutualistic networks, we compiled a set of 10 pollination and 15 frugivory networks totaling 25 mutualistic networks (see Table S1 in supporting information). These networks ranged from networks with small species richness (animal species richness, A = 14; plant species richness, P = 11) to species-rich networks (A = 64; P = 43) and from loosely connected networks (connectance, C = 0.07) to highly connected networks (C = 0.47). For each of those networks, we generated an ensemble of 1000 matrices using four different models to test model performance. Whenever a model generated a network with disconnected species or with a C value 3% larger or smaller than the real one, we discarded that network before running the model again [5], [33].

In most mutualistic relationships, interactions can only occur between species in two well-defined sets (e.g., animals and plants), but food webs do not have this two-mode structure. In this sense, in food web models, all species but producers can be both predator and prey; in contrast, animals in the mutualistic networks studied here (pollination and seed dispersal) act as foragers by feeding on fruits and nectar provided by plants. Therefore, we adapted all food web models used to the two-mode nature of mutualistic networks. Our objective was to make as few changes as possible in the original models. We used the same set of simple rules of food web models, but interactions only occurred among species of different sets. As a result, all of the models used the input parameters A and P as well as the connectance, which is defined as C = E/AP_, where E is the number of recorded interactions. Below, we first describe each model in detail and then the adaptations we made to deal with the two-mode nature of mutualistic networks. We recognize that the models used in this study only represent a subset of the available food web models (e.g., [6], [13], [34], [35]), but we consider this to be a representative set of models that encompass a wide range of candidate rules for how food webs are built up. Moreover, several models were proposed to explain the structure of mutualistic networks (e.g., [2], [14], [36]). However, because our focus is to build a bridge between models describing antagonistic and mutualistic relationships, we chose to compare food web model performance with that of a recent proposed model that was directly inspired by food web models and has been shown to successfully reproduce the structure of mutualistic networks [15].

The cascade model.

The cascade model was the first of a series of static models that were capable of reproducing some of the structural properties of real food webs [4]. The cascade model is based on the assumption of hierarchical feeding, assigning each of the S species in the community a random value that is uniformly drawn from the interval [0,1], which represents species position along a one-dimensional feeding hierarchy (Fig. 1A). Each species has a probability q = 2CS/(S – 1) of consuming those species whose values are smaller than its own [5]. In our effort to adapt the cascade model to the two-mode nature of mutualistic networks, the position of species are assigned independently for animals and plants so that instead of ordering all species along an axis there are two axes: one for animals and the other for plants (Fig. 1B). Animals can potentially interact with plants whose values are smaller than their own but can never interact with other animals. The probability q of the original model was not valid for the two-mode version; we defined it as q = E/T, in which T is the number of possible interactions after species positions are defined. This approach ensures that the model creates networks with connectance that closely resembles the connectance of the empirical food web.

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Figure 1. Diagrams comparing original food web models and their two-mode version.

(A) the cascade model: each species (represented as an inverted triangle) is assigned a random value being placed along an axis. A given species i (gray) potentially interacts with those species whose values are less than the value assigned to i (as indicated by arrows); (B) the cascade model for two-mode networks: species that pertain to different sets (e.g. plants and animals) are randomly placed along two separate axes. The upper axis represents the axis of consumers. Therefore a given species i in the upper axis potentially interacts with those species in the lower axis whose values are lower than the value assigned to i. (C) The niche model: Each species is assigned a random value n_i and consume all species within a range of niche values r_i. (D) The niche model for two-mode networks: species that pertain to different sets (e.g. plants and animals) are placed along two separate axes according to their n_i. Each species in the upper axis consume all species in the lower axis that fall within a range of niche values r_i.

https://doi.org/10.1371/journal.pone.0027280.g001

The niche model.

The niche model addresses some of the limitations of the cascade model; in particular, it addresses the lack of feeding cycles and cannibalism. However, the niche model retains much of the simplicity and tractability embodied by the earlier model [5], [12]. As in the cascade model, the original niche model [5] assigns a position (n_i) taken from a uniform distribution on the interval [0,1] for all S species and places each of them along a gradient (Fig. 1C). For each consumer i, a niche range r_i = xn_i, where 0 ≤ x ≤ 1 is a random variable with a beta-distributed probability density function p(x) = β(1 – x)( β – 1) with β = (1/2C) – 1 is then defined. This causes species with higher n_i to tend to eat more species and ensures that the average of all species' r equals C [33]. The range center (c_i) is a uniformly random number between r_i/2 and min (n_i, 1−r_i/2). A consumer i eats all species j whose n_j fall within its range (Fig. 1C). Hence, a diet interval I(D_i) = [c_i−r_i/2, c_i+r_i/2] is defined for all species. As in the cascade model, to adapt the niche model to mutualistic networks, we defined n for plants and animals within two separate axes and diet ranges were defined only for animals and projected in the plants axis, such that animals always behaved as consumers and plants always behaved as food resources (Fig. 1D). Although we recognize that in many cases plant traits are responsible for selecting their interaction partners and thus network assembly could occur from the perspective of plants (e.g., [36]) we opted for an approach that is similar to the original models in which basal species have no defined ranges [5]. To obtain I(D) for animals, we used functions that are identical to those used in the original model (see Text S1 for reasoning).

In addition to having a more complex set of rules, the niche model differs from the cascade model because it imposes intervality in how links to resource species are assigned. Intervality means that all of the species in a food web can be placed in a fixed order on a line such that each consumer's set of resources forms a single contiguous segment of that line. Therefore, intervality suggests that trophic niche space can be represented by a single dimension [12], [37].

The minimum potential model.

Even though the niche model seemed to perform fairly well in reproducing most of the features of empirical food webs, food webs often do not show intervality for all species [34]. The minimum potential niche model [7] is a niche-based model that relaxes the interval feeding constraint of the niche model in a similar way to the relaxed niche model [33]. In the minimum potential niche model (hereafter MPN), forbidden interactions lead to gaps in consumers' diets [7]. The MPN model can be seen as a way of embedding multidimensional niches into a one-dimensional context [7]. The MPN model is similar to the niche model in that at first, the positions along the niche axis and diet interval I(D_i) of each species are defined. However a consumer eats species that fall within its diet interval with probability 1 – f, where f is the probability of having forbidden links in the diet (see Text S2). To adapt the MPN model to mutualisms two axes are defined and only animals posses I(D) as in the niche model.

Model of bipartite cooperation networks.

The model of bipartite cooperation (hereafter the BC model) was conceived for two-mode cooperation networks such as mutualistic networks and was directly inspired by food web models [15]. Here, we used a slightly different version of the model described by Saavedra et al. [15], following the authors' suggestion. In this model, plants are treated as members of class P and animals as members of class A. The model consists of two mechanisms: specialization and interaction. The specialization rule determines the number of interaction partners, l_pi, of each species p ∈ P. This number is determined by the interaction among two values: the reward trait, t_Rp, a number randomly drawn from an uniform distribution [0,1], which is attenuated or amplified by an external factor λ_p that is randomly drawn from an exponential distribution, which accounts for effects such as population density. The higher the reward value of plant p_i, the higher is the number of potential interactions established by p_i. The interaction rule determines which species a ∈ A interacts with each species p ∈ P. Interactions are limited by the complementarity between the reward traits, t_Rp, for p ∈ P and foraging traits, t_Fa, for a ∈ A. The foraging trait t_Fa, which are also uniformly drawn from [0,1], limits the range of possible partners for each member of class A, but again, interactions are affected by external factors λ_lp, which could represent, for instance, temporal variation and population density that are randomly drawn from an exponential distribution for each interaction.

Interactions are distributed to plants sequentially, in ascending order, according to their foraging traits t_Rp. Whenever t_Rpi>λ_lpi each link l_pi is connected to the first node a′ ∈ A′, where A′ is the subset of nodes in A that have not already been linked to by another node p ≠ p_i. Conversely, if t_Rpi ≤ λ_lpi, interactions of p_i are distributed using a mechanism similar to that proposed by Cattin et al. [6], i.e., a plant p ∈ P with lower trait value is randomly selected, and an interaction is established with an animal randomly chosen among its partners a'' ∈ _A'' where A'' is the subset of nodes in A that have been linked in a previous time step. If the supply of nodes in either A′ or A'' is exhausted before all l_pi links have been allocated, then nodes in the other subset are linked to instead. For additional detailed information on the model we refer readers to Saavedra et al. [15].

Results

All models performed remarkably well in reproducing both the nestedness and modularity of the mutualistic networks. The percentage of networks whose metrics were reproduced by each model varied between nearly 50% and 95% (Table 1). The models that reproduced the properties in the largest proportion of networks were the two-mode cascade model and the BC model (Table 1). When we directly compared the models as competing hypotheses using the likelihood approach, the outcome of the model comparison depended on the property being analyzed (Table 1). The cascade and niche models were among the most likely models for 84% of the networks considering nestedness. This result held when using a different nestedness metric, the matrix temperature, which indicates that these results are not affected by metric choice (Text S4). Similarly, when considering modularity, the cascade model was among the most likely models for 84% of the networks. However, the BC model instead of the niche model was the second best model in terms of reproducing modularity (Table 1). Regarding degree distributions, the results are less straightforward. All four models reproduced degree distributions for nearly all analyzed networks according to the Kolmogorov-Smirnov test results (Table 1). Nonetheless, the model comparison suggested that the cascade model was usually among the best models in reproducing plants degree distributions, whereas the niche and BC models outperformed the others more often in reproducing the degree distribution of animals (Table 1).

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Table 1. Proportion of mutualistic networks (N = 25) whose properties were reproduced by each model (NME<1; P_KS<0.05)/proportion of networks in which each model was among the most likely.

https://doi.org/10.1371/journal.pone.0027280.t001

The sign of NME indicates whether the model overestimates or underestimates a property value for a given network. Therefore, an excess of negative values of NME indicates that a model often underestimates a given property, whereas positive values suggest that the model has a tendency to overestimate it. The niche and MPN models tended to generate networks with lower degrees of nestedness and higher degrees of modularity than real networks (Fig. 2). The cascade and BC models were more balanced and showed fewer signs of systematic biases in one direction or another (Fig. 2). However, the degree of fit of models was associated with basic topological features of networks (see Table 2). Noteworthy network basic features explained between 70 and 95% of variation in model fit regarding nestedness and modularity. All models tended to underestimate nestedness as the degree of relative nestedness observed increased (P<0.01; Table 2, Fig. 2A). The degree of relative nestedness also affected the ability of the cascade, niche and MPN models to reproduce modularity. These models tended to overestimate network modularity for networks that had a high degree of relative nestedness (Table 2). The degree of relative modularity had the opposite effect for the cascade, MPN and BC models. When reproducing networks with high relative modularity, these models were more prone to underestimate modularity (Table 2). Connectance also affected model fit. Networks with larger connectance tended to have their degrees of modularity and nestedness overestimated by the cascade (only for nestedness) and BC models, whereas modularity NME decreased with increasing connectance for the niche model (Table 2).

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Figure 2. Normalized error (NME) of each model in reproducing nestedness (A) and modularity (B) for each of the 25 analyzed networks.

In (A) networks are sorted in increasing order of relative nestedness. Notice nestedness tend to be underestimated for networks with large nestedness degrees as suggested by partial regression coefficients (Table 2). In (B) networks are sorted in increasing order of relative modularity.

https://doi.org/10.1371/journal.pone.0027280.g002

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Table 2. Effects of basic real network features in model degree of fit as expressed by the NME.

https://doi.org/10.1371/journal.pone.0027280.t002

Discussion

Our results show that all four models performed fairly well in reproducing the properties of empirical mutualistic networks. However, the cascade and BC models more often generated theoretical networks that were in agreement with the structure of real mutualistic networks. Moreover, the cascade model was frequently among the most likely candidate models in reproducing the structure of mutualistic networks. Although the performance of the cascade and BC models was similar, the cascade model is much simpler than the BC model. In addition to attributing a value to each species as done in the cascade model, the BC model has many other free parameters that act as external factors that affect interactions. Therefore, the good performance of the cascade model appears even better when model complexity is taken into account.

In food webs, the cascade model also reproduced some aspects of the structure of interactions between consumers and resources [5]. Nevertheless, other models such as the niche and MPN models often outperformed the cascade model in reproducing food web structure [5], [7], [33]. The niche model was mainly proposed as a solution that included the possibilities of feeding loops and cannibalism, which were not allowed by the minimal rules of the cascade model [12]. In plant-animal mutualisms, on the other hand, interactions only occur between species in different trophic levels (plants and animals that forage in plant resources). Therefore, as we dealt with this two-mode structure of mutualisms, feeding loops were not a problem. This may partially explain the success of the cascade model for mutualisms in spite of being outperformed by niche model derivatives in the context of food webs [5], [7], [33]. In addition to the two-mode structure, other biological aspects of mutualisms might explain why the strict feeding hierarchy generated by the cascade model suffices to reproduce much of the structure of mutualistic networks.

Hierarchy is also an essential component in the BC model, which was directly inspired by the set of rules of food web models [15]. The success of the BC model in reproducing network structural patterns in a previous work [15] already suggested that such hierarchical processes should play a crucial role in organizing mutualistic networks. Because all models considered here encompass hierarchical processes our results reinforce their relevance in mutualisms. Moreover, the similar success in reproducing the structure of real networks of both BC and the much simpler food web models suggest that the feeding hierarchy by itself is enough to capture much of the structure of mutualistic networks. Although multiple processes may generate similar patterns in ecological systems, our results at least indicate possible mechanisms shaping the organization of mutualistic interactions in networks of interacting species.

The most compelling biological basis proposed for the ordering dimension that induces a feeding hierarchy in food web models is body size [12], [37], [43], [46]. In this sense, in the context of food webs, the hierarchical ordering in the cascade model would lead to larger species interacting with smaller species. Similarly, in niche models, larger species would tend to have wider trophic niches [43]. In the case of pollination and frugivory networks, such hierarchy could refer to any measurable traits related to the feeding interaction among fruiting/nectar-producing plants and fruit/nectar consumers such as bill diameter, bill or mouthparts length, and fruit size or corolla depth. Such traits would be represented in the adapted models as the two independent axes in which animals and plants are ordered. Indeed in a series of studies, Stang et al. [47], [48] showed that structural patterns of pollination networks such as nestedness could be reproduced by incorporating size thresholds imposed by floral morphology on nectar-feeding animals. Moreover, body size was found to predict the number of interactions of ants in ant-plant mutualisms [49]. Finally, larger frugivores are often able to eat a large variation in fruit sizes than smaller frugivores, leading to hierarchical ordering in frugivory [31]. From an evolutionary perspective trait based feeding hierarchies can emerge as a consequence of natural selection favoring particular high profitable resource combinations [17].

The way each model encompass feeding hierarchies may also partially explain differences in model performance. Species-rich mutualisms often form networks modules of interacting species based on shared phenotypic traits such as fruit color, flower shape, animal body mass [27], [50]. Nevertheless, modularity in mutualisms such as pollination and seed dispersal is often smaller than observed in antagonistic interactions [18] or in symbiotic mutualisms [2]. The strict feeding hierarchy imposed by the cascade model causes high overlap in the set of interaction partners among consumer species, leading to low modularity. Conversely the set of rules in other food web models, such as niche and MPN models, that partially relax the cascade hierarchy [9] might favor higher modularity. In niche and MPN model, species whose feeding ranges overlap may form network modules that differ from modules formed by species whose feeding ranges overlap farther in the niche axis. In fact, both niche and MPN models were outperformed by the cascade and BC models in reproducing the low degree of modularity in mutualistic networks, especially because they usually generated networks that were more modular than the empirical ones. This may also partially explain the superior performance of both the niche and MPN models in comparison with the cascade model in generating the more modular structure of food webs [5], [33].

The degree of relative nestedness and relative modularity of the real network were the main features of real networks affecting model fit; for networks with higher relative nestedness, the cascade, niche and MPN models tended to underestimate nestedness and overestimate the modularity of real networks. Conversely for networks with higher relative modularity, real modularity was usually underestimated. The sensitivity of the models accuracy to the degree of nestedness and modularity in the real networks indicates that the high degrees of nestedness or modularity observed in some mutualistic networks are not completely explained by the processes incorporated in food web models analyzed here. Stouffer et al. [13] showed analytically that a food web model should satisfy two criteria in order to reproduce most empirical food web properties: niche values should form a totally ordered set, and each species has a specific, exponentially decaying probability of preying on a fraction of the species with lower niche-values. In the context of mutualisms, it seems that a model's ability to reproduce empirical networks is not only a matter of reproducing the functional forms for the distributions of numbers of prey, predators and links per species, but also of reproducing the relationship between nestedness and modularity. Many mechanisms have been proposed for the occurrence of the nested pattern, namely, differences in abundance among species [26], [51], low interaction intimacy [2], trait complementarity and/or exploitation barriers coupled with coevolutionary convergence [14], [17], [48] and frequent extinctions of specialist-specialist interactions [52]. Along the same lines, trait matching along with phylogenetic constraints [20] and high interaction intimacy [2] are regarded as the main mechanisms that could lead to a modular structure in mutualistic networks [27]. The rules of the cascade, niche and MPN models can be interpreted as a form of encompassing trait complementarity and exploitation barriers among interacting species. Similarly, the BC model is based on the complementarity among plants reward traits and animals foraging traits. Although they do incorporate complementarity, they do not explicitly consider other mechanisms shaping network structure such as interaction intimacy, differential extinction and phylogenetic constraints. Evolving network models, models in which the number of species and interactions change over time, have also been shown to partially explain the structure of mutualistic networks [2], [36]. Future studies combining the mechanisms present in these two different classes of models might provide additional insights in the organization of mutualistic networks.

To sum up, food web minimal models were capable of reproducing most of the mutualistic networks analyzed. Noteworthy, even the cascade model, the simplest among the models considered here, reproduced the structure of nearly the whole set of networks. Such results open the possibility that the assembly of networks that describe mutualisms and antagonisms obey a similar simple set of rules and reinforce that feeding hierarchy might be a fundamental piece in this puzzle. Therefore, despite the differences in ecology and evolution of mutualisms and antagonisms [17], [18], they seem to share some key aspects. Our knowledge of the assembly of natural communities would benefit from future studies that scrutinize those commonalities and differences and attempt to sort out the evolutionary and ecological mechanisms that are responsible for each.

Supporting Information

Acknowledgments

We thank R. Guimerà for providing us with the modularity algorithm, S. Saavedra for providing the updated version of the BC model, S. Allesina and two anonymous reviewers for comments and suggestions.