Complexity-stability trade-off in empirical microbial ecosystems

May's stability theory, which holds that large ecosystems can be stable up to a critical level of complexity, a product of the number of resident species and the intensity of their interactions, has been a central paradigm in theoretical ecology. So far, however, empirically demonstrating this theory in real ecological systems has been a long-standing challenge with inconsistent results. Especially, it is unknown whether this theory is pertinent in the rich and complex communities of natural microbiomes, mainly due to the challenge of reliably reconstructing such large ecological interaction networks. Here we introduce a computational framework for estimating an ecosystem's complexity without relying on a priori knowledge of its underlying interaction network. By applying this method to human-associated microbial communities from different body sites and sponge-associated microbial communities from different geographical locations, we found that in both cases the communities display a pronounced trade-off between the number of species and their effective connectance. These results suggest that natural microbiomes are shaped by stability constraints, which limit their complexity.

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.

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.

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.

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.

Species interactions in an Andean bird-flowering plant network: phenology is more important than abundance or morphology

Biological constraints and neutral processes have been proposed to explain the properties of plant–pollinator networks. Using interactions between nectarivorous birds (hummingbirds and flowerpiercers) and flowering plants in high elevation forests (i.e., “elfin” forests) of the Andes, we explore the importance of biological constraints and neutral processes (random interactions) to explain the observed species interactions and network metrics, such as connectance, specialization, nestedness and asymmetry. In cold environments of elfin forests, which are located at the top of the tropical montane forest zone, many plants are adapted for pollination by birds, making this an ideal system to study plant–pollinator networks. To build the network of interactions between birds and plants, we used direct field observations. We measured abundance of birds using mist-nets and flower abundance using transects, and phenology by scoring presence of birds and flowers over time. We compared the length of birds’ bills to flower length to identify “forbidden interactions”—those interactions that could not result in legitimate floral visits based on mis-match in morphology. Diglossa flowerpiercers, which are characterized as “illegitimate” flower visitors, were relatively abundant. We found that the elfin forest network was nested with phenology being the factor that best explained interaction frequencies and nestedness, providing support for biological constraints hypothesis. We did not find morphological constraints to be important in explaining observed interaction frequencies and network metrics. Other network metrics (connectance, evenness and asymmetry), however, were better predicted by abundance (neutral process) models. Flowerpiercers, which cut holes and access flowers at their base and, consequently, facilitate nectar access for other hummingbirds, explain why morphological mis-matches were relatively unimportant in this system. Future work should focus on how changes in abundance and phenology, likely results of climate change and habitat fragmentation, and the role of nectar robbers impact ecological and evolutionary dynamics of plant–pollinator (or flower-visitor) interactions.

Characterizing Species Interactions to Understand Press Perturbations: What Is the Community Matrix?

The community matrix is among ecology's most important mathematical abstractions, formally encapsulating the interconnected network of effects that species have on one another's populations. Despite its importance, the term “community matrix” has been applied to multiple types of matrices that have differing interpretations. This has hindered the application of theory for understanding community structure and perturbation responses. Here, we clarify the correspondence and distinctions among the Interaction matrix, the Alpha matrix, and the Jacobian matrix, terms that are frequently used interchangeably as well as synonymously with the term “community matrix.” We illustrate how these matrices correspond to different ways of characterizing interaction strengths, how they permit insights regarding different types of press perturbations, and how these are related by a simple scaling relationship. Connections to additional interaction strength characterizations encapsulated by the Beta matrix, the Gamma matrix, and the Removal matrix are also discussed. Our synthesis highlights the empirical challenges that remain in using these tools to understand actual communities.

Beyond connectedness: why pairwise metrics cannot capture community stability

The connectedness of species in a trophic web has long been a key structural characteristic for both theoreticians and empiricists in their understanding of community stability. In the past decades, there has been a shift from focussing on determining the number of interactions to taking into account their relative strengths. The question is: How do the strengths of the interactions determine the stability of a community? Recently, a metric has been proposed which compares the stability of observed communities in terms of the strength of three‐ and two‐link feedback loops (cycles of interaction strengths). However, it has also been suggested that we do not need to go beyond the pairwise structure of interactions to capture stability. Here, we directly compare the performance of the feedback and pairwise metrics. Using observed food‐web structures, we show that the pairwise metric does not work as a comparator of stability and is many orders of magnitude away from the actual stability values. We argue that metrics based on pairwise‐strength information cannot capture the complex organization of strong and weak links in a community, which is essential for system stability.

No complexity-stability relationship in empirical ecosystems

Understanding the mechanisms responsible for stability and persistence of ecosystems is one of the greatest challenges in ecology. Robert May showed that, contrary to intuition, complex randomly built ecosystems are less likely to be stable than simpler ones. Few attempts have been tried to test May's prediction empirically, and we still ignore what is the actual complexity–stability relationship in natural ecosystems. Here we perform a stability analysis of 116 quantitative food webs sampled worldwide. We find that classic descriptors of complexity (species richness, connectance and interaction strength) are not associated with stability in empirical food webs. Further analysis reveals that a correlation between the effects of predators on prey and those of prey on predators, combined with a high frequency of weak interactions, stabilize food web dynamics relative to the random expectation. We conclude that empirical food webs have several non-random properties contributing to the absence of a complexity–stability relationship.

A long-standing ecological hypothesis is that complexity should decrease stability in food webs. Here, Jacquet and colleagues analyse over 100 real-world food webs and show that complexity does not decrease stability, but that a high frequency of weak species interactions stabilizes complex food webs.

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.

Nonlinear analogue of the May-Wigner instability transition

We study a system of [Formula: see text] degrees of freedom coupled via a smooth homogeneous Gaussian vector field with both gradient and divergence-free components. In the absence of coupling, the system is exponentially relaxing to an equilibrium with rate μ We show that, while increasing the ratio of the coupling strength to the relaxation rate, the system experiences an abrupt transition from a topologically trivial phase portrait with a single equilibrium into a topologically nontrivial regime characterized by an exponential number of equilibria, the vast majority of which are expected to be unstable. It is suggested that this picture provides a global view on the nature of the May-Wigner instability transition originally discovered by local linear stability analysis.