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.food webs | May's paradox | diversity-stability debate | dynamical stability | complex networks I n the early seventies, Robert May addressed the question of whether a generic system of coupled dynamical elements randomly connected to each other would be stable. He found that the larger and more interconnected the system, the more difficult it would be to stabilize (1, 2). His deduction followed from the behavior of the leading eigenvalue of the interaction matrix, which, in a randomly wired system, grows with the square root of the mean number of links per element. This result clashed with the received wisdom in ecology-that large, complex ecosystems were particularly stable-and initiated the "diversity-stability debate" (3-6). Indeed, Charles Elton had expressed the prevailing view in 1958: "the balance of relatively simple communities of plants and animals is more easily upset than that of richer ones; that is, more subject to destructive oscillations in populations, especially of animals, and more vulnerable to invasions" (7). Even if this description were not accurate, the mere existence of rainforests and coral reefs seems incongruous with a general mathematical principle that "complexity begets instability," and has become known as May's paradox.One solution might be that the linear stability analysis used by May and many subsequent studies does not capture essential characteristics of ecosystem dynamics, and much work has gone into exploring how more accurate dynamical descriptions might enhance stability (5,8,9). However, as ever-better ecological data are gathered, it is becoming apparent that the leading eigenvalues of matrices related to food webs (networks in which the species are nodes and the links represent predation) do not exhibit the expected dependence on size or link density (10). Food webs must, therefore, have some unknown structural feature that accounts for this deviation from randomness-irrespectively of other stabilizing factors.We show here that a network feature we call trophic coherence accounts for much of the varianc...

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.

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.

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.

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.

Molecular valves are nanostructured materials that are becoming popular, due to their potential use in bio-medical applications. However, little is known concerning their performance when dealing with small molecules, which are of interest in energy and environmental areas. It has been observed experimentally that zeolite RHO shows unique pore deformations upon changes in hydration, cation siting, cation type, and/or temperature-pressure conditions. By varying the level of distortion of double 8-rings it is possible to control the adsorption properties, which confers a molecular valve behavior to this material. We have employed interatomic potentials-based simulations to obtain a detailed atomistic view of the structural distortion mechanisms of zeolite RHO, in contrast with the averaged and space group restricted information that can be retrieved from diffraction studies. We have modeled the pure silica zeolite RHO as well as four aluminosilicate structures, containing Li + , Na + , K + , Ca 2+ and Sr 2+ cations. It has been found that the distortions of the three zeolite rings are coupled, although the four-membered rings are rather rigid and both six-and eight-membered rings are largely flexible. A large dependence on the polarizing power of the extra-framework cations and with the loading of water has been found for the minimum aperture of the eight-membered rings that control the nanovalve effect. The energy barriers needed to move the cations across the eightmembered rings are calculated to be very high, which explains the origin of the experimentally observed slow kinetics of the phase transition, as well as the appearance of metastable phases.

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.

Understanding the causes and effects of spatial vegetation patterns is a fundamental problem in ecology, especially because these can be used as early predictors of catastrophic shifts such as desertification processes. Empirical studies of the vegetation cover in some areas such as drylands and semiarid regions have revealed the existence of vegetation patches of broadly diverse sizes. In particular, the probability distribution of patch sizes can be fitted by a power law, i.e. vegetation patches are approximately scale free up to some maximum size. Different explanatory mechanisms, such as plant–plant interactions and plant-water feedback loops have been proposed to rationalize the emergence of such scale-free patterns, yet a full understanding has not been reached. Using a simple model for vegetation dynamics, we show that environmental temporal variability—a well-recognized feature of semiarid environments—promotes in a robust way (i.e. for a wide range of parameter values) the emergence of vegetation patches with broadly distributed cluster sizes. Furthermore, this result is related to a percolation phenomenon that occurs in an intermittent or fluctuating way. The model also reveals that the power-law exponents fitting the tails of the probability distributions depend on the overall vegetation-cover density, in agreement with empirical observations. This supports the idea that environmental variability plays a key role in the formation of scale-free vegetation patterns. From a practical viewpoint, this may be of importance to predict the effects that changes in environmental conditions may have in real ecosystems. From a theoretical side, our study sheds new light on a novel type of percolation phenomena occurring under temporally-varying external conditions, that still needs further work to be fully characterized.