Feasibility and coexistence of large ecological communities

Diverse communities behave like typical random ecosystems

In 1972, Robert May triggered a worldwide research program studying ecological communities using random matrix theory. Yet, it remains unclear if and when we can treat real communities as random ecosystems. Here, we draw on recent progress in random matrix theory and statistical physics to extend May's approach to generalized consumer-resource models. We show that in diverse ecosystems adding even modest amounts of noise to consumer preferences results in a transition to "typicality," where macroscopic ecological properties of communities are indistinguishable from those of random ecosystems, even when resource preferences have prominent designed structures. We test these ideas using numerical simulations on a wide variety of ecological models. Our work offers an explanation for the success of random consumer resource models in reproducing experimentally observed ecological patterns in microbial communities and highlights the difficulty of scaling up bottom-up approaches in synthetic ecology to diverse communities.

Understanding the emergence of contingent and deterministic exclusion in multispecies communities

Competitive exclusion can be classified as deterministic or as historically contingent. While competitive exclusion is common in nature, it has remained unclear when multispecies communities formed by more than two species should be dominated by deterministic or contingent exclusion. Here, we take a fully parameterised model of an empirical competitive system between invasive annual and native perennial plant species to explain both the emergence and sources of competitive exclusion in multispecies communities. Using a structural approach to understand the range of parameters promoting deterministic and contingent exclusions, we then find heuristic theoretical support for the following three general conclusions. First, we find that the life-history of perennial species increases the probability of observing contingent exclusion by increasing their effective intrinsic growth rates. Second, we find that the probability of observing contingent exclusion increases with weaker intraspecific competition, and not with the level of hierarchical competition. Third, we find a shift from contingent exclusion to deterministic exclusion with increasing numbers of competing species. Our work provides a heuristic framework to increase our understanding about the predictability of species persistence within multispecies communities.

The ratio of single to co-colonization is key to complexity in interacting systems with multiple strains

The high number and diversity of microbial strains circulating in host populations have motivated extensive research on the mechanisms that maintain biodiversity. However, much of this work focuses on strain-specific and cross-immunity interactions. Another less explored mode of pairwise interaction is via altered susceptibilities to co-colonization in hosts already colonized by one strain. Diversity in such interaction coefficients enables strains to create dynamically their niches for growth and persistence, and "engineer" their common environment. How such a network of interactions with others mediates collective coexistence remains puzzling analytically and computationally difficult to simulate. Furthermore, the gradients modulating stability-complexity regimes in such multi-player endemic systems remain poorly understood. In a recent study (Madec & Gjini, Bulletin of Mathematical Biology, 82), we obtained an analytic representation for N-type coexistence in an SIS epidemiological model with co-colonization. We mapped multi-strain dynamics to a replicator equation using timescale separation. Here, we examine what drives coexistence regimes in such co-colonization system. We find the ratio of single to co-colonization, µ, critically determines the type of equilibrium and number of coexisting strains, and encodes a trade-off between overall transmission intensity R 0 and mean interaction coefficient in strain space, k. Preserving a given coexistence regime, under fixed trait variation, requires balancing between higher mean competition in favorable environments, and higher cooperation in harsher environments, and is consistent with the stress gradient hypothesis. Multi-strain coexistence tends to steady-state attractors for small µ, whereas as µ increases, dynamics tend to more complex attractors. Following strain frequencies, evolutionary dynamics in the system also display contrasting patterns with µ, interpolating between multi-stable and fluctuating selection for cooperation and mean invasion fitness, in the two extremes. This co-colonization framework could be applied more generally, to study invariant principles in collective coexistence, and to quantify how critical shifts in community dynamics get potentiated by mean-field and environmental gradients.

Microfluidic and mathematical modeling of aquatic microbial communities

Aquatic microbial communities contribute fundamentally to biogeochemical transformations in natural ecosystems, and disruption of these communities can lead to ecological disasters such as harmful algal blooms. Microbial communities are highly dynamic, and their composition and function are tightly controlled by the biophysical (e.g., light, fluid flow, and temperature) and biochemical (e.g., chemical gradients and cell concentration) parameters of the surrounding environment. Due to the large number of environmental factors involved, a systematic understanding of the microbial community-environment interactions is lacking. In this article, we show that microfluidic platforms present a unique opportunity to recreate well-defined environmental factors in a laboratory setting in a high throughput way, enabling quantitative studies of microbial communities that are amenable to theoretical modeling. The focus of this article is on aquatic microbial communities, but the microfluidic and mathematical models discussed here can be readily applied to investigate other microbiomes.

Dimensionality reduction of complex dynamical systems

One of the outstanding problems in complexity science and engineering is the study of high-dimensional networked systems and of their susceptibility to transitions to undesired states as a result of changes in external drivers or in the structural properties. Because of the incredibly large number of parameters controlling the state of such complex systems and the heterogeneity of its components, the study of their dynamics is extremely difficult. Here we propose an analytical framework for collapsing complex N-dimensional networked systems into an S+1-dimensional manifold as a function of S effective control parameters with S << N. We test our approach on a variety of real-world complex problems showing how this new framework can approximate the system's response to changes and correctly identify the regions in the parameter space corresponding to the system's transitions. Our work offers an analytical method to evaluate optimal strategies in the design or management of networked systems.

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We analytically collapse N-dimensional networked dynamics in low-dimensional manifolds

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We test this approach on a variety of real-world complex problems

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We accurately predict the system's response to changes in parameter values

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We identify regions in parameter space corresponding to system's critical transitions

Merging dynamical and structural indicators to measure resilience in multispecies systems

Resilience is broadly understood as the ability of an ecological system to resist and recover from perturbations acting on species abundances and on the system's structure. However, one of the main problems in assessing resilience is to understand the extent to which measures of recovery and resistance provide complementary information about a system. While recovery from abundance perturbations has a strong tradition under the analysis of dynamical stability, it is unclear whether this same formalism can be used to measure resistance to structural perturbations (e.g. perturbations to model parameters). Here, we provide a framework grounded on dynamical and structural stability in Lotka-Volterra systems to link recovery from small perturbations on species abundances (i.e. dynamical indicators) with resistance to parameter perturbations of any magnitude (i.e. structural indicators). We use theoretical and experimental multispecies systems to show that the faster the recovery from abundance perturbations, the higher the resistance to parameter perturbations. We first use theoretical systems to show that the return rate along the slowest direction after a small random abundance perturbation (what we call full recovery) is negatively correlated with the largest random parameter perturbation that a system can withstand before losing any species (what we call full resistance). We also show that the return rate along the second fastest direction after a small random abundance perturbation (what we call partial recovery) is negatively correlated with the largest random parameter perturbation that a system can withstand before at most one species survives (what we call partial resistance). Then, we use a dataset of experimental microbial systems to confirm our theoretical expectations and to demonstrate that full and partial components of resilience are complementary. Our findings reveal that we can obtain the same level of information about resilience by measuring either a dynamical (i.e. recovery) or a structural (i.e. resistance) indicator. Irrespective of the chosen indicator (dynamical or structural), our results show that we can obtain additional information by separating the indicator into its full and partial components. We believe these results can motivate new theoretical approaches and empirical analyses to increase our understanding about risk in ecological systems.

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.

Behaviour moderates the impacts of food-web structure on species coexistence

How species coexistence (mathematical 'feasibility') in food webs emerges from species' trophic interactions remains a long-standing open question. Here we investigate how structure (network topology and body-size structure) and behaviour (foraging strategy and spatial dimensionality of interactions) interactively affect feasibility in food webs. Metabolically-constrained modelling of food-web dynamics based on whole-organism consumption revealed that feasibility is promoted in systems dominated by large-eat-small foraging (consumers eating smaller resources) whenever (1) many top consumers are present, (2) grazing or sit-and-wait foraging strategies are common, and (3) species engage in two-dimensional interactions. Congruently, the first two conditions were associated with dominance of large-eat-small foraging in 74 well-resolved (primarily aquatic) real-world food webs. Our findings provide a new, mechanistic understanding of how behavioural properties can modulate the effects of structural properties on species coexistence in food webs, and suggest that 'being feasible' constrains the spectra of behavioural and structural properties seen in natural food webs.

Data-Driven Corrections of Partial Lotka-Volterra Models

In many applications of interacting systems, we are only interested in the dynamic behavior of a subset of all possible active species. For example, this is true in combustion models (many transient chemical species are not of interest in a given reaction) and in epidemiological models (only certain subpopulations are consequential). Thus, it is common to use greatly reduced or partial models in which only the interactions among the species of interest are known. In this work, we explore the use of an embedded, sparse, and data-driven discrepancy operator to augment these partial interaction models. Preliminary results show that the model error caused by severe reductions-e.g., elimination of hundreds of terms-can be captured with sparse operators, built with only a small fraction of that number. The operator is embedded within the differential equations of the model, which allows the action of the operator to be interpretable. Moreover, it is constrained by available physical information and calibrated over many scenarios. These qualities of the discrepancy model-interpretability, physical consistency, and robustness to different scenarios-are intended to support reliable predictions under extrapolative conditions.

Ecological models: higher complexity in, higher feasibility out

Finding a compromise between tractability and realism has always been at the core of ecological modelling. The introduction of nonlinear functional responses in two-species models has reconciled part of this compromise. However, it remains unclear whether this compromise can be extended to multispecies models. Yet, answering this question is necessary in order to differentiate whether the explanatory power of a model comes from the general form of its polynomial or from a more realistic description of multispecies systems. Here, we study the probability of feasibility (the existence of at least one positive real equilibrium) in complex models by adding higher-order interactions and nonlinear functional responses to the linear Lotka-Volterra model. We characterize complexity by the number of free-equilibrium points generated by a model, which is a function of the polynomial degree and system's dimension. We show that the probability of generating a feasible system in a model is an increasing function of its complexity, regardless of the specific mechanism invoked. Furthermore, we find that the probability of feasibility in a model will exceed that of the linear Lotka-Volterra model when a minimum level of complexity is reached. Importantly, this minimum level is modulated by parameter restrictions, but can always be exceeded via increasing the polynomial degree or system's dimension. Our results reveal that conclusions regarding the relevance of mechanisms embedded in complex models must be evaluated in relation to the expected explanatory power of their polynomial forms.

An individual-based model for the eco-evolutionary emergence of bipartite interaction networks

How ecological interaction networks emerge on evolutionary time scales remains unclear. Here we build an individual-based eco-evolutionary model for the emergence of mutualistic, antagonistic and neutral bipartite interaction networks. Exploring networks evolved under these scenarios, we find three main results. First, antagonistic interactions tend to foster species and trait diversity, while mutualistic interactions reduce diversity. Second, antagonistic interactors evolve higher specialisation, which results in networks that are often more modular than neutral ones; resource species in these networks often display phylogenetic conservatism in interaction partners. Third, mutualistic interactions lead to networks that are more nested than neutral ones, with low phylogenetic conservatism in interaction partners. These results tend to match overall empirical trends, demonstrating that structures of empirical networks that have most often been explained by ecological processes can result from an evolutionary emergence. Our model contributes to the ongoing effort of better integrating ecological interactions and macroevolution.

Estimating the risk of species interaction loss in mutualistic communities

Interactions between species generate the functions on which ecosystems and humans depend. However, we lack an understanding of the risk that interaction loss poses to ecological communities. Here, we quantify the risk of interaction loss for 4,330 species interactions from 41 empirical pollination and seed dispersal networks across 6 continents. We estimate risk as a function of interaction vulnerability to extinction (likelihood of loss) and contribution to network feasibility, a measure of how much an interaction helps a community tolerate environmental perturbations. Remarkably, we find that more vulnerable interactions have higher contributions to network feasibility. Furthermore, interactions tend to have more similar vulnerability and contribution to feasibility across networks than expected by chance, suggesting that vulnerability and feasibility contribution may be intrinsic properties of interactions, rather than only a function of ecological context. These results may provide a starting point for prioritising interactions for conservation in species interaction networks in the future.

A study of 4,330 species interactions from 41 empirical pollination and seed dispersal networks across six continents reveals that species interactions which are most vulnerable to extinction are also the most important for ecological community stability; moreover, vulnerable interactions that are important for stability tend to be important and vulnerable wherever they occur.

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.

Predicting collapse of complex ecological systems: quantifying the stability-complexity continuum

Dynamical shifts between the extremes of stability and collapse are hallmarks of ecological systems. These shifts are limited by and change with biodiversity, complexity, and the topology and hierarchy of interactions. Most ecological research has focused on identifying conditions for a system to shift from stability to any degree of instability-species abundances do not return to exact same values after perturbation. Real ecosystems likely have a continuum of shifting between stability and collapse that depends on the specifics of how the interactions are structured, as well as the type and degree of disturbance due to environmental change. Here we map boundaries for the extremes of strict stability and collapse. In between these boundaries, we find an intermediate regime that consists of single-species extinctions, which we call the extinction continuum. We also develop a metric that locates the position of the system within the extinction continuum-thus quantifying proximity to stability or collapse-in terms of ecologically measurable quantities such as growth rates and interaction strengths. Furthermore, we provide analytical and numerical techniques for estimating our new metric. We show that our metric does an excellent job of capturing the system's behaviour in comparison with other existing methods-such as May's stability criteria or critical slowdown. Our metric should thus enable deeper insights about how to classify real systems in terms of their overall dynamics and their limits of stability and collapse.

Complex interactions can create persistent fluctuations in high-diversity ecosystems

When can ecological interactions drive an entire ecosystem into a persistent non-equilibrium state, where many species populations fluctuate without going to extinction? We show that high-diversity spatially heterogeneous systems can exhibit chaotic dynamics which persist for extremely long times. We develop a theoretical framework, based on dynamical mean-field theory, to quantify the conditions under which these fluctuating states exist, and predict their properties. We uncover parallels with the persistence of externally-perturbed ecosystems, such as the role of perturbation strength, synchrony and correlation time. But uniquely to endogenous fluctuations, these properties arise from the species dynamics themselves, creating feedback loops between perturbation and response. A key result is that fluctuation amplitude and species diversity are tightly linked: in particular, fluctuations enable dramatically more species to coexist than at equilibrium in the very same system. Our findings highlight crucial differences between well-mixed and spatially-extended systems, with implications for experiments and their ability to reproduce natural dynamics. They shed light on the maintenance of biodiversity, and the strength and synchrony of fluctuations observed in natural systems.

Unveiling dimensions of stability in complex ecological networks

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

Multistability and regime shifts in microbial communities explained by competition for essential nutrients

Microbial communities routinely have several possible species compositions or community states observed for the same environmental parameters. Changes in these parameters can trigger abrupt and persistent transitions (regime shifts) between such community states. Yet little is known about the main determinants and mechanisms of multistability in microbial communities. Here, we introduce and study a consumer-resource model in which microbes compete for two types of essential nutrients each represented by multiple different metabolites. We adapt game-theoretical methods of the stable matching problem to identify all possible species compositions of such microbial communities. We then classify them by their resilience against three types of perturbations: fluctuations in nutrient supply, invasions by new species, and small changes of abundances of existing ones. We observe multistability and explore an intricate network of regime shifts between stable states in our model. Our results suggest that multistability requires microbial species to have different stoichiometries of essential nutrients. We also find that a balanced nutrient supply promotes multistability and species diversity, yet make individual community states less stable.

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.

Reconciling cooperation, biodiversity and stability in complex ecological communities

Empirical evidences show that ecosystems with high biodiversity can persist in time even in the presence of few types of resources and are more stable than low biodiverse communities. This evidence is contrasted by the conventional mathematical modeling, which predicts that the presence of many species and/or cooperative interactions are detrimental for ecological stability and persistence. Here we propose a modelling framework for population dynamics, which also include indirect cooperative interactions mediated by other species (e.g. habitat modification). We show that in the large system size limit, any number of species can coexist and stability increases as the number of species grows, if mediated cooperation is present, even in presence of exploitative or harmful interactions (e.g. antibiotics). Our theoretical approach thus shows that appropriate models of mediated cooperation naturally lead to a solution of the long-standing question about complexity-stability paradox and on how highly biodiverse communities can coexist.

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.

Warnings and caveats in brain controllability

A recent article by Gu et al. (Nat. Commun. 6, 2015) proposed to characterize brain networks, quantified using anatomical diffusion imaging, in terms of their "controllability", drawing on concepts and methods of control theory. They reported that brain activity is controllable from a single node, and that the topology of brain networks provides an explanation for the types of control roles that different regions play in the brain. In this work, we first briefly review the framework of control theory applied to complex networks. We then show contrasting results on brain controllability through the analysis of five different datasets and numerical simulations. We find that brain networks are not controllable (in a statistical significant way) by one single region. Additionally, we show that random null models, with no biological resemblance to brain network architecture, produce the same type of relationship observed by Gu et al. between the average/modal controllability and weighted degree. Finally, we find that resting state networks defined with fMRI cannot be attributed specific control roles. In summary, our study highlights some warning and caveats in the brain controllability framework.

Stability criteria for complex microbial communities

Competition and mutualism are inevitable processes in microbial ecology, and a central question is which and how many taxa will persist in the face of these interactions. Ecological theory has demonstrated that when direct, pairwise interactions among a group of species are too numerous, or too strong, then the coexistence of these species will be unstable to any slight perturbation. Here, we refine and to some extent overturn that understanding, by considering explicitly the resources that microbes consume and produce. In contrast to more complex organisms, microbial cells consume primarily abiotic resources, and mutualistic interactions are often mediated through the mechanism of crossfeeding. We show that if microbes consume, but do not produce resources, then any positive equilibrium will always be stable to small perturbations. We go on to show that in the presence of crossfeeding, stability is no longer guaranteed. However, positive equilibria remain stable whenever mutualistic interactions are either sufficiently weak, or when all pairs of taxa reciprocate each other's assistance.

Microbial communities as dynamical systems

Nowadays, microbial communities are frequently monitored over long periods of time and the interactions between their members are explored in vitro. This development has opened the way to apply mathematical models to characterize community structure and dynamics, to predict responses to perturbations and to explore general dynamical properties such as stability, alternative stable states and periodicity. Here, we highlight the role of dynamical systems theory in the exploration of microbial communities, with a special emphasis on the generalized Lotka-Volterra (gLV) equations. In particular, we discuss applications, assumptions and limitations of the gLV model, mention modifications to address these limitations and review stochastic extensions. The development of dynamical models, together with the generation of time series data, can improve the design and control of microbial communities.

Coexistence of many species in random ecosystems

Rich ecosystems harbour thousands of species interacting in tangled networks encompassing predation, mutualism and competition. Such widespread biodiversity is puzzling, because in ecological models it is exceedingly improbable for large communities to stably coexist. One aspect rarely considered in these models, however, is that coexisting species in natural communities are a selected portion of a much larger pool, which has been pruned by population dynamics. Here we compute the distribution of the number of species that can coexist when we start from a pool of species interacting randomly, and show that even in this case we can observe rich, stable communities. Interestingly, our results show that, once stability conditions are met, network structure has very little influence on the level of biodiversity attained. Our results identify the main drivers responsible for widespread coexistence in natural communities, providing a baseline for determining which structural aspects of empirical communities promote or hinder coexistence.

Structural stability as a consistent predictor of phenological events

The timing of the first and last seasonal appearance of a species in a community typically follows a pattern that is governed by temporal factors. While it has been shown that changes in the environment are linked to phenological changes, the direction of this link appears elusive and context-dependent. Thus, finding consistent predictors of phenological events is of central importance for a better assessment of expected changes in the temporal dynamics of ecological communities. Here we introduce a measure of structural stability derived from species interaction networks as an estimator of the expected range of environmental conditions compatible with the existence of a community. We test this measure as a predictor of changes in species richness recorded on a daily basis in a high-arctic plant-pollinator community during two spring seasons. We find that our measure of structural stability is the only consistent predictor of changes in species richness among different ecological and environmental variables. Our findings suggest that measures based on the notion of structural stability can synthesize the expected variation of environmental conditions tolerated by a community, and explain more consistently the phenological changes observed in ecological communities.

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.

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.

Will a small randomly assembled community be feasible and stable?

How likely is it that few species can randomly assemble into a feasible and stable community? Some studies have answered that as long as the community is feasible, it will nearly always be stable. In contrast, other studies have answered that the likelihood is almost null. Here, we show that the origin of this debate has been the underestimation of the association of the parameter space of intrinsic growth rates with the feasibility and stability properties of small randomly-assembled communities. In particular, we demonstrate that not all parameterizations and sampling distributions of intrinsic growth rates lead to the same probabilities of stability and feasibility, which could mistakenly lead to under- or overestimate the stability properties of feasible communities. Additionally, we find that stability imposes a filtering of species abundances "towards" more even distributions in small feasible randomly-assembled communities. This indicates that the stability of feasible communities is inherently linked to the starting distribution of species abundances, a characteristic that many times has been ignored, but should be incorporated in manageable lab and field experiments. Overall, the return to this debate is a central reminder that a more systematic exploration of the feasible parameter space is necessary to derive general conclusions about the stability properties of ecological communities.

Mapping the ecological networks of microbial communities

Mapping the ecological networks of microbial communities is a necessary step toward understanding their assembly rules and predicting their temporal behavior. However, existing methods require assuming a particular population dynamics model, which is not known a priori. Moreover, those methods require fitting longitudinal abundance data, which are often not informative enough for reliable inference. To overcome these limitations, here we develop a new method based on steady-state abundance data. Our method can infer the network topology and inter-taxa interaction types without assuming any particular population dynamics model. Additionally, when the population dynamics is assumed to follow the classic Generalized Lotka-Volterra model, our method can infer the inter-taxa interaction strengths and intrinsic growth rates. We systematically validate our method using simulated data, and then apply it to four experimental data sets. Our method represents a key step towards reliable modeling of complex, real-world microbial communities, such as the human gut microbiota.

Unifying relationships between complexity and stability in mutualistic ecological communities

Conserving ecosystem function and associated services requires deep understanding of the underlying basis of system stability. While the study of ecological dynamics is a mature and diverse field, the lack of a general model that predicts a broad range of theoretical and empirical observations has allowed unresolved contradictions to persist. Here we provide a general model of mutualistic ecological interactions between two groups and show for the first time how the conditions for bi-stability, the nature of critical transitions, and identifiable leading indicators in time-series can be derived from the basic parameters describing the underlying ecological interactions. Strong mutualism and nonlinearity in handling-time are found to be necessary conditions for the occurrence of critical transitions. We use the model to resolve open questions concerning the effects of heterogeneity in inter-species interactions on both resilience and abundance, and discuss these in terms of potential trade-offs in real systems. This framework provides a basis for rich investigations of ecological system dynamics, and may be generalizable across many ecological contexts.

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.

Explorability and the origin of network sparsity in living systems

The increasing volume of ecologically and biologically relevant data has revealed a wide collection of emergent patterns in living systems. Analysing different data sets, ranging from metabolic gene-regulatory to species interaction networks, we find that these networks are sparse, i.e. the percentage of the active interactions scales inversely proportional to the system size. To explain the origin of this puzzling common characteristic, we introduce the new concept of explorability: a measure of the ability of an interacting system to adapt to newly intervening changes. We show that sparsity is an emergent property resulting from optimising both explorability and dynamical robustness, i.e. the capacity of the system to remain stable after perturbations of the underlying dynamics. Networks with higher connectivities lead to an incremental difficulty to find better values for both the explorability and dynamical robustness, associated with the fine-tuning of the newly added interactions. A relevant characteristic of our solution is its scale invariance, i.e., it remains optimal when several communities are assembled together. Connectivity is also a key ingredient in determining ecosystem stability and our proposed solution contributes to solving May's celebrated complexity-stability paradox.

Collapse of resilience patterns in generalized Lotka-Volterra dynamics and beyond

Recently, a theoretical framework aimed at separating the roles of dynamics and topology in multidimensional systems has been developed [Gao et al., Nature (London) 530, 307 (2016)10.1038/nature16948]. The validity of their method is assumed to hold depending on two main hypotheses: (i) The network determined by the the interaction between pairs of nodes has negligible degree correlations; (ii) the node activities are uniform across nodes on both the drift and the pairwise interaction functions. Moreover, the authors consider only positive (mutualistic) interactions. Here we show the conditions proposed by Gao and collaborators [Nature (London) 530, 307 (2016)10.1038/nature16948] are neither sufficient nor necessary to guarantee that their method works in general and validity of their results are not independent of the model chosen within the class of dynamics they considered. Indeed we find that a new condition poses effective limitations to their framework and we provide quantitative predictions of the quality of the one-dimensional collapse as a function of the properties of interaction networks and stable dynamics using results from random matrix theory. We also find that multidimensional reduction may work also for an interaction matrix with a mixture of positive and negative signs, opening up an application of the framework to food webs, neuronal networks, and social and economic interactions.