Predicting the stability of large structured food webs

Hierarchy of Chaotic Dynamics in Random Modular Networks

We introduce a model of randomly connected neural populations and study its dynamics by means of the dynamical mean-field theory and simulations. Our analysis uncovers a rich phase diagram, featuring high- and low-dimensional chaotic phases, separated by a crossover region characterized by low values of the maximal Lyapunov exponent and participation ratio dimension, but with high values of the Lyapunov dimension that change significantly across the region. Counterintuitively, chaos can be attenuated by either adding noise to strongly modular connectivity or by introducing modularity into random connectivity. Extending the model to include a multilevel, hierarchical connectivity reveals that a loose balance between activities across levels drives the system towards the edge of chaos.

Microbial populations hardly ever grow logistically and never sublinearly

We investigate the growth dynamics of microbial populations, challenging the conventional logistic model. By analyzing empirical data from various biomes, we demonstrate that microbial growth is better described by a generalized logistic model, the θ-logistic model. This accounts for different growth mechanisms and environmental fluctuations, leading to a generalized gamma distribution of abundance fluctuations. Our findings reveal that microbial growth is never sublinear, so they cannot endorse-at least in the microbial world-the recent proposal of this mechanism as a stability enhancer of highly diverse communities. These results have significant implications for understanding macroecological patterns and the stability of microbial ecosystems.

Path-integral approach to sparse non-Hermitian random matrices

The theory of large random matrices has proved an invaluable tool for the study of systems with disordered interactions in many quite disparate research areas. Widely applicable results, such as the celebrated elliptic law for dense random matrices, allow one to deduce the statistical properties of the interactions in a complex dynamical system that permit stability. However, such simple and universal results have so far proved difficult to come by in the case of sparse random matrices. Here, we perform an expansion in the inverse connectivity, and thus derive general modified versions of the classic elliptic and semicircle laws, taking into account the sparse correction. This is accomplished using a dynamical approach, which maps the hermitized resolvent of a random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling one to construct Feynman diagrams, which facilitate the perturbative analysis. Additionally, in order to demonstrate the broad utility of the path integral framework, we derive a generic non-Hermitian generalization of the Marchenko-Pastur law, and we also show how one can handle non-negligible higher-order statistics (i.e., non-Gaussian statistics) in dense ensembles.

Interaction networks in persistent Lotka-Volterra communities

A central concern of community ecology is the interdependence between interaction strengths and the underlying structure of the network upon which species interact. In this work we present a solvable example of such a feedback mechanism in a generalized Lotka-Volterra dynamical system. Beginning with a community of species interacting on a network with arbitrary degree distribution, we provide an analytical framework from which properties of the eventual "surviving community" can be derived. We find that highly connected species are less likely to survive than their poorly connected counterparts, which skews the eventual degree distribution towards a preponderance of species with lower degrees. Furthermore, the average abundance of the neighbors of a species in the surviving community is lower than the community average (reminiscent of the famed friendship paradox). Finally, we show that correlations emerge between the connectivity of a species and its interactions with its neighbors. More precisely, we find that highly connected species tend to benefit from their neighbors more than their neighbors benefit from them. These correlations are not present in the initial pool of species and are a result of the dynamics.

Symmetry-based approach to species-rich ecological communities

Disordered systems theory provides powerful tools to analyze the generic behaviors of high-dimensional systems, such as species-rich ecological communities or neural networks. By assuming randomness in their interactions, universality ensures that many microscopic details are irrelevant to system-wide dynamics; but the choice of a random ensemble still limits the generality of results. We show here, in the context of ecological dynamics, that these analytical tools do not require a specific choice of ensemble and that solutions can be found based only on a fundamental rotational symmetry in the interactions, encoding the idea that traits can be recombined into new species without altering global features. Dynamical outcomes then depend on the spectrum of the interaction matrix as a free parameter, allowing us to bridge between results found in different models of interactions and extend beyond them to previously unidentified behaviors. The distinctive feature of ecological models is the possibility of species extinctions, which leads to an increased universality of dynamics as the fraction of extinct species increases. We expect these findings can inform new developments in theoretical ecology as well as other families of complex systems.

Ecosystem stability relies on diversity difference between trophic levels

The stability of ecological communities has a profound impact on humans, ranging from individual health influenced by the microbiome to ecosystem services provided by fisheries. A long-standing goal of ecology is the elucidation of the interplay between biodiversity and ecosystem stability, with some ecologists warning of instability due to loss of species diversity while others arguing that greater diversity will instead lead to instability. Here, by considering a minimal two-level ecosystem with multiple predator and prey species, we show that stability does not depend on absolute diversity but rather on diversity differences between levels. We found that increasing diversity in either level first destabilizes but then stabilizes the community (i.e., a reentrant stability transition). We therefore find that it is the diversity difference between levels that is the key to stability, with the least stable communities having similar diversities in different levels. An analytical stability criterion is derived, demonstrating quantitatively that the critical diversity difference is determined by the correlation between how one level affects another and how it is affected in turn. Our stability criterion also applies to consumer-resource models with other forms of interaction such as cross-feeding. Finally, we show that stability depends on diversity differences in ecosystems with three trophic levels. Our finding of a nonmonotonic dependence of stability on diversity provides a natural explanation for the variety of diversity-stability relationships reported in the literature, and emphasizes the significance of level structure in predicting complex community behaviors.

Stability of Ecological Systems: A Theoretical Review

The stability of ecological systems is a fundamental concept in ecology, which offers profound insights into species coexistence, biodiversity, and community persistence. In this article, we provide a systematic and comprehensive review on the theoretical frameworks for analyzing the stability of ecological systems. Notably, we survey various stability notions, including linear stability, sign stability, diagonal stability, D-stability, total stability, sector stability, and structural stability. For each of these stability notions, we examine necessary or sufficient conditions for achieving such stability and demonstrate the intricate interplay of these conditions on the network structures of ecological systems. We further discuss the stability of ecological systems with higher-order interactions.

The Interplay of Binary and Quantitative Structure on the Stability of Mutualistic Networks

Understanding how the structure of biological systems impacts their resilience (broadly defined) is a recurring question across multiple levels of biological organization. In ecology, considerable effort has been devoted to understanding how the structure of interactions between species in ecological networks is linked to different broad resilience outcomes, especially local stability. Still, nearly all of that work has focused on interaction structure in presence-absence terms and has not investigated quantitative structure, i.e., the arrangement of interaction strengths in ecological networks. We investigated how the interplay between binary and quantitative structure impacts stability in mutualistic interaction networks (those in which species interactions are mutually beneficial), using community matrix approaches. We additionally examined the effects of network complexity and within-guild competition for context. In terms of structure, we focused on understanding the stability impacts of nestedness, a structure in which more-specialized species interact with smaller subsets of the same species that more-generalized species interact with. Most mutualistic networks in nature display binary nestedness, which is puzzling because both binary and quantitative nestedness are known to be destabilizing on their own. We found that quantitative network structure has important consequences for local stability. In more-complex networks, binary-nested structures were the most stable configurations, depending on the quantitative structures, but which quantitative structure was stabilizing depended on network complexity and competitive context. As complexity increases and in the absence of within-guild competition, the most stable configurations have a nested binary structure with a complementary (i.e., anti-nested) quantitative structure. In the presence of within-guild competition, however, the most stable networks are those with a nested binary structure and a nested quantitative structure. In other words, the impact of interaction overlap on community persistence is dependent on the competitive context. These results help to explain the prevalence of binary-nested structures in nature and underscore the need for future empirical work on quantitative structure.

Unraveling emergent network indeterminacy in complex ecosystems: A random matrix approach

Indeterminacy of ecological networks-the unpredictability of ecosystem responses to persistent perturbations-is an emergent property of indirect effects a species has on another through interaction chains. Thus, numerous indirect pathways in large, complex ecological communities could make forecasting the long-term outcomes of environmental changes challenging. However, a comprehensive understanding of ecological structures causing indeterminacy has not yet been reached. Here, using random matrix theory (RMT), we provide mathematical criteria determining whether network indeterminacy emerges across various ecological communities. Our analytical and simulation results show that indeterminacy intricately depends on the characteristics of species interaction. Specifically, contrary to conventional wisdom, network indeterminacy is unlikely to emerge in large competitive and mutualistic communities, while it is a common feature in top-down regulated food webs. Furthermore, we found that predictable and unpredictable perturbations can coexist in the same community and that indeterminate responses to environmental changes arise more frequently in networks where predator-prey relationships predominate than competitive and mutualistic ones. These findings highlight the importance of elucidating direct species relationships and analyzing them with an RMT perspective on two fronts: It aids in 1) determining whether the network's responses to environmental changes are ultimately indeterminate and 2) identifying the types of perturbations causing less predictable outcomes in a complex ecosystem. In addition, our framework should apply to the inverse problem of network identification, i.e., determining whether observed responses to sustained perturbations can reconstruct their proximate causalities, potentially impacting other fields such as microbial and medical sciences.

Structural dynamics of plant-pollinator mutualistic networks

The discourse surrounding the structural organization of mutualistic interactions mostly revolves around modularity and nestedness. The former is known to enhance the stability of communities, while the latter is related to their feasibility, albeit compromising the stability. However, it has recently been shown that the joint emergence of these structures poses challenges that can eventually lead to limitations in the dynamic properties of mutualistic communities. We hypothesize that considering compound arrangements-modules with internal nested organization-can offer valuable insights in this debate. We analyze the temporal structural dynamics of 20 plant-pollinator interaction networks and observe large structural variability throughout the year. Compound structures are particularly prevalent during the peak of the pollination season, often coexisting with nested and modular arrangements in varying degrees. Motivated by these empirical findings, we synthetically investigate the dynamics of the structural patterns across two control parameters-community size and connectance levels-mimicking the progression of the pollination season. Our analysis reveals contrasting impacts on the stability and feasibility of these mutualistic communities. We characterize the consistent relationship between network structure and stability, which follows a monotonic pattern. But, in terms of feasibility, we observe nonlinear relationships. Compound structures exhibit a favorable balance between stability and feasibility, particularly in mid-sized ecological communities, suggesting they may effectively navigate the simultaneous requirements of stability and feasibility. These findings may indicate that the assembly process of mutualistic communities is driven by a delicate balance among multiple properties, rather than the dominance of a single one.

Eigenvalue spectra of finely structured random matrices

Random matrix theory allows for the deduction of stability criteria for complex systems using only a summary knowledge of the statistics of the interactions between components. As such, results like the well-known elliptical law are applicable in a myriad of different contexts. However, it is often assumed that all components of the complex system in question are statistically equivalent, which is unrealistic in many applications. Here we introduce the concept of a finely structured random matrix. These are random matrices with element-specific statistics, which can be used to model systems in which the individual components are statistically distinct. By supposing that the degree of "fine structure" in the matrix is small, we arrive at a succinct "modified" elliptical law. We demonstrate the direct applicability of our results to the niche and cascade models in theoretical ecology, as well as a model of a neural network, and a directed network with arbitrary degree distribution. The simple closed form of our central results allow us to draw broad qualitative conclusions about the effect of fine structure on stability.

Individual diet variability shapes the architecture of Antarctic benthic food webs

Antarctic biodiversity is affected by seasonal sea-ice dynamics driving basal resource availability. To (1) determine the role of intraspecific dietary variability in structuring benthic food webs sustaining Antarctic biodiversity, and (2) understand how food webs and the position of topologically central species vary with sea-ice cover, single benthic individuals' diets were studied by isotopic analysis before sea-ice breakup and afterwards. Isotopic trophospecies (or Isotopic Trophic Units) were investigated and food webs reconstructed using Bayesian Mixing Models. As nodes, these webs used either ITUs regardless of their taxonomic membership (ITU-webs) or ITUs assigned to species (population-webs). Both were compared to taxonomic-webs based on taxa and their mean isotopic values. Higher resource availability after sea-ice breakup led to simpler community structure, with lower connectance and linkage density. Intra-population diet variability and compartmentalisation were crucial in determining community structure, showing population-webs to be more complex, stable and robust to biodiversity loss than taxonomic-webs. The core web, representing the minimal community 'skeleton' that expands opportunistically while maintaining web stability with changing resource availability, was also identified. Central nodes included the sea-urchin Sterechinus neumayeri and the bivalve Adamussium colbecki, whose diet is described in unprecedented detail. The core web, compartmentalisation and topologically central nodes represent crucial factors underlying Antarctica's rich benthic food web persistence.

Trophic tug-of-war: Coexistence mechanisms within and across trophic levels

Ecological communities encompass rich diversity across multiple trophic levels. While modern coexistence theory has been widely applied to understand community assembly, its traditional formalism only allows assembly within a single trophic level. Here, using an expanded definition of niche and fitness differences applicable to multitrophic communities, we study how diversity within and across trophic levels affects species coexistence. If each trophic level is analysed separately, both lower- and higher trophic levels are governed by the same coexistence mechanisms. In contrast, if the multitrophic community is analysed as a whole, different trophic levels are governed by different coexistence mechanisms: coexistence at lower trophic levels is predominantly limited by fitness differences, whereas coexistence at higher trophic levels is predominantly limited by niche differences. This dichotomy in coexistence mechanisms is supported by theoretical derivations, simulations of phenomenological and trait-based models, and a case study of a primeval forest ecosystem. Our work provides a general and testable prediction of coexistence mechanism operating in multitrophic communities.

Diversity begets stability: Sublinear growth and competitive coexistence across ecosystems

The worldwide loss of species diversity brings urgency to understanding how diverse ecosystems maintain stability. Whereas early ecological ideas and classic observations suggested that stability increases with diversity, ecological theory makes the opposite prediction, leading to the long-standing "diversity-stability debate." Here, we show that this puzzle can be resolved if growth scales as a sublinear power law with biomass (exponent <1), exhibiting a form of population self-regulation analogous to models of individual ontogeny. We show that competitive interactions among populations with sublinear growth do not lead to exclusion, as occurs with logistic growth, but instead promote stability at higher diversity. Our model realigns theory with classic observations and predicts large-scale macroecological patterns. However, it makes an unsettling prediction: Biodiversity loss may accelerate the destabilization of ecosystems.

Horizontal gene transfer is predicted to overcome the diversity limit of competing microbial species

Natural microbial ecosystems harbor substantial diversity of competing species. Explaining such diversity is challenging, because in classic theories it is extremely infeasible for a large community of competing species to stably coexist in homogeneous environments. One important aspect mostly overlooked in these theories, however, is that microbes commonly share genetic materials with their neighbors through horizontal gene transfer (HGT), which enables the dynamic change of species growth rates due to the fitness effects of the mobile genetic elements (MGEs). Here, we establish a framework of species competition by accounting for the dynamic gene flow among competing microbes. Combining theoretical derivation and numerical simulations, we show that in many conditions HGT can surprisingly overcome the biodiversity limit predicted by the classic model and allow the coexistence of many competitors, by enabling dynamic neutrality of competing species. In contrast with the static neutrality proposed by previous theories, the diversity maintained by HGT is highly stable against random perturbations of microbial fitness. Our work highlights the importance of considering gene flow when addressing fundamental ecological questions in the world of microbes and has broad implications for the design and engineering of complex microbial consortia.

Adaptive rewiring shapes structure and stability in a three-guild herbivore-plant-pollinator network

Animal species, encompassing both pollinators and herbivores, exhibit a preference for plants based on optimal foraging theory. Understanding the intricacies of these adaptive plant-animal interactions in the context of community assembly poses a main challenge in ecology. This study delves into the impact of adaptive interaction rewiring between species belonging to different guilds on the structure and stability of a 3-guild ecological network, incorporating both mutualistic and antagonistic interactions. Our findings reveal that adaptive rewiring results in sub-networks becoming more nested and compartmentalized. Furthermore, the rewiring of interactions uncovers a positive correlation between a plant's generalism concerning both pollinators and herbivores. Additionally, there is a positive correlation between a plant's degree centrality and its energy budget. Although network stability does not exhibit a clear relationship with non-random structures, it is primarily influenced by the balance of multiple interaction strengths. In summary, our results underscore the significance of adaptive interaction rewiring in shaping the structure of 3-guild networks. They emphasize the importance of considering the balance of multiple interactions for the stability of adaptive networks, providing valuable insights into the complex dynamics of ecological communities.

Reactivity of complex communities can be more important than stability

Understanding stability-whether a community will eventually return to its original state after a perturbation-is a major focus in the study of various complex systems, particularly complex ecosystems. Here, we challenge this focus, showing that short-term dynamics can be a better predictor of outcomes for complex ecosystems. Using random matrix theory, we study how complex ecosystems behave immediately after small perturbations. Our analyses show that many communities are expected to be 'reactive', whereby some perturbations will be amplified initially and generate a response that is directly opposite to that predicted by typical stability measures. In particular, we find reactivity is prevalent for complex communities of mixed interactions and for structured communities, which are both expected to be common in nature. Finally, we show that reactivity can be a better predictor of extinction risk than stability, particularly when communities face frequent perturbations, as is increasingly common. Our results suggest that, alongside stability, reactivity is a fundamental measure for assessing ecosystem health.

Time delays modulate the stability of complex ecosystems

What drives the stability, or instability, of complex ecosystems? This question sits at the heart of community ecology and has motivated a large body of theoretical work exploring how community properties shape ecosystem dynamics. However, the overwhelming majority of current theory assumes that species interactions are instantaneous, meaning that changes in the abundance of one species will lead to immediate changes in the abundances of its partners. In practice, time delays in how species respond to one another are widespread across ecological contexts, yet the impact of these delays on ecosystems remains unclear. Here we derive a new body of theory to comprehensively study the impact of time delays on ecological stability. We find that time delays are important for ecosystem stability. Large delays are typically destabilizing but, surprisingly, short delays can substantially increase community stability. Moreover, in stark contrast to delay-free systems, delays dictate that communities with more abundant species can be less stable than ones with less abundant species. Finally, we show that delays fundamentally shift how species interactions impact ecosystem stability, with communities of mixed interaction types becoming the most stable class of ecosystem. Our work demonstrates that time delays can be critical for the stability of complex ecosystems.

Changes in vertical and horizontal diversities mediated by the size structure of introduced fish collectively shape food-web stability

Species introductions can alter local food-web structure by changing the vertical or horizontal diversity within communities, largely driven by their body size distributions. Increasing vertical and horizontal diversities is predicted to have opposing effects on stability. However, their interactive effects remain largely overlooked. We investigated the independent and collective effects of vertical and horizontal diversities on food-web stability in alpine lakes stocked with variable body size distributions of introduced fish species. Introduced predators destabilize food-webs by increasing vertical diversity through food chain lengthening. Alternatively, increasing horizontal diversity results in more stable food-web topologies. A non-linear interaction between vertical and horizontal diversities suggests that increasing vertical diversity is most destabilizing when horizontal diversity is low. Our findings suggest that the size structure of introduced predators drives their impacts on stability by modifying the structure of food-webs, and highlights the interactive effects of vertical and horizontal diversities on stability.

Spatial scaling of soil microbial co-occurrence networks in a fragmented landscape

Habitat loss has been a primary threat to biodiversity. However, species do not function in isolation but often associate with each other and form complex networks. Thus, revealing how the network complexity and stability scale with habitat area will give us more insights into the effects of habitat loss on ecosystems. In this study, we explored the relationships between the island area and the network complexity and stability of soil microbes. We found that the complexity and stability of soil microbial co-occurrence networks scale positively with island area, indicating that habitat loss will potentially simplify and destabilize soil microbial networks.

Temperature, productivity, and habitat characteristics collectively drive lake food web structure

While many efforts have been devoted to understand variations in food web structure among terrestrial and aquatic ecosystems, the environmental factors influencing food web structure at large spatial scales remain hardly explored. Here, we compiled biodiversity inventories to infer food web structure of 67 French lakes using an allometric niche-based model and tested how environmental variables (temperature, productivity, and habitat) influence them. By applying a multivariate analysis on 20 metrics of food web topology, we found that food web structural variations are represented by two distinct complementary and independent structural descriptors. The first is related to the overall trophic diversity, whereas the second is related to the vertical structure. Interestingly, the trophic diversity descriptor was mostly explained by habitat size (26.7% of total deviance explained) and habitat complexity (20.1%) followed by productivity (dissolved organic carbon: 16.4%; nitrate: 9.1%) and thermal variations (10.7%). Regarding the vertical structure descriptor, it was mostly explained by water thermal seasonality (39.0% of total deviance explained) and habitat depth (31.9%) followed by habitat complexity (8.5%) and size (5.5%) as well as annual mean temperature (5.6%). Overall, we found that temperature, productivity, and habitat characteristics collectively shape lake food web structure. We also found that intermediate levels of productivity, high levels of temperature (mean and seasonality), as well as large habitats are associated with the largest and most complex food webs. Our findings, therefore, highlight the importance of focusing on these three components especially in the context of global change, as significant structural changes in aquatic food webs could be expected under increased temperature, pollution, and habitat alterations.

Eukaryotes contribute more than bacteria to the recovery of freshwater ecosystem functions under different drought durations

Global climate change mostly impacts river ecosystems by affecting microbial biodiversity and ecological functions. Considering the high functional redundancy of microorganisms, the unknown relationship between biodiversity and ecosystem functions obstructs river ecological research, especially under the influence of increasing weather extremes, such as in intermittent rivers and ephemeral streams (IRES). Herein, dry-wet alternation experiments were conducted in artificial stream channels for 25 and 90 days of drought, both followed by 20 days of rewetting. The dynamic recovery of microbial biodiversity and ecosystem functions (represented by ecosystem metabolism and denitrification rate) were determined to analyse biodiversity-ecosystem-function (BEF) relationships after different drought durations. There was a significant difference between bacterial and eukaryotic biodiversity recovery after drought. Eukaryotic biodiversity was more sensitive to drought duration than bacterial, and the eukaryotic network was more stable under dry-wet alternations. Based on the establishment of partial least squares path models, we found that eukaryotic biodiversity has a stronger effect on ecosystem functions than bacteria after long-term drought. Indeed, this work represents a significant step forward for further research on the ecosystem functions of IRES, especially emphasizing the importance of eukaryotic biodiversity in the BEF relationship.

Maximal ecological diversity exceeds evolutionary diversity in model ecosystems

Understanding community saturation is fundamental to ecological theory. While investigations of the diversity of evolutionary stable states (ESSs) are widespread, the diversity of communities that have yet to reach an evolutionary endpoint is poorly understood. We use Lotka-Volterra dynamics and trait-based competition to compare the diversity of randomly assembled communities to the diversity of the ESS. We show that, with a large enough founding diversity (whether assembled at once or through sequential invasions), the number of long-time surviving species exceeds that of the ESS. However, the excessive founding diversity required to assemble a saturated community increases rapidly with the dimension of phenotype space. Additionally, traits present in communities resulting from random assembly are more clustered in phenotype space compared to random, although still markedly less ordered than the ESS. By combining theories of random assembly and ESSs we bring a new viewpoint to both the saturation and random assembly literature.

Generalized Lotka-Volterra model with hierarchical interactions

In the analysis of complex ecosystems it is common to use random interaction coefficients, which are often assumed to be such that all species are statistically equivalent. In this work we relax this assumption by imposing hierarchical interspecies interactions. These are incorporated into a generalized Lotka-Volterra dynamical system. In a hierarchical community species benefit more, on average, from interactions with species further below them in the hierarchy than from interactions with those above. Using dynamic mean-field theory, we demonstrate that a strong hierarchical structure is stabilizing, but that it reduces the number of species in the surviving community, as well as their abundances. Additionally, we show that increased heterogeneity in the variances of the interaction coefficients across positions in the hierarchy is destabilizing. We also comment on the structure of the surviving community and demonstrate that the abundance and probability of survival of a species are dependent on its position in the hierarchy.

Stability of multi-layer ecosystems

Community structure is reported to play a critical role in ecosystem stability, which indicates the ability of a system to return to equilibrium after perturbations. However, current studies rely on the assumption that the community consists of only a single-layer structure. In practice, multi-layer structures are common in ecosystems, e.g. the distributions of both microorganisms in strata and fish in the ocean usually stratify into multi-layer structures. Here we use multi-layer networks to model species interactions within each layer and between different layers, and study the stability of multi-layer ecosystems under different interaction types. We show that competitive interactions within each layer have a more substantial stabilizing effect in multi-layer ecosystems relative to their impact in single-layer ecosystems. Surprisingly, between different layers, we find that competition between species destabilizes the ecosystem. We further provide a theoretical analysis of the stability of multi-layer ecosystems and confirm the robustness of our findings for different connectances between layers, numbers of species in each layer, and numbers of layers. Our work provides a general framework for investigating the stability of multi-layer ecosystems and uncovers the double-sided role of competitive interactions in the stability of multi-layer ecosystems.

Eigenvalue spectra and stability of directed complex networks

Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction network between components has on the eigenvalue spectrum. We build on previous results, which usually only take into account the mean degree of the network, by allowing for nontrivial network degree heterogeneity. We derive closed-form expressions for the eigenvalue spectrum of the adjacency matrix of a general weighted and directed network. Using these results, which are valid for any large well-connected complex network, we then derive compact formulas for the corrections (due to nonzero network heterogeneity) to well-known results in random matrix theory. Specifically, we derive modified versions of the Wigner semicircle law, the Girko circle law, and the elliptic law and any outlier eigenvalues. We also derive a surprisingly neat analytical expression for the eigenvalue density of a directed Barabási-Albert network. We are thus able to deduce that network heterogeneity is mostly a destabilizing influence in complex dynamical systems.

Structured interactions as a stabilizing mechanism for competitive ecological communities

How large ecosystems can create and maintain the remarkable biodiversity we see in nature is probably one of the biggest open questions in science, attracting attention from different fields, from theoretical ecology to mathematics and physics. In this context, modeling the stable coexistence of species competing for limited resources is a particularly challenging task. From a mathematical point of view, coexistence in competitive dynamics can be achieved when dominance among species forms intransitive loops. However, these relationships usually lead to species' relative abundances neutrally cycling without converging to a stable equilibrium. Although in recent years several mechanisms have been proposed, models able to explain species coexistence in competitive communities are still limited. Here we identify locality in the interactions as one of the simplest mechanisms leading to stable species coexistence. We consider a simplified ecosystem where individuals of each species lay on a spatial network and interactions are possible only between nodes within a certain distance. Varying such distance allows to interpolate between local and global competition. Our results demonstrate, within the scope of our model, that species coexist reaching a stable equilibrium when two conditions are met: individuals are embedded in space and can only interact with other individuals within a short distance. On the contrary, when one of these ingredients is missing, large oscillations and neutral cycles emerge.

The evolution of trait variance creates a tension between species diversity and functional diversity

It seems intuitively obvious that species diversity promotes functional diversity: communities with more plant species imply more varied plant leaf chemistry, more species of crops provide more kinds of food, etc. Recent literature has nuanced this view, showing how the relationship between the two can be modulated along latitudinal or environmental gradients. Here we show that even without such effects, the evolution of functional trait variance can erase or even reverse the expected positive relationship between species- and functional diversity. We present theory showing that trait-based eco-evolutionary processes force species to evolve narrower trait breadths in more tightly packed, species-rich communities, in their effort to avoid competition with neighboring species. This effect is so strong that it leads to an overall reduction in trait space coverage whenever a new species establishes. Empirical data from land snail communities on the Galápagos Islands are consistent with this claim. The finding that the relationship between species- and functional diversity can be negative implies that trait data from species-poor communities may misjudge functional diversity in species-rich ones, and vice versa.

The positive relationship between species diversity and functional diversity has been shown to vary. Here, the authors use theoretical models and data from Galápagos land snail communities to show how eco-evolutionary processes can force species to evolve narrower trait breadths in more species-rich communities to avoid competition, creating a negative relationship.

Eigenvalues of Random Matrices with Generalized Correlations: A Path Integral Approach

Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical systems. In this Letter, we study the eigenvalue spectrum of an ensemble of random matrices with correlations between any pair of elements. To this end, we introduce an analytical method that maps the resolvent of the random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling us to make deductions about the eigenvalue spectrum. Our central result is a simple, closed-form expression for the leading eigenvalue of a large random matrix with generalized correlations. This formula demonstrates that correlations between matrix elements that are not diagonally opposite, which are often neglected, can have a significant impact on stability.

Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations

We analyze the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modeling the stability of fixed points in large systems defined on complex networks, such as ecosystems consisting of a large number of species that interact through a food web. We develop an exact theory for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices. This theory reveals that the nature of local interactions has a strong influence on a system's stability. We show that, in general, linear dynamical systems defined on random graphs with a prescribed degree distribution of unbounded support are unstable if they are large enough, implying a tradeoff between stability and diversity. Remarkably, in contrast to the generic case, antagonistic systems that contain only interactions of the predator-prey type can be stable in the infinite size limit. This feature for antagonistic systems is accompanied by a peculiar oscillatory behavior of the dynamical response of the system after a perturbation, when the mean degree of the graph is small enough. Moreover, for antagonistic systems we also find that there exist a dynamical phase transition and critical mean degree above which the response becomes nonoscillatory.

High-order correlations in species interactions lead to complex diversity-stability relationships for ecosystems

How ecosystems maintain stability is an active area of research. Inspired by applications of random matrix theory in nuclear physics, May showed decades ago that in an ecosystem model with many randomly interacting species, increasing species diversity decreases the stability of the ecosystem. There have since been many additions to May's efforts, one being an improved understanding the effect of mutualistic, competitive, or predator-prey-like correlations between pairs of species. Here we extend a random matrix technique developed in the context of spin-glass theory to study the effect of high-order correlations among species interactions. The resulting analytically solvable models include next-to-nearest-neighbor correlations in the species interaction network, such as the enemy of my enemy is my friend, as well as higher-order correlations. We find qualitative differences from May and others' models, including nonmonotonic diversity-stability relationships. Furthermore, inclusion of particular next-to-nearest-neighbor correlations in predator-prey as opposed to mutualist-competitive networks causes the former to transition to being more stable at higher species diversity. We discuss potential applicability of our results to microbiota engineering and to the ecology of interpredator interactions, such as cub predation between lions and hyenas as well as companionship between humans and dogs.

Local stability properties of complex, species-rich soil food webs with functional block structure

Ecologists have long debated the properties that confer stability to complex, species-rich ecological networks. Species-level soil food webs are large and structured networks of central importance to ecosystem functioning. Here, we conducted an analysis of the stability properties of an up-to-date set of theoretical soil food web models that account both for realistic levels of species richness and the most recent views on the topological structure (who is connected to whom) of these food webs. The stability of the network was best explained by two factors: strong correlations between interaction strengths and the blocked, nonrandom trophic structure of the web. These two factors could stabilize our model food webs even at the high levels of species richness that are typically found in soil, and that would make random systems very unstable. Also, the stability of our soil food webs is well-approximated by the cascade model. This result suggests that stability could emerge from the hierarchical structure of the functional organization of the web. Our study shows that under the assumption of equilibrium and small perturbations, theoretical soil food webs possess a topological structure that allows them to be complex yet more locally stable than their random counterpart. In particular, results strongly support the general hypothesis that the stability of rich and complex soil food webs is mostly driven by correlations in interaction strength and the organization of the soil food web into functional groups. The implication is that in real-world food web, any force disrupting the functional structure and distribution pattern of interaction strengths (i.e., energy fluxes) of the soil food webs will destabilize the dynamics of the system, leading to species extinction and major changes in the relative abundances of species.

Fluctuation spectra of large random dynamical systems reveal hidden structure in ecological networks

Understanding the relationship between complexity and stability in large dynamical systems-such as ecosystems-remains a key open question in complexity theory which has inspired a rich body of work developed over more than fifty years. The vast majority of this theory addresses asymptotic linear stability around equilibrium points, but the idea of 'stability' in fact has other uses in the empirical ecological literature. The important notion of 'temporal stability' describes the character of fluctuations in population dynamics, driven by intrinsic or extrinsic noise. Here we apply tools from random matrix theory to the problem of temporal stability, deriving analytical predictions for the fluctuation spectra of complex ecological networks. We show that different network structures leave distinct signatures in the spectrum of fluctuations, and demonstrate the application of our theory to the analysis of ecological time-series data of plankton abundances.

A modified niche model for generating food webs with stage-structured consumers: The stabilizing effects of life-history stages on complex food webs

Almost all organisms grow in size during their lifetime and switch diets, trophic positions, and interacting partners as they grow. Such ontogenetic development introduces life-history stages and flows of biomass between the stages through growth and reproduction. However, current research on complex food webs rarely considers life-history stages. The few previously proposed methods do not take full advantage of the existing food web structural models that can produce realistic food web topologies.We extended the niche model developed by Williams and Martinez (Nature, 2000, 404, 180-183) to generate food webs that included trophic species with a life-history stage structure. Our method aggregated trophic species based on niche overlap to form a life-history structured population; therefore, it largely preserved the topological structure of food webs generated by the niche model. We applied the theory of allometric predator-prey body mass ratio and parameterized an allometric bioenergetic model augmented with biomass flow between stages via growth and reproduction to study the effects of a stage structure on the stability of food webs.When life-history stages were linked via growth and reproduction, more food webs persisted, and persisting food webs tended to retain more trophic species. Topological differences between persisting linked and unlinked food webs were small to modest. The slopes of biomass spectra were lower, and weak interaction links were more prevalent in the linked food webs than the unlinked ones, suggesting that a life-history stage structure promotes characteristics that can enhance stability of complex food webs.Our results suggest a positive relationship between the complexity and stability of complex food webs. A life-history stage structure in food webs may play important roles in dynamics of and diversity in food webs.

Stability of generalized ecological-network models

The stability of ecological networks of varying topologies and predator-prey relationships is explored by applying the concept of generalized modeling. The effects of omnivory, complexity, enrichment, number of top predators, and predatory response are discussed. The degree of omnivory plays a large role in governing web stability at steady state. Complexity as measured from connectance and network size is not a perfect indicator of stability; large, highly connected webs can be just as stable as smaller, less connected ones. Learning behavior as expressed in Holling's type III predatory response is stabilizing for food webs and provides exceptions to the paradox of enrichment for some topologies.

Localization and Universality of Eigenvectors in Directed Random Graphs

Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, if the fourth moment of the degree distribution is finite, then the critical mean degree of the localization transition is independent of the degree fluctuations, which is different from localization in undirected graphs that is governed by degree fluctuations. We also show that in the high connectivity limit the distribution of the right eigenvector components is solely determined by the degree distribution. For delocalized eigenvectors, we recover in this limit the universal results from standard random matrix theory that are independent of the degree distribution, while for localized eigenvectors the eigenvector distribution depends on the degree distribution.

Dispersal-induced instability in complex ecosystems

In his seminal work in the 1970s, Robert May suggested that there is an upper limit to the number of species that can be sustained in stable equilibrium by an ecosystem. This deduction was at odds with both intuition and the observed complexity of many natural ecosystems. The so-called stability-diversity debate ensued, and the discussion about the factors contributing to ecosystem stability or instability continues to this day. We show in this work that dispersal can be a destabilising influence. To do this, we combine ideas from Alan Turing's work on pattern formation with May's random-matrix approach. We demonstrate how a stable equilibrium in a complex ecosystem with trophic structure can become unstable with the introduction of dispersal in space, and we discuss the factors which contribute to this effect. Our work highlights that adding more details to the model of May can give rise to more ways for an ecosystem to become unstable. Making May's simple model more realistic is therefore unlikely to entirely remove the upper bound on complexity.

The multiple meanings of omnivory influence empirical, modular theory and whole food web stability relationships

The persistence of whole communities hinges on the presence of select interactions which act to stabilize communities making the identification of these keystone interactions critical. One potential candidate is omnivory, yet theoretical research on omnivory thus far has been dominated by a modular theory approach whereby an omnivore and consumer compete for a shared resource. Empirical research, however, has highlighted the presence of a broader suite of omnivory modules. Here, we integrate empirical data analysis and mathematical models to explore the influence of both omnivory module (including classic, multi-resource, higher level, mutual predation and cannibalism) and omnivore-resource interaction type on food web stability. We use six classic empirical food webs to examine the prevalence of the different types of omnivory, a multi-species consumer-resource model to determine the stability of these different kinds of omnivory within a module context, and finally extend these models to a 50 species, whole food web model to examine the influence of omnivory on whole food web persistence. Our results challenge the concept that omnivory is broadly stabilizing. In particular, we demonstrate that the impact of omnivory depends on the type of omnivory being examined with multi-resource omnivory having the largest correlation with whole food web persistence. Moreover, our results highlight that we need to exercise caution when scaling modular theory to whole food web theory. Cannibalism, for example, was the most persistent and stable omnivory module in the modular theory analysis, but only demonstrated a weak correlation with whole food web persistence. Lastly, our results demonstrate that the frequency of omnivory modules are more important for whole food web persistence than the frequency of omnivore-resource interactions. Together, these results demonstrate that the role of omnivory often depends both on the type of omnivory being examined and the food web within which it is nested. In whole food web models, omnivory acts less as a keystone interaction, rather, specific types of omnivory, particularly multi-resource omnivory, act as keystone modules. Future work integrating module and whole food web theory is critical for resolving the role of key interactions in food webs.

Field distribution patterns of pests are asymmetrically affected by the presence of other herbivores

Because plant phenotypes can change in response to attacks by herbivores in highly variable ways, the distribution of herbivores depends on the occurrence of other herbivore species on the same plant. We carried out a field study to evaluate the co-occurrence of three coconut pests, the mites Aceria guerreronis (Acari: Eriophyidae), Steneotarsonemus concavuscutum (Acari: Tarsonemidae) and the moth Atheloca bondari (Lepidoptera: Pyralidae). The eriophyid mite Ac. guerreronis is the most important coconut pest around the world, whereas S. concavuscutum and At. bondari are economically important only in some areas along the Brazilian coast. A previous study suggested that the necrosis caused by Ac. guerreronis facilitates the infestation of At. bondari larvae. Because all three species infest the area under the perianths on coconuts and S. concavuscutum also causes necrosis that could facilitate At. bondari, we evaluated the co-occurrence of all three species. We found that the occurrence of At. bondari was positively associated with Ac. guerreronis, but negatively associated with S. concavuscutum. In addition, the two mite species showed negative co-occurrence. Atheloca bondari was found on nuts of all ages, but more on nuts that had fallen than on those on the trees, suggesting that nuts infested by At. bondari tend to fall more frequently. We discuss the status of At. bondari as a pest and discuss experiments to test the causes of these co-occurrence patterns.

Periodic density as an endpoint of customized plankton community responses to petroleum hydrocarbons: A level of toxic effect should be matched with a suitable time scale

As an endpoint of community response to contaminants, average periodic density of populations (APDP) has been introduced to model species interactions in a community with 4 planktonic species. An ecological model for the community was developed by means of interspecific relationship including competition and predation to calculate the APDP. As a case study, we reported here the ecotoxicological effects of petroleum hydrocarbons (PHC) collected from Bohai oil field on densities of two algae, Platymonas subcordiformis and Isochrysis galbana, a rotifer, Brachionus plicatilis, and of a cladocera, Penilia avirostris, in single species and a microcosm experiment. Time scales expressing toxic effect increased with increasing levels of toxic effect from molecule to community. Remarkable periodic changes in densities were found during the tests in microcosm experiment, revealing a strong species reaction. The minimum time scale characterizing toxic effect at a community level should be the common cycle of population densities of the microcosm. In addition, the cycles of plankton densities shortened in general with increasing PHC, showing an evident toxic effect on the microcosm. Using APDP as the endpoint, a threshold concentration for the modeled microcosm was calculated to be 0.404 mg-PHC L^-1. The APDP was found to be more sensitive and reliable than the standing crops of populations as the endpoint. This indicated that the APDP, an endpoint at the community level, could be quantitatively related to the endpoints at the population level, and led to the quantitative concentration-toxic effect relationship at the community level.

Component response rate variation underlies the stability of highly complex finite systems

The stability of a complex system generally decreases with increasing system size and interconnectivity, a counterintuitive result of widespread importance across the physical, life, and social sciences. Despite recent interest in the relationship between system properties and stability, the effect of variation in response rate across system components remains unconsidered. Here I vary the component response rates (γ) of randomly generated complex systems. I use numerical simulations to show that when component response rates vary, the potential for system stability increases. These results are robust to common network structures, including small-world and scale-free networks, and cascade food webs. Variation in γ is especially important for stability in highly complex systems, in which the probability of stability would otherwise be negligible. At such extremes of simulated system complexity, the largest stable complex systems would be unstable if not for variation in γ. My results therefore reveal a previously unconsidered aspect of system stability that is likely to be pervasive across all realistic complex systems.

Macro- and mesoscale pattern interdependencies in complex networks

Identifying and explaining the structure of complex networks at different scales has become an important problem across disciplines. At the mesoscale, modular architecture has attracted most of the attention. At the macroscale, other arrangements-e.g. nestedness or core-periphery-have been studied in parallel, but to a much lesser extent. However, empirical evidence increasingly suggests that characterizing a network with a unique pattern typology may be too simplistic, since a system can integrate properties from distinct organizations at different scales. Here, we explore the relationship between some of these different organizational patterns: two at the mesoscale (modularity and in-block nestedness); and one at the macroscale (nestedness). We show experimentally and analytically that nestedness imposes bounds to modularity, with exact analytical results in idealized scenarios. Specifically, we show that nestedness and modularity are interdependent. Furthermore, we analytically evidence that in-block nestedness provides a natural combination between nested and modular networks, taking structural properties of both. Far from a mere theoretical exercise, understanding the boundaries that discriminate each architecture is fundamental, to the extent that modularity and nestedness are known to place heavy dynamical effects on processes, such as species abundances and stability in ecology.

An Approach to Study Species Persistence in Unconstrained Random Networks

The connection between structure and stability of ecological networks has been widely studied in the last fifty years. A challenge that scientists continue to face is that in-depth mathematical model analysis is often difficult, unless the considered systems are specifically constrained. This makes it challenging to generalize results. Therefore, methods are needed that relax the required restrictions. Here, we introduce a novel heuristic approach that provides persistence estimates for random systems without limiting the admissible parameter range and system behaviour. We apply our approach to study persistence of species in random generalized Lotka-Volterra systems and present simulation results, which confirm the accuracy of our predictions. Our results suggest that persistence is mainly driven by the linkage density, whereby additional links can both favour and hinder persistence. In particular, we observed "persistence bistability", a rarely studied feature of random networks, leading to a dependency of persistence on initial species densities. Networks with this property exhibit tipping points, in which species loss can lead to a cascade of extinctions. The methods developed in this paper may facilitate the study of more general models and thereby provide a step forward towards a unifying framework of network architecture and stability.

Antarctic food web architecture under varying dynamics of sea ice cover

In the Ross Sea, biodiversity organisation is strongly influenced by sea-ice cover, which is characterised by marked spatio-temporal variations. Expected changes in seasonal sea-ice dynamics will be reflected in food web architecture, providing a unique opportunity to study effects of climate change. Based on individual stable isotope analyses and the high taxonomic resolution of sampled specimens, we described benthic food webs in contrasting conditions of seasonal sea-ice persistence (early vs. late sea-ice break up) in medium-depth waters in Terra Nova Bay (Ross Sea). The architecture of biodiversity was reshaped by the pulsed input of sympagic food sources following sea-ice break up, with food web simplification, decreased intraguild predation, potential disturbance propagation and increased vulnerability to biodiversity loss. Following our approach, it was possible to describe in unprecedented detail the complex structure of biodiverse communities, emphasising the role of sympagic inputs, regulated by sea-ice dynamics, in structuring Antarctic medium-depth benthic food webs.

Understanding the Mechanisms Behind the Response to Environmental Perturbation in Microbial Mats: A Metagenomic-Network Based Approach

To date, it remains unclear how anthropogenic perturbations influence the dynamics of microbial communities, what general patterns arise in response to disturbance, and whether it is possible to predict them. Here, we suggest the use of microbial mats as a model of study to reveal patterns that can illuminate the ecological processes underlying microbial dynamics in response to stress. We traced the responses to anthropogenic perturbation caused by water depletion in microbial mats from Cuatro Cienegas Basin (CCB), Mexico, by using a time-series spatially resolved analysis in a novel combination of three computational approaches. First, we implemented MEBS (Multi-genomic Entropy-Based Score) to evaluate the dynamics of major biogeochemical cycles across spatio-temporal scales with a single informative value. Second, we used robust Time Series-Ecological Networks (TS-ENs) to evaluate the total percentage of interactions at different taxonomic levels. Lastly, we utilized network motifs to characterize specific interaction patterns. Our results indicate that microbial mats from CCB contain an enormous taxonomic diversity with at least 100 phyla, mainly represented by members of the rare biosphere (RB). Statistical ecological analyses point out a clear involvement of anaerobic guilds related to sulfur and methane cycles during wet versus dry conditions, where we find an increase in fungi, photosynthetic, and halotolerant taxa. TS-ENs indicate that in wet conditions, there was an equilibrium between cooperation and competition (positive and negative relationships, respectively), while under dry conditions there is an over-representation of negative relationships. Furthermore, most of the keystone taxa of the TS-ENs at family level are members of the RB and the microbial mat core highlighting their crucial role within the community. Our results indicate that microbial mats are more robust to perturbation due to redundant functions that are likely shared among community members in the highly connected TS-ENs with density values close to one (≈0.9). Finally, we provide evidence that suggests that a large taxonomic diversity where all community members interact with each other (low modularity), the presence of permanent of low-abundant taxa, and an increase in competition can be potential buffers against environmental disturbance in microbial mats.

The effect of multiple biotic interaction types on species persistence

No species can persist in isolation from other species, but how biotic interactions affect species persistence is still a matter of inquiry. Is persistence more likely in communities with higher proportion of competing species, or in communities with more positive interactions? How do different components of community structure mediate this relationship? We address these questions using a novel simulation framework that generates realistic communities with varying numbers of species and different proportions of biotic interaction types within and across trophic levels. We show that when communities have fewer species, persistence is more likely if positive interactions-such as mutualism and commensalism-are prevalent. In species-rich communities, the disproportionate effect of positive interactions on persistence is diluted and different combinations of biotic interaction types can coexist without affecting persistence significantly. We present the first theoretical examination of how multiple-interaction networks with varying architectures relate to local species persistence, and provide insight about the underlying causes of stability in communities.

Effect of population abundances on the stability of large random ecosystems

Random matrix theory successfully connects the structure of interactions of large ecological communities to their ability to respond to perturbations. One of the most debated aspects of this approach is that so far studies have neglected the role of population abundances on stability. While species abundances are well studied and empirically accessible, studies on stability have so far failed to incorporate this information. Here we tackle this question by explicitly including population abundances in a random matrix framework. We derive an analytical formula that describes the spectrum of a large community matrix for arbitrary feasible species abundance distributions. The emerging picture is remarkably simple: while population abundances affect the rate to return to equilibrium after a perturbation, the stability of large ecosystems is uniquely determined by the interaction matrix. We confirm this result by showing that the likelihood of having a feasible and unstable solution in the Lotka-Volterra system of equations decreases exponentially with the number of species for stable interaction matrices.