Predicting the stability of large structured food webs

Identifying a common backbone of interactions underlying food webs from different ecosystems

Although the structure of empirical food webs can differ between ecosystems, there is growing evidence of multiple ways in which they also exhibit common topological properties. To reconcile these contrasting observations, we postulate the existence of a backbone of interactions underlying all ecological networks—a common substructure within every network comprised of species playing similar ecological roles—and a periphery of species whose idiosyncrasies help explain the differences between networks. To test this conjecture, we introduce a new approach to investigate the structural similarity of 411 food webs from multiple environments and biomes. We first find significant differences in the way species in different ecosystems interact with each other. Despite these differences, we then show that there is compelling evidence of a common backbone of interactions underpinning all food webs. We expect that identifying a backbone of interactions will shed light on the rules driving assembly of different ecological communities.

The structure of ecological networks can vary dramatically, yet there may be common features across networks from different ecosystem types. Here, Bramon Mora et al. use network alignment to demonstrate that there is a common backbone of interactions underlying empirical food webs.

Potential oscillators and keystone modules in food webs

Food web theory suggests that the placement of a weak interaction is critical such that under some conditions even one well-placed weak interaction can stabilise multiple strong interactions. This theory suggests that complex stable webs may be built from pivotal weak interactions such that the removal of even one to a few keystone interactions can have significant cascading impacts on whole system diversity and structure. However, the connection between weak interactions, derived from the theory of modular food web components, and keystone species, derived from empirical results, is not yet well understood. Here, we develop numerical techniques to detect potential oscillators hidden in complex food webs, and show that, both in random and real food webs, keystone consumer-resource interactions often operate to stabilise them. Alarmingly, this result suggests that nature frequently may be dangerously close to precipitous change with even the loss of one or a few weakly interacting species.

The feasibility and stability of large complex biological networks: a random matrix approach

In the 70's, Robert May demonstrated that complexity creates instability in generic models of ecological networks having random interaction matrices A. Similar random matrix models have since been applied in many disciplines. Central to assessing stability is the "circular law" since it describes the eigenvalue distribution for an important class of random matrices A. However, despite widespread adoption, the "circular law" does not apply for ecological systems in which density-dependence operates (i.e., where a species growth is determined by its density). Instead one needs to study the far more complicated eigenvalue distribution of the community matrix S = DA, where D is a diagonal matrix of population equilibrium values. Here we obtain this eigenvalue distribution. We show that if the random matrix A is locally stable, the community matrix S = DA will also be locally stable, providing the system is feasible (i.e., all species have positive equilibria D > 0). This helps explain why, unusually, nearly all feasible systems studied here are locally stable. Large complex systems may thus be even more fragile than May predicted, given the difficulty of assembling a feasible system. It was also found that the degree of stability, or resilience of a system, depended on the minimum equilibrium population.

Fluctuating interaction network and time-varying stability of a natural fish community

Ecological theory suggests that large-scale patterns such as community stability can be influenced by changes in interspecific interactions that arise from the behavioural and/or physiological responses of individual species varying over time. Although this theory has experimental support, evidence from natural ecosystems is lacking owing to the challenges of tracking rapid changes in interspecific interactions (known to occur on timescales much shorter than a generation time) and then identifying the effect of such changes on large-scale community dynamics. Here, using tools for analysing nonlinear time series and a 12-year-long dataset of fortnightly collected observations on a natural marine fish community in Maizuru Bay, Japan, we show that short-term changes in interaction networks influence overall community dynamics. Among the 15 dominant species, we identify 14 interspecific interactions to construct a dynamic interaction network. We show that the strengths, and even types, of interactions change with time; we also develop a time-varying stability measure based on local Lyapunov stability for attractor dynamics in non-equilibrium nonlinear systems. We use this dynamic stability measure to examine the link between the time-varying interaction network and community stability. We find seasonal patterns in dynamic stability for this fish community that broadly support expectations of current ecological theory. Specifically, the dominance of weak interactions and higher species diversity during summer months are associated with higher dynamic stability and smaller population fluctuations. We suggest that interspecific interactions, community network structure and community stability are dynamic properties, and that linking fluctuating interaction networks to community-level dynamic properties is key to understanding the maintenance of ecological communities in nature.

The feasibility of equilibria in large ecosystems: A primary but neglected concept in the complexity-stability debate

The consensus that complexity begets stability in ecosystems was challenged in the seventies, a result recently extended to ecologically-inspired networks. The approaches assume the existence of a feasible equilibrium, i.e. with positive abundances. However, this key assumption has not been tested. We provide analytical results complemented by simulations which show that equilibrium feasibility vanishes in species rich systems. This result leaves us in the uncomfortable situation in which the existence of a feasible equilibrium assumed in local stability criteria is far from granted. We extend our analyses by changing interaction structure and intensity, and find that feasibility and stability is warranted irrespective of species richness with weak interactions. Interestingly, we find that the dynamical behaviour of ecologically inspired architectures is very different and richer than that of unstructured systems. Our results suggest that a general understanding of ecosystem dynamics requires focusing on the interplay between interaction strength and network architecture.

The structure of plant spatial association networks is linked to plant diversity in global drylands

1. Despite commonly used to unveil the complex structure of interactions within ecological communities and their value to assess their resilience against external disturbances, network analyses have seldom been applied in plant communities. We evaluated how plant-plant spatial association networks vary in global drylands, and assessed whether network structure was related to plant diversity in these ecosystems. 2. We surveyed 185 dryland ecosystems from all continents except Antarctica and built networks using the local spatial association between all the perennial plants species present in the communities studied. Then, for each network we calculated four descriptors of network structure (link density, link weight mean and heterogeneity, and structural balance), and evaluated their significance with null models. Finally, we used structural equation models to evaluate how abiotic factors (including geography, topography, climate and soil conditions) and network descriptors influenced plant species richness and evenness. 3. Plant networks were highly variable worldwide, but at most study sites (72%) presented common structures such as a higher link density than expected. We also find evidence of the presence of high structural balance in the networks studied. Moreover, all network descriptors considered had a positive and significant effect on plant diversity, and on species richness in particular. Synthesis. Our results constitute the first empirical evidence showing the existence of common network architectures structuring dryland plant communities at the global scale, and suggest a relationship between the structure of spatial networks and plant diversity. They also highlight the importance of system-level approaches to explain the diversity and structure of interactions in plant communities, two major drivers of terrestrial ecosystem functioning.

Self-regulation and the stability of large ecological networks

The stability of complex ecological networks depends both on the interactions between species and the direct effects of the species on themselves. These self-effects are known as 'self-regulation' when an increase in a species' abundance decreases its per-capita growth rate. Sources of self-regulation include intraspecific interference, cannibalism, time-scale separation between consumers and their resources, spatial heterogeneity and nonlinear functional responses coupling predators with their prey. The influence of self-regulation on network stability is understudied and in addition, the empirical estimation of self-effects poses a formidable challenge. Here, we show that empirical food web structures cannot be stabilized unless the majority of species exhibit substantially strong self-regulation. We also derive an analytical formula predicting the effect of self-regulation on network stability with high accuracy and show that even for random networks, as well as networks with a cascade structure, stability requires negative self-effects for a large proportion of species. These results suggest that the aforementioned potential mechanisms of self-regulation are probably more important in contributing to the stability of observed ecological networks than was previously thought.

Phenotypic variation explains food web structural patterns

Food webs (i.e., networks of species and their feeding interactions) share multiple structural features across ecosystems. The factors explaining such similarities are still debated, and the role played by most organismal traits and their intraspecific variation is unknown. Here, we assess how variation in traits controlling predator-prey interactions (e.g., body size) affects food web structure. We show that larger phenotypic variation increases connectivity among predators and their prey as well as total food intake rate. For predators able to eat only a few species (i.e., specialists), low phenotypic variation maximizes intake rates, while the opposite is true for consumers with broader diets (i.e., generalists). We also show that variation sets predator trophic level by determining interaction strengths with prey at different trophic levels. Merging these results, we make two general predictions about the structure of food webs: (i) trophic level should increase with predator connectivity, and (ii) interaction strengths should decrease with prey trophic level. We confirm these predictions empirically using a global dataset of well-resolved food webs. Our results provide understanding of the processes structuring food webs that include functional traits and their naturally occurring variation.

Alternative steady states in ecological networks

In many natural situations, one observes a local system with many competing species that is coupled by weak immigration to a regional species pool. The dynamics of such a system is dominated by its stable and uninvadable (SU) states. When the competition matrix is random, the number of SUs depends on the average value and variance of its entries. Here we consider the problem in the limit of weak competition and large variance. Using a yes-no interaction model, we show that the number of SUs corresponds to the number of maximum cliques in an Erdös-Rényi network. The number of SUs grows exponentially with the number of species in this limit, unless the network is completely asymmetric. In the asymmetric limit, the number of SUs is O(1). Numerical simulations suggest that these results are valid for models with a continuous distribution of competition terms.

Evaluation of an automated protocol for efficient and reliable DNA extraction of dietary samples

Molecular techniques have become an important tool to empirically assess feeding interactions. The increased usage of next‐generation sequencing approaches has stressed the need of fast DNA extraction that does not compromise DNA quality. Dietary samples here pose a particular challenge, as these demand high‐quality DNA extraction procedures for obtaining the minute quantities of short‐fragmented food DNA. Automatic high‐throughput procedures significantly decrease time and costs and allow for standardization of extracting total DNA. However, these approaches have not yet been evaluated for dietary samples. We tested the efficiency of an automatic DNA extraction platform and a traditional CTAB protocol, employing a variety of dietary samples including invertebrate whole‐body extracts as well as invertebrate and vertebrate gut content samples and feces. Extraction efficacy was quantified using the proportions of successful PCR amplifications of both total and prey DNA, and cost was estimated in terms of time and material expense. For extraction of total DNA, the automated platform performed better for both invertebrate and vertebrate samples. This was also true for prey detection in vertebrate samples. For the dietary analysis in invertebrates, there is still room for improvement when using the high‐throughput system for optimal DNA yields. Overall, the automated DNA extraction system turned out as a promising alternative to labor‐intensive, low‐throughput manual extraction methods such as CTAB. It is opening up the opportunity for an extensive use of this cost‐efficient and innovative methodology at low contamination risk also in trophic ecology.

Degree heterogeneity and stability of ecological networks

A classic measure of ecological stability describes the tendency of a community to return to equilibrium after small perturbations. While many advances show how the network architecture of these communities severely constrains such tendencies, one of the most fundamental properties of network structure, i.e. degree heterogeneity-the variability of the number of links associated with each species, deserves further study. Here we show that the effects of degree heterogeneity on stability vary with different types of interspecific interactions. Degree heterogeneity consistently destabilizes ecological networks with both competitive and mutualistic interactions, while its effects on networks of predator-prey interactions such as food webs depend on prey contiguity, i.e. the extent to which the species consume an unbroken sequence of prey in community niche space. Increasing degree heterogeneity tends to stabilize food webs except those with the highest prey contiguity. These findings help explain why food webs are highly but not completely interval and, more broadly, deepen our understanding of the stability of complex ecological networks.

Feasibility and coexistence of large ecological communities

The role of species interactions in controlling the interplay between the stability of ecosystems and their biodiversity is still not well understood. The ability of ecological communities to recover after small perturbations of the species abundances (local asymptotic stability) has been well studied, whereas the likelihood of a community to persist when the conditions change (structural stability) has received much less attention. Our goal is to understand the effects of diversity, interaction strengths and ecological network structure on the volume of parameter space leading to feasible equilibria. We develop a geometrical framework to study the range of conditions necessary for feasible coexistence. We show that feasibility is determined by few quantities describing the interactions, yielding a nontrivial complexity–feasibility relationship. Analysing more than 100 empirical networks, we show that the range of coexistence conditions in mutualistic systems can be analytically predicted. Finally, we characterize the geometric shape of the feasibility domain, thereby identifying the direction of perturbations that are more likely to cause extinctions.

A central question in theoretical ecology is how diverse species can coexist in communities, and how that coexistence depends on network properties. Here, Grilli et al. quantify the extent of feasible coexistence of empirical networks, showing that it is smaller for trophic than mutualism networks.

Eigenvalue spectra of large correlated random matrices

Using the diagrammatic method, we derive a set of self-consistent equations that describe eigenvalue distributions of large correlated asymmetric random matrices. The matrix elements can have different variances and be correlated with each other. The analytical results are confirmed by numerical simulations. The results have implications for the dynamics of neural and other biological networks where plasticity induces correlations in the connection strengths within the network. We find that the presence of correlations can have a major impact on network stability.

The robustness of ecosystems to the species loss of community

To study the robustness of ecosystems is crucial to promote the sustainable development of human society. This paper aims to analyze the robustness of ecosystems from an interesting viewpoint of the species loss of community. Unlike the existing definitions, we first introduce the notion of a community as a population of species belonging to the same trophic level. We then put forward a novel multiobjective optimization model which can be utilized to discover community structures from arbitrary unipartite networks. Because an ecosystem is commonly represented as a multipartite network, we further introduce a mechanism of competition among species whereby a multipartite network is transformed into a unipartite signed network without loss of species interaction information. Finally, we examine three strategies to test the robustness of an ecosystem. Our experiments indicate that ecosystems are robust to random species loss of community but fragile to target ones. We also investigate the relationships between the robustness of an ecosystem and that of its community composed network both to species loss. Our experiments indicate that the robustness analysis of a large-scale ecosystem to species loss may be akin to that of its community composed network which is usually small in size.

The Google matrix controls the stability of structured ecological and biological networks

May's celebrated theoretical work of the 70's contradicted the established paradigm by demonstrating that complexity leads to instability in biological systems. Here May's random-matrix modelling approach is generalized to realistic large-scale webs of species interactions, be they structured by networks of competition, mutualism or both. Simple relationships are found to govern these otherwise intractable models, and control the parameter ranges for which biological systems are stable and feasible. Our analysis of model and real empirical networks is only achievable on introducing a simplifying Google-matrix reduction scheme, which in the process, yields a practical ecological eigenvalue stability index. These results provide an insight into how network topology, especially connectance, influences species stable coexistence. Constraints controlling feasibility (positive equilibrium populations) in these systems are found more restrictive than those controlling stability, helping explain the enigma of why many classes of feasible ecological models are nearly always stable.

High-order species interactions shape ecosystem diversity

Classical theory shows that large communities are destabilized by random interactions among species pairs, creating an upper bound on ecosystem diversity. However, species interactions often occur in high-order combinations, whereby the interaction between two species is modulated by one or more other species. Here, by simulating the dynamics of communities with random interactions, we find that the classical relationship between diversity and stability is inverted for high-order interactions. More specifically, while a community becomes more sensitive to pairwise interactions as its number of species increases, its sensitivity to three-way interactions remains unchanged, and its sensitivity to four-way interactions actually decreases. Therefore, while pairwise interactions lead to sensitivity to the addition of species, four-way interactions lead to sensitivity to species removal, and their combination creates both a lower and an upper bound on the number of species. These findings highlight the importance of high-order species interactions in determining the diversity of natural ecosystems.

Modularity and stability in ecological communities

Networks composed of distinct, densely connected subsystems are called modular. In ecology, it has been posited that a modular organization of species interactions would benefit the dynamical stability of communities, even though evidence supporting this hypothesis is mixed. Here we study the effect of modularity on the local stability of ecological dynamical systems, by presenting new results in random matrix theory, which are obtained using a quaternionic parameterization of the cavity method. Results show that modularity can have moderate stabilizing effects for particular parameter choices, while anti-modularity can greatly destabilize ecological networks.

Modularity in food webs can be caused by spatial and temporal mismatches in interactions. Here, Jacopo Grilli, Tim Rogers and Stefano Allesina show that modularity, contrary to expectations, does not generally help stabilizing ecological communities.

Low-dimensional dynamics of structured random networks

Using a generalized random recurrent neural network model, and by extending our recently developed mean-field approach [J. Aljadeff, M. Stern, and T. Sharpee, Phys. Rev. Lett. 114, 088101 (2015)], we study the relationship between the network connectivity structure and its low-dimensional dynamics. Each connection in the network is a random number with mean 0 and variance that depends on pre- and postsynaptic neurons through a sufficiently smooth function g of their identities. We find that these networks undergo a phase transition from a silent to a chaotic state at a critical point we derive as a function of g. Above the critical point, although unit activation levels are chaotic, their autocorrelation functions are restricted to a low-dimensional subspace. This provides a direct link between the network's structure and some of its functional characteristics. We discuss example applications of the general results to neuroscience where we derive the support of the spectrum of connectivity matrices with heterogeneous and possibly correlated degree distributions, and to ecology where we study the stability of the cascade model for food web structure.