Interaction network structure explains species' temporal persistence in empirical plant-pollinator communities.

Despite clear evidence that some pollinator populations are declining, our ability to predict pollinator communities prone to collapse or species at risk of local extinction is remarkably poor. Here, we develop a model grounded in the structuralist approach that allows us to draw sound predictions regarding the temporal persistence of species in mutualistic networks. Using high-resolution data from a six-year study following 12 independent plant-pollinator communities, we confirm that pollinator species with more persistent populations in the field are theoretically predicted to tolerate a larger range of environmental changes. Persistent communities are not necessarily more diverse, but are generally located in larger habitat patches, and present a distinctive combination of generalist and specialist species resulting in a more nested structure, as predicted by previous theoretical work. Hence, pollinator interactions directly inform about their ability to persist, opening the door to use theoretically informed models to predict species' fate within the ongoing global change.

The structure and robustness of ecological networks with two interaction types.

Until recently, most ecological network analyses investigating the effects of species' declines and extinctions have focused on a single type of interaction (e.g. feeding). In nature, however, diverse interactions co-occur, each of them forming a layer of a 'multilayer' network. Data including information on multiple interaction types has recently started to emerge, giving us the opportunity to have a first glance at possible commonalities in the structure of these networks. We studied the structural features of 44 tripartite ecological networks from the literature, each composed of two layers of interactions (e.g. herbivory and pollination), and investigated their robustness to species losses. Considering two interactions simultaneously, we found that the robustness of the whole community is a combination of the robustness of the two ecological networks composing it. The way in which the layers of interactions are connected to each other affects the interdependence of their robustness. In many networks, this interdependence is low, suggesting that restoration efforts would not automatically propagate through the whole community. Our results highlight the importance of considering multiple interactions simultaneously to better gauge the robustness of ecological communities to species loss and to more reliably identify key species that are important for the persistence of ecological communities.

Unveiling dimensions of stability in complex ecological networks.

Understanding the stability of ecological communities is a matter of increasing importance in the context of global environmental change. Yet it has proved to be a challenging task. Different metrics are used to assess the stability of ecological systems, and the choice of one metric over another may result in conflicting conclusions. Although each of the multitude of metrics is useful for answering a specific question about stability, the relationship among metrics is poorly understood. Such lack of understanding prevents scientists from developing a unified concept of stability. Instead, by investigating these relationships we can unveil how many dimensions of stability there are (i.e., in how many independent components stability metrics can be grouped), which should help build a more comprehensive concept of stability. Here we simultaneously measured 27 stability metrics frequently used in ecological studies. Our approach is based on dynamical simulations of multispecies trophic communities under different perturbation scenarios. Mapping the relationships between the metrics revealed that they can be lumped into 3 main groups of relatively independent stability components: early response to pulse, sensitivities to press, and distance to threshold. Selecting metrics from each of these groups allows a more accurate and comprehensive quantification of the overall stability of ecological communities. These results contribute to improving our understanding and assessment of stability in ecological communities.

Advancing our understanding of ecological stability.

The concept of ecological stability occupies a prominent place in both fundamental and applied ecological research. We review decades of work on the topic and examine how our understanding has progressed. We show that our understanding of stability has remained fragmented and is limited largely to simple or simplified systems. There has been a profusion of metrics proposed to quantify stability, of which only a handful are used commonly. Furthermore, studies typically quantify one to two metrics of stability at a time and in response to a single perturbation, with some of the main environmental pressures of today being the least studied. We argue that we need to build on the existing consensus and strong theoretical foundation of the stability concept to better understand its multidimensionality and the interdependencies between metrics, levels of organisation and types of perturbations. Only by doing so can we make progress in the quantification of stability in theory and in practice, and eventually build a more comprehensive understanding of how ecosystems will respond to ongoing environmental change.

Intervality and coherence in complex networks.

Food webs-networks of predators and prey-have long been known to exhibit "intervality": species can generally be ordered along a single axis in such a way that the prey of any given predator tend to lie on unbroken compact intervals. Although the meaning of this axis-usually identified with a "niche" dimension-has remained a mystery, it is assumed to lie at the basis of the highly non-trivial structure of food webs. With this in mind, most trophic network modelling has for decades been based on assigning species a niche value by hand. However, we argue here that intervality should not be considered the cause but rather a consequence of food-web structure. First, analysing a set of 46 empirical food webs, we find that they also exhibit predator intervality: the predators of any given species are as likely to be contiguous as the prey are, but in a different ordering. Furthermore, this property is not exclusive of trophic networks: several networks of genes, neurons, metabolites, cellular machines, airports, and words are found to be approximately as interval as food webs. We go on to show that a simple model of food-web assembly which does not make use of a niche axis can nevertheless generate significant intervality. Therefore, the niche dimension (in the sense used for food-web modelling) could in fact be the consequence of other, more fundamental structural traits. We conclude that a new approach to food-web modelling is required for a deeper understanding of ecosystem assembly, structure, and function, and propose that certain topological features thought to be specific of food webs are in fact common to many complex networks.

Ranking species in mutualistic networks.

Understanding the architectural subtleties of ecological networks, believed to confer them enhanced stability and robustness, is a subject of outmost relevance. Mutualistic interactions have been profusely studied and their corresponding bipartite networks, such as plant-pollinator networks, have been reported to exhibit a characteristic "nested" structure. Assessing the importance of any given species in mutualistic networks is a key task when evaluating extinction risks and possible cascade effects. Inspired in a recently introduced algorithm--similar in spirit to Google's PageRank but with a built-in non-linearity--here we propose a method which--by exploiting their nested architecture--allows us to derive a sound ranking of species importance in mutualistic networks. This method clearly outperforms other existing ranking schemes and can become very useful for ecosystem management and biodiversity preservation, where decisions on what aspects of ecosystems to explicitly protect need to be made.

Inherent directionality explains the lack of feedback loops in empirical networks.

We explore the hypothesis that the relative abundance of feedback loops in many empirical complex networks is severely reduced owing to the presence of an inherent global directionality. Aimed at quantifying this idea, we propose a simple probabilistic model in which a free parameter γ controls the degree of inherent directionality. Upon strengthening such directionality, the model predicts a drastic reduction in the fraction of loops which are also feedback loops. To test this prediction, we extensively enumerated loops and feedback loops in many empirical biological, ecological and socio-technological directed networks. We show that, in almost all cases, empirical networks have a much smaller fraction of feedback loops than network randomizations. Quite remarkably, this empirical finding is quantitatively reproduced, for all loop lengths, by our model by fitting its only parameter γ. Moreover, the fitted value of γ correlates quite well with another direct measurement of network directionality, performed by means of a novel algorithm. We conclude that the existence of an inherent network directionality provides a parsimonious quantitative explanation for the observed lack of feedback loops in empirical networks.

Trophic coherence determines food-web stability.

Why are large, complex ecosystems stable? Both theory and simulations of current models predict the onset of instability with growing size and complexity, so for decades it has been conjectured that ecosystems must have some unidentified structural property exempting them from this outcome. We show that trophic coherence--a hitherto ignored feature of food webs that current structural models fail to reproduce--is a better statistical predictor of linear stability than size or complexity. Furthermore, we prove that a maximally coherent network with constant interaction strengths will always be linearly stable. We also propose a simple model that, by correctly capturing the trophic coherence of food webs, accurately reproduces their stability and other basic structural features. Most remarkably, our model shows that stability can increase with size and complexity. This suggests a key to May's paradox, and a range of opportunities and concerns for biodiversity conservation.

Factors determining nestedness in complex networks.

Understanding the causes and effects of network structural features is a key task in deciphering complex systems. In this context, the property of network nestedness has aroused a fair amount of interest as regards ecological networks. Indeed, Bastolla et al. introduced a simple measure of network nestedness which opened the door to analytical understanding, allowing them to conclude that biodiversity is strongly enhanced in highly nested mutualistic networks. Here, we suggest a slightly refined version of such a measure of nestedness and study how it is influenced by the most basic structural properties of networks, such as degree distribution and degree-degree correlations (i.e. assortativity). We find that most of the empirically found nestedness stems from heterogeneity in the degree distribution. Once such an influence has been discounted - as a second factor - we find that nestedness is strongly correlated with disassortativity and hence - as random networks have been recently found to be naturally disassortative - they also tend to be naturally nested just as the result of chance.