Feasibility and coexistence of large ecological communities

Random evolutionary dynamics in predator-prey systems yields large, clustered ecosystems

We study the effect of introducing new species through evolution into communities. We use the setting of predator-prey systems. Predator-prey dynamics is classically well modeled by Lotka-Volterra (LV) equations, also when multiple predator and prey species co-exist. We use a stochastic method to introduce new species in a two-trophic LV system. We find that introducing random evolving species leads to robust ecosystems in which large numbers of species coexist. Crucially, in these large ecosystems an emergent clustering of species is observed, tying functional differences to phylogenetic history.

Emergence of alternative stable states in microbial communities undergoing horizontal gene transfer

Microbial communities living in the same environment often display alternative stable states, each characterized by a unique composition of species. Understanding the origin and determinants of microbiome multistability has broad implications in environments, human health, and microbiome engineering. However, despite its conceptual importance, how multistability emerges in complex communities remains largely unknown. Here, we focused on the role of horizontal gene transfer (HGT), one important aspect mostly overlooked in previous studies, on the stability landscape of microbial populations. Combining mathematical modeling and numerical simulations, we demonstrate that, when mobile genetic elements (MGEs) only affect bacterial growth rates, increasing HGT rate in general promotes multistability of complex microbiota. We further extend our analysis to scenarios where HGT changes interspecies interactions, microbial communities are subjected to strong environmental selections and microbes live in metacommunities consisting of multiple local habitats. We also discuss the role of different mechanisms, including interspecies interaction strength, the growth rate effects of MGEs, MGE epistasis and microbial death rates in shaping the multistability of microbial communities undergoing HGT. These results reveal how different dynamic processes collectively shape community multistability and diversity. Our results provide key insights for the predictive control and engineering of complex microbiota.

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.

Ecosystem Knowledge Should Replace Coexistence and Stability Assumptions in Ecological Network Modelling

Quantitative population modelling is an invaluable tool for identifying the cascading effects of conservation on an ecosystem. When population data from monitoring programs is not available, deterministic ecosystem models have often been calibrated using the theoretical assumption that ecosystems have a stable, coexisting equilibrium. However, a growing body of literature suggests these theoretical assumptions are inappropriate for conservation contexts. Here, we develop an alternative for data-free population modelling that relies on expert-elicited knowledge of species populations. Our new Bayesian algorithm systematically removes model parameters that lead to impossible predictions, as defined by experts, without incurring excessive computational costs. We demonstrate our framework on an ordinary differential equation model by limiting predicted population sizes and their ability to change rapidly, utilising readily available knowledge from field observations and experts rather than relying on theoretical ecosystem properties. Our results show that using only coexistence and stability requirements can lead to unrealistic population dynamics, which can be avoided by switching to expert-derived information. We demonstrate how this change can dramatically impact population predictions, expected responses to management, conservation decision-making, and long-term ecosystem behaviour. Without data, we argue that field observations and expert knowledge are more trustworthy for representing ecosystems observed in nature, improving the precision and confidence in predictions.

Landscape and environmental heterogeneity support coexistence in competitive metacommunities

Metapopulation models have been instrumental in quantifying the ecological impact of landscape structure on the survival of a focal species. However, extensions to multiple species with arbitrary dispersal networks often rely on phenomenological assumptions that inevitably limit their scope. Here, we propose a multilayer network model of competitive dispersing metacommunities to investigate how spatially structured environments impact species coexistence and ecosystem stability. We introduce the concept of landscape-mediated fitness, quantifying how fit a species is in a given environment in terms of colonization and extinction. We show that, when all environments are equivalent, one species excludes all the others-except the marginal case where species fitnesses are in exact trade-off. However, we prove that stable coexistence becomes possible in sufficiently heterogeneous environments by introducing spatial disorder in the model and solving it exactly in the mean-field limit. Crucially, coexistence is supported by the spontaneous localization of species through the emergence of ecological niches. We show that our results remain qualitatively valid in arbitrary dispersal networks, where topological features can improve species coexistence by buffering competition. Finally, we employ our model to study how correlated heterogeneity promotes spatial ecological patterns in realistic terrestrial and riverine landscapes. Our work provides a framework to understand how landscape structure enables coexistence in metacommunities by acting as the substrate for ecological interactions.

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.

Variation in thermal physiology can drive the temperature-dependence of microbial community richness

Predicting how species diversity changes along environmental gradients is an enduring problem in ecology. In microbes, current theories tend to invoke energy availability and enzyme kinetics as the main drivers of temperature-richness relationships. Here, we derive a general empirically-grounded theory that can explain this phenomenon by linking microbial species richness in competitive communities to variation in the temperature-dependence of their interaction and growth rates. Specifically, the shape of the microbial community temperature-richness relationship depends on how rapidly the strength of effective competition between species pairs changes with temperature relative to the variance of their growth rates. Furthermore, it predicts that a thermal specialist-generalist tradeoff in growth rates alters coexistence by shifting this balance, causing richness to peak at relatively higher temperatures. Finally, we show that the observed patterns of variation in thermal performance curves of metabolic traits across extant bacterial taxa is indeed sufficient to generate the variety of community-level temperature-richness responses observed in the real world. Our results provide a new and general mechanism that can help explain temperature-diversity gradients in microbial communities, and provide a quantitative framework for interlinking variation in the thermal physiology of microbial species to their community-level diversity.

Biodiversity in changing environments: An external-driver internal-topology framework to guide intervention

Accompanying the climate crisis is the more enigmatic biodiversity crisis. Rapid reorganization of biodiversity due to global environmental change has defied prediction and tested the basic tenets of conservation and restoration. Conceptual and practical innovation is needed to support decision making in the face of these unprecedented shifts. Critical questions include: How can we generalize biodiversity change at the community level? When are systems able to reorganize and maintain integrity, and when does abiotic change result in collapse or restructuring? How does this understanding provide a template to guide when and how to intervene in conservation and restoration? To this end, we frame changes in community organization as the modulation of external abiotic drivers on the internal topology of species interactions, using plant-plant interactions in terrestrial communities as a starting point. We then explore how this framing can help translate available data on species abundance and trait distributions to corresponding decisions in management. Given the expectation that community response and reorganization are highly complex, the external-driver internal-topology (EDIT) framework offers a way to capture general patterns of biodiversity that can help guide resilience and adaptation in changing environments.

Interspecific dispersal constraints suppress pattern formation in metacommunities

Decisions to disperse from a habitat stand out among organismal behaviours as pivotal drivers of ecosystem dynamics across scales. Encounters with other species are an important component of adaptive decision-making in dispersal, resulting in widespread behaviours like tracking resources or avoiding consumers in space. Despite this, metacommunity models often treat dispersal as a function of intraspecific density alone. We show, focusing initially on three-species network motifs, that interspecific dispersal rules generally drive a transition in metacommunities from homogeneous steady states to self-organized heterogeneous spatial patterns. However, when ecologically realistic constraints reflecting adaptive behaviours are imposed-prey tracking and predator avoidance-a pronounced homogenizing effect emerges where spatial pattern formation is suppressed. We demonstrate this effect for each motif by computing master stability functions that separate the contributions of local and spatial interactions to pattern formation. We extend this result to species-rich food webs using a random matrix approach, where we find that eventually, webs become large enough to override the homogenizing effect of adaptive dispersal behaviours, leading once again to predominately pattern-forming dynamics. Our results emphasize the critical role of interspecific dispersal rules in shaping spatial patterns across landscapes, highlighting the need to incorporate adaptive behavioural constraints in efforts to link local species interactions and metacommunity structure. This article is part of the theme issue 'Diversity-dependence of dispersal: interspecific interactions determine spatial dynamics'.

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.

Exploring spatial segregation induced by competition avoidance as driving mechanism for emergent coexistence in microbial communities

This study investigates the role of spatial segregation, prompted by competition avoidance, as a key mechanism for emergent coexistence within microbial communities. Recognizing these communities as complex adaptive systems, we challenge the sufficiency of mean-field pairwise interaction models, and we consider the impact of spatial dynamics. We developed an individual-based spatial simulation depicting bacterial movement through a pattern of random walks influenced by competition avoidance, leading to the formation of spatially segregated clusters. This model was integrated with a Lotka-Volterra metapopulation framework focused on competitive interactions. Our findings reveal that spatial segregation combined with low diffusion rates and high compositional heterogeneity among patches can lead to emergent coexistence in microbial communities. This reveals a novel mechanism underpinning the formation of stable, coexisting microbe clusters, which is nonetheless incapable of promoting coexistence in the case of isolated pairs of species. This study underscores the importance of considering spatial factors in understanding the dynamics of microbial ecosystems.

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.

When can higher-order interactions produce stable coexistence?

Most ecological models are based on the assumption that species interact in pairs. Diverse communities, however, can have higher-order interactions, in which two or more species jointly impact the growth of a third species. A pitfall of the common pairwise approach is that it misses the higher-order interactions potentially responsible for maintaining natural diversity. Here, we explore the stability properties of systems where higher-order interactions guarantee that a specified set of abundances is a feasible equilibrium of the dynamics. Even these higher-order interactions which lead to equilibria do not necessarily produce stable coexistence. Instead, these systems are more likely to be stable when the pairwise interactions are weak or facilitative. Correlations between the pairwise and higher-order interactions, however, do permit robust coexistence even in diverse systems. Our work not only reveals the challenges in generating stable coexistence through higher-order interactions but also uncovers interaction patterns that can enable diversity.

Sediment DNA Records the Critical Transition of Bacterial Communities in the Arid Lake

It is necessary to predict the critical transition of lake ecosystems due to their abrupt, non-linear effects on social-economic systems. Given the promising application of paleolimnological archives to tracking the historical changes of lake ecosystems, it is speculated that they can also record the lake's critical transition. We studied Lake Dali-Nor in the arid region of Inner Mongolia because of the profound shrinking the lake experienced between the 1300 s and the 1600 s. We reconstructed the succession of bacterial communities from a 140-cm-long sediment core at 4-cm intervals and detected the critical transition. Our results showed that the historical trajectory of bacterial communities from the 1200 s to the 2010s was divided into two alternative states: state1 from 1200 to 1300 s and state2 from 1400 to 2010s. Furthermore, in the late 1300 s, the appearance of a tipping point and critical slowing down implied the existence of a critical transition. By using a multi-decadal time series from the sedimentary core, with general Lotka-Volterra model simulations, local stability analysis found that bacterial communities were the most unstable as they approached the critical transition, suggesting that the collapse of stability triggers the community shift from an equilibrium state to another state. Furthermore, the most unstable community harbored the strongest antagonistic and mutualistic interactions, which may imply the detrimental role of interaction strength on community stability. Collectively, our study showed that sediment DNA can be used to detect the critical transition of lake ecosystems.

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.

Unlocking ensemble ecosystem modelling for large and complex networks

The potential effects of conservation actions on threatened species can be predicted using ensemble ecosystem models by forecasting populations with and without intervention. These model ensembles commonly assume stable coexistence of species in the absence of available data. However, existing ensemble-generation methods become computationally inefficient as the size of the ecosystem network increases, preventing larger networks from being studied. We present a novel sequential Monte Carlo sampling approach for ensemble generation that is orders of magnitude faster than existing approaches. We demonstrate that the methods produce equivalent parameter inferences, model predictions, and tightly constrained parameter combinations using a novel sensitivity analysis method. For one case study, we demonstrate a speed-up from 108 days to 6 hours, while maintaining equivalent ensembles. Additionally, we demonstrate how to identify the parameter combinations that strongly drive feasibility and stability, drawing ecological insight from the ensembles. Now, for the first time, larger and more realistic networks can be practically simulated and analysed.

Emergent competition shapes top-down versus bottom-up control in multi-trophic ecosystems

Ecosystems are commonly organized into trophic levels-organisms that occupy the same level in a food chain (e.g., plants, herbivores, carnivores). A fundamental question in theoretical ecology is how the interplay between trophic structure, diversity, and competition shapes the properties of ecosystems. To address this problem, we analyze a generalized Consumer Resource Model with three trophic levels using the zero-temperature cavity method and numerical simulations. We derive the corresponding mean-field cavity equations and show that intra-trophic diversity gives rise to an effective "emergent competition" term between species within a trophic level due to feedbacks mediated by other trophic levels. This emergent competition gives rise to a crossover from a regime of top-down control (populations are limited by predators) to a regime of bottom-up control (populations are limited by primary producers) and is captured by a simple order parameter related to the ratio of surviving species in different trophic levels. We show that our theoretical results agree with empirical observations, suggesting that the theoretical approach outlined here can be used to understand complex ecosystems with multiple trophic levels.

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.

How collectively integrated are ecological communities?

Beyond abiotic conditions, do population dynamics mostly depend on a species' direct predators, preys and conspecifics? Or can indirect feedback that ripples across the whole community be equally important? Determining where ecological communities sit on the spectrum between these two characterizations requires a metric able to capture the difference between them. Here we show that the spectral radius of a community's interaction matrix provides such a metric, thus a measure of ecological collectivity, which is accessible from imperfect knowledge of biotic interactions and related to observable signatures. This measure of collectivity integrates existing approaches to complexity, interaction structure and indirect interactions. Our work thus provides an original perspective on the question of to what degree communities are more than loose collections of species or simple interaction motifs and explains when pragmatic reductionist approaches ought to suffice or fail when applied to ecological communities.

Stochastic model corrections for reduced Lotka-Volterra models exhibiting mutual, commensal, competitive, and predatory interactions

We explore model-form error and how to correct it in systems of ordinary differential equations. In particular, we focus on the Lotka-Volterra equations, which are used broadly in fields such as ecology, biology, economics, chemistry, and physics. Accounting for every object and their complex interactions with a complete model often becomes infeasible, thereby requiring reduced models. However, reduced models may omit vital relationships, resulting in discrepancies between reduced model predictions and observations from the true system. In this work, we propose a model correction framework for decreasing such discrepancies. Specifically, we embed a stochastic enrichment operator into the reduced model's system of equations. The enrichment operator is theory-informed, calibrated with observations from the complete model, and extended to extrapolative combinations of parameters and initial conditions. The complete model involves N species, while the reduced and enriched models only track M Collapse

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Affiliation(s)

R Bandy Department of Computer Science, University of Colorado Boulder, Boulder, Colorado 80309, USA
R Morrison Department of Computer Science, University of Colorado Boulder, Boulder, Colorado 80309, USA

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Sparsity of higher-order landscape interactions enables learning and prediction for microbiomes

Microbiome engineering offers the potential to leverage microbial communities to improve outcomes in human health, agriculture, and climate. To translate this potential into reality, it is crucial to reliably predict community composition and function. But a brute force approach to cataloging community function is hindered by the combinatorial explosion in the number of ways we can combine microbial species. An alternative is to parameterize microbial community outcomes using simplified, mechanistic models, and then extrapolate these models beyond where we have sampled. But these approaches remain data-hungry, as well as requiring an a priori specification of what kinds of mechanisms are included and which are omitted. Here, we resolve both issues by introducing a mechanism-agnostic approach to predicting microbial community compositions and functions using limited data. The critical step is the identification of a sparse representation of the community landscape. We then leverage this sparsity to predict community compositions and functions, drawing from techniques in compressive sensing. We validate this approach on in silico community data, generated from a theoretical model. By sampling just [Formula: see text]1% of all possible communities, we accurately predict community compositions out of sample. We then demonstrate the real-world application of our approach by applying it to four experimental datasets and showing that we can recover interpretable, accurate predictions on composition and community function from highly limited data.

Structural asymmetry in biotic interactions as a tool to understand and predict ecological persistence

A universal feature of ecological systems is that species do not interact with others with the same sign and strength. Yet, the consequences of this asymmetry in biotic interactions for the short- and long-term persistence of individual species and entire communities remains unclear. Here, we develop a set of metrics to evaluate how asymmetric interactions among species translate to asymmetries in their individual vulnerability to extinction under changing environmental conditions. These metrics, which solve previous limitations of how to independently quantify the size from the shape of the so-called feasibility domain, provide rigorous advances to understand simultaneously why some species and communities present more opportunities to persist than others. We further demonstrate that our shape-related metrics are useful to predict short-term changes in species' relative abundances during 7 years in a Mediterranean grassland. Our approach is designed to be applied to any ecological system regardless of the number of species and type of interactions. With it, we show that is possible to obtain both mechanistic and predictive information on ecological persistence for individual species and entire communities, paving the way for a stronger integration of theoretical and empirical research.

Mean species responses predict effects of environmental change on coexistence

Environmental change research is plagued by the curse of dimensionality: the number of communities at risk and the number of environmental drivers are both large. This raises the pressing question if a general understanding of ecological effects is achievable. Here, we show evidence that this is indeed possible. Using theoretical and simulation-based evidence for bi- and tritrophic communities, we show that environmental change effects on coexistence are proportional to mean species responses and depend on how trophic levels on average interact prior to environmental change. We then benchmark our findings using relevant cases of environmental change, showing that means of temperature optima and of species sensitivities to pollution predict concomitant effects on coexistence. Finally, we demonstrate how to apply our theory to the analysis of field data, finding support for effects of land use change on coexistence in natural invertebrate communities.

Stability of human gut microbiome: Comparison of ecological modelling and observational approaches

The gut microbiome plays a pivotal role in the human body, and perturbations in its composition have been linked to various disorders. Stability is an essential property of a healthy human gut microbiome, which allows it to maintain its functional richness under the external influences. This property has been explored through two distinct methodologies - mathematical modelling based on ecological principles and statistical analysis drawn from observations in interventional studies. Here we conducted a meta-analysis aimed to compare the two approaches utilising the data from 9 interventional and time series studies encompassing 3512 gut microbiome profiles obtained via 16S rRNA gene sequencing. By employing the previously published compositional Lotka-Volterra method, we modelled the dynamics of the microbial community and evaluated ecological stability measures. These measures were compared to those based on observed microbiome changes. There was a substantial correlation between the outcomes of the two approaches. Particularly, local stability assessed within the ecological paradigm was positively correlated with observational stability measures accounting for the compositional nature of microbiome data. Additionally, we were able to reproduce the previously reported inverse relationship between the community's robustness to microorganism loss and local stability, attributed to the distinct impacts of coefficient characterising the network decomposition on these two stability assessments. Our findings demonstrate harmonisation between the ecological and observational approaches to microbiome analysis, advancing the understanding of healthy gut microbiome concept. This paves the way to develop efficient microbiome-targeting interventions for disease prevention and treatment.

Quasi-neutral dynamics in a coinfection system with N strains and asymmetries along multiple traits

Understanding the interplay of different traits in a co-infection system with multiple strains has many applications in ecology and epidemiology. Because of high dimensionality and complex feedback between traits manifested in infection and co-infection, the study of such systems remains a challenge. In the case where strains are similar (quasi-neutrality assumption), we can model trait variation as perturbations in parameters, which simplifies analysis. Here, we apply singular perturbation theory to many strain parameters simultaneously and advance analytically to obtain their explicit collective dynamics. We consider and study such a quasi-neutral model of susceptible-infected-susceptible (SIS) dynamics among N strains, which vary in 5 fitness dimensions: transmissibility, clearance rate of single- and co-infection, transmission probability from mixed coinfection, and co-colonization vulnerability factors encompassing cooperation and competition. This quasi-neutral system is analyzed with a singular perturbation method through an appropriate slow-fast decomposition. The fast dynamics correspond to the embedded neutral system, while the slow dynamics are governed by an N-dimensional replicator equation, describing the time evolution of strain frequencies. The coefficients of this replicator system are pairwise invasion fitnesses between strains, which, in our model, are an explicit weighted sum of pairwise asymmetries along all trait dimensions. Remarkably these weights depend only on the parameters of the neutral system. Such model reduction highlights the centrality of the neutral system for dynamics at the edge of neutrality and exposes critical features for the maintenance of diversity.

Stable diverse food webs become more common when interactions are more biologically constrained

Ecologists have long sought to understand how diversity and structure mediate the stability of whole ecosystems. For high-diversity food webs, the interactions between species are typically represented using matrices with randomly chosen interaction strengths. Unfortunately, this procedure tends to produce ecological systems with no underlying equilibrium solution, and so ecological inferences from this approach may be biased by nonbiological outcomes. Using recent computationally efficient methodological advances from metabolic networks, we employ for the first time an inverse approach to diversity-stability research. We compare classical random interaction matrices of realistic food web topology (hereafter the classical model) to feasible, biologically constrained, webs produced using the inverse approach. We show that an energetically constrained feasible model yields a far higher proportion of stable high-diversity webs than the classical random matrix approach. When we examine the energetically constrained interaction strength distributions of these matrix models, we find that although these diverse webs have consistent negative self-regulation, they do not require strong self-regulation to persist. These energetically constrained diverse webs instead show an increasing preponderance of weak interactions that are known to increase local stability. Further examination shows that some of these weak interactions naturally appear to arise in the model food webs from a constraint-generated realistic generalist-specialist trade-off, whereby generalist predators have weaker interactions than more specialized species. Additionally, the inverse technique we present here has enormous promise for understanding the role of the biological structure behind stable high-diversity webs and for linking empirical data to the theory.

Equilibrium and surviving species in a large Lotka-Volterra system of differential equations

Lotka-Volterra (LV) equations play a key role in the mathematical modeling of various ecological, biological and chemical systems. When the number of species (or, depending on the viewpoint, chemical components) becomes large, basic but fundamental questions such as computing the number of surviving species still lack theoretical answers. In this paper, we consider a large system of LV equations where the interactions between the various species are a realization of a random matrix. We provide conditions to have a unique equilibrium and present a heuristics to compute the number of surviving species. This heuristics combines arguments from Random Matrix Theory, mathematical optimization (LCP), and standard extreme value theory. Numerical simulations, together with an empirical study where the strength of interactions evolves with time, illustrate the accuracy and scope of the results.

Rapid monitoring of ecological persistence

Effective conservation of ecological communities requires accurate and up-to-date information about whether species are persisting or declining to extinction. The persistence of an ecological community is supported by its underlying network of species interactions. While the persistence of the network supporting the whole community is the most relevant scale for conservation, in practice, only small subsets of these networks can be monitored. There is therefore an urgent need to establish links between the small snapshots of data conservationists can collect, and the "big picture" conclusions about ecosystem health demanded by policymakers, scientists, and societies. Here, we show that the persistence of small subnetworks (motifs) in isolation-that is, their persistence when considered separately from the larger network of which they are a part-is a reliable probabilistic indicator of the persistence of the network as a whole. Our methods show that it is easier to detect if an ecological community is not persistent than if it is persistent, allowing for rapid detection of extinction risk in endangered systems. Our results also justify the common practice of predicting ecological persistence from incomplete surveys by simulating the population dynamics of sampled subnetworks. Empirically, we show that our theoretical predictions are supported by data on invaded networks in restored and unrestored areas, even in the presence of environmental variability. Our work suggests that coordinated action to aggregate information from incomplete sampling can provide a means to rapidly assess the persistence of entire ecological networks and the expected success of restoration strategies.

Feasibility and stability in large Lotka Volterra systems with interaction structure

Complex system stability can be studied via linear stability analysis using random matrix theory (RMT) or via feasibility (requiring positive equilibrium abundances). Both approaches highlight the importance of interaction structure. Here we show, analytically and numerically, how RMT and feasibility approaches can be complementary. In generalized Lotka-Volterra (GLV) models with random interaction matrices, feasibility increases when predator-prey interactions increase; increasing competition/mutualism has the opposite effect. These changes have crucial impact on the stability of the GLV model.

Breakdown of Random-Matrix Universality in Persistent Lotka-Volterra Communities

The eigenvalue spectrum of a random matrix often only depends on the first and second moments of its elements, but not on the specific distribution from which they are drawn. The validity of this universality principle is often assumed without proof in applications. In this Letter, we offer a pertinent counterexample in the context of the generalized Lotka-Volterra equations. Using dynamic mean-field theory, we derive the statistics of the interactions between species in an evolved ecological community. We then show that the full statistics of these interactions, beyond those of a Gaussian ensemble, are required to correctly predict the eigenvalue spectrum and therefore stability. Consequently, the universality principle fails in this system. We thus show that the eigenvalue spectra of random matrices can be used to deduce the stability of "feasible" ecological communities, but only if the emergent non-Gaussian statistics of the interactions between species are taken into account.

Non-random interactions within and across guilds shape the potential to coexist in multi-trophic ecological communities

Theory posits that the persistence of species in ecological communities is shaped by their interactions within and across trophic guilds. However, we lack empirical evaluations of how the structure, strength and sign of biotic interactions drive the potential to coexist in diverse multi-trophic communities. Here, we model community feasibility domains, a theoretically informed measure of multi-species coexistence probability, from grassland communities comprising more than 45 species on average from three trophic guilds (plants, pollinators and herbivores). Contrary to our hypothesis, increasing community complexity, measured either as the number of guilds or community richness, did not decrease community feasibility. Rather, we observed that high degrees of species self-regulation and niche partitioning allow for maintaining larger levels of community feasibility and higher species persistence in more diverse communities. Our results show that biotic interactions within and across guilds are not random in nature and both structures significantly contribute to maintaining multi-trophic diversity.

Intraspecific variation in species interactions promotes the feasibility of mutualistic assemblages

Patterns of resource use observed at the species level emerge from the way individuals exploit the range of available resources. Hence, accounting for interindividual differences in resource use, such as pollinator use by plants, is essential to advance our understanding of community assembly and persistence. By using finely resolved data on plant-pollinator interactions, we evaluated how interindividual plant variation in pollinator use scales up to affect community structure and dynamics. All co-occurring plant species comprised specialists interacting with proper subsets of pollinators that visited generalists, and differences in interaction patterns were driven by among-individual trait variation. Furthermore, the nested structure and feasibility of plant-pollinator communities were maximised at higher levels of interindividual plant variation in traits and pollinator use. Our study sheds light on how pervasive properties of community structure arise from individual-level processes and contributes to elucidate the importance of preserving intraspecific variation in traits and resource use within populations.

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.

The Resilience of Plant-Pollinator Networks

There is growing awareness of pollinator declines worldwide. Conservation efforts have mainly focused on finding the direct causes, while paying less attention to building a systemic understanding of the fragility of these communities of pollinators. To fill this gap, we need operational measures of network resilience that integrate two different approaches in theoretical ecology. First, we should consider the range of conditions compatible with the stable coexistence of all of the species in a community. Second, we should address the rate and shape of network collapse once this safe operational space is exited. In this review, we describe this integrative approach and consider several mechanisms that may enhance the resilience of pollinator communities, chiefly rewiring the network of interactions, increasing heterogeneity, allowing variance, and enhancing coevolution. The most pressing need is to develop ways to reduce the gap between these theoretical recommendations and practical applications. This perspective shifts the emphasis from traditional approaches focusing on the equilibrium states to strategies that allow pollination networks to cope with global environmental change.

Feasibility of sparse large Lotka-Volterra ecosystems

Consider a large ecosystem (foodweb) with n species, where the abundances follow a Lotka-Volterra system of coupled differential equations. We assume that each species interacts with [Formula: see text] other species and that their interaction coefficients are independent random variables. This parameter d reflects the connectance of the foodweb and the sparsity of its interactions especially if d is much smaller that n. We address the question of feasibility of the foodweb, that is the existence of an equilibrium solution of the Lotka-Volterra system with no vanishing species. We establish that for a given range of d, namely [Formula: see text] or [Formula: see text] with an extra condition on the sparsity structure, there exists an explicit threshold depending on n and d and reflecting the strength of the interactions, which guarantees the existence of a positive equilibrium as the number of species n gets large. From a mathematical point of view, the study of feasibility is equivalent to the existence of a positive solution [Formula: see text] (component-wise) to the equilibrium linear equation: [Formula: see text]where [Formula: see text] is the [Formula: see text] vector with components 1 and [Formula: see text] is a large sparse random matrix, accounting for the interactions between species. The analysis of such positive solutions essentially relies on large random matrix theory for sparse matrices and Gaussian concentration of measure. The stability of the equilibrium is established. The results in this article extend to a sparse setting the results obtained by Bizeul and Najim in Bizeul and Najim (2021).

Coexistence in diverse communities with higher-order interactions

A central assumption in most ecological models is that the interactions in a community operate only between pairs of species. However, two species may interactively affect the growth of a focal species. Although interactions among three or more species, called higher-order interactions, have the potential to modify our theoretical understanding of coexistence, ecologists lack clear expectations for how these interactions shape community structure. Here we analytically predict and numerically confirm how the variability and strength of higher-order interactions affect species coexistence. We found that as higher-order interaction strengths became more variable across species, fewer species could coexist, echoing the behavior of pairwise models. If interspecific higher-order interactions became too harmful relative to self-regulation, coexistence in diverse communities was destabilized, but coexistence was also lost when these interactions were too weak and mutualistic higher-order effects became prevalent. This behavior depended on the functional form of the interactions as the destabilizing effects of the mutualistic higher-order interactions were ameliorated when their strength saturated with species' densities. Last, we showed that more species-rich communities structured by higher-order interactions lose species more readily than their species-poor counterparts, generalizing classic results for community stability. Our work provides needed theoretical expectations for how higher-order interactions impact species coexistence in diverse communities.

The geometry of evolved community matrix spectra

Random matrix theory has been applied to food web stability for decades, implying elliptical eigenvalue spectra and that large food webs should be unstable. Here we allow feasible food webs to self-assemble within an evolutionary process, using simple Lotka–Volterra equations and several elementary interaction types. We show that, as complex food webs evolve under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${10^5}$$\end{document}105 invasion attempts, the community matrix spectra become bi-modal, rather than falling onto elliptical geometries. Our results raise questions as to the applicability of random matrix theory to the analysis of food web steady states.

Mean-field theory for double-well systems on degree-heterogeneous networks

Many complex dynamical systems in the real world, including ecological, climate, financial and power-grid systems, often show critical transitions, or tipping points, in which the system’s dynamics suddenly transit into a qualitatively different state. In mathematical models, tipping points happen as a control parameter gradually changes and crosses a certain threshold. Tipping elements in such systems may interact with each other as a network, and understanding the behaviour of interacting tipping elements is a challenge because of the high dimensionality originating from the network. Here, we develop a degree-based mean-field theory for a prototypical double-well system coupled on a network with the aim of understanding coupled tipping dynamics with a low-dimensional description. The method approximates both the onset of the tipping point and the position of equilibria with a reasonable accuracy. Based on the developed theory and numerical simulations, we also provide evidence for multistage tipping point transitions in networks of double-well systems.

Effect of delay on the emergent stability patterns in generalized Lotka-Volterra ecological dynamics

Understanding the conditions of feasibility and stability in ecological systems is a major challenge in theoretical ecology. The seminal work of May in 1972 and recent developments based on the theory of random matrices have shown the existence of emergent universal patterns of both stability and feasibility in ecological dynamics. However, only a few studies have investigated the role of delay coupled with population dynamics in the emergence of feasible and stable states. In this work, we study the effects of delay on generalized Loka-Volterra population dynamics of several interacting species in closed ecological environments. First, we investigate the relation between feasibility and stability of the modelled ecological community in the absence of delay and find a simple analytical relation when intra-species interactions are dominant. We then show how, by increasing the time delay, there is a transition in the stability phases of the population dynamics: from an equilibrium state to a stable non-point attractor phase. We calculate analytically the critical delay of that transition and show that it is in excellent agreement with numerical simulations. Finally, following a similar approach to characterizing stability in empirical studies, we investigate the coefficient of variation, which quantifies the magnitude of population fluctuations. We show that in the oscillatory regime induced by the delay, the variability at community level decreases for increasing diversity. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Spatial coherence and the persistence of high diversity in spatially heterogeneous landscapes

Our planet hosts a variety of highly diverse ecosystems. The persistence of high diversity is generally attributed to factors such as the structure of interactions among species and the dispersal of species in metacommunities. Here, we show that large contiguous landscapes-that are characterized by high dispersal-facilitate high species richness due to the spatial heterogeneity in interspecies interactions. We base our analysis on metacommunities under high dispersal where species densities become equal across habitats (spatially coherent). We find that the spatially coherent metacommunity can be represented by an effective species interaction-web that has a significantly lower complexity than the constituent habitats. Our framework also explains how spatial heterogeneity eliminates differences in the effective interaction-web, providing a basis for deviations from the area-heterogeneity tradeoff. These results highlight the often-overlooked case of high dispersal where spatial coherence provides a novel mechanism for supporting high diversity in large heterogeneous landscapes.

Global structural stability and the role of cooperation in mutualistic systems

Dynamical systems on graphs allow to describe multiple phenomena from different areas of Science. In particular, many complex systems in Ecology are studied by this approach. In this paper we analize the mathematical framework for the study of the structural stability of each stationary point, feasible or not, introducing a generalization for this concept, defined as Global Structural Stability. This approach would fit with the proper mathematical concept of structural stability, in which we find a full description of the complex dynamics on the phase space due to nonlinear dynamics. This fact can be analyzed as an informational field grounded in a global attractor whose structure can be completely characterized. These attractors are stable under perturbation and suppose the minimal structurally stable sets. We also study in detail, mathematically and computationally, the zones characterizing different levels of biodiversity in bipartite graphs describing mutualistic antagonistic systems of population dynamics. In particular, we investigate the dependence of the region of maximal biodiversity of a system on its connectivity matrix. On the other hand, as the network topology does not completely determine the robustness of the dynamics of a complex network, we study the correlation between structural stability and several graph measures. A systematic study on synthetic and biological graphs is presented, including 10 mutualistic networks of plants and seed-dispersal and 1000 random synthetic networks. We compare the role of centrality measures and modularity, concluding the importance of just cooperation strength among nodes when describing areas of maximal biodiversity. Indeed, we show that cooperation parameters are the central role for biodiversity while other measures act as secondary supporting functions.

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.

Accuracy of a one-dimensional reduction of dynamical systems on networks

Resilience is an ability of a system with which the system can adjust its activity to maintain its functionality when it is perturbed. To study resilience of dynamics on networks, Gao et al. [Nature (London) 530, 307 (2016)0028-083610.1038/nature16948] proposed a theoretical framework to reduce dynamical systems on networks, which are high dimensional in general, to one-dimensional dynamical systems. The accuracy of this one-dimensional reduction relies on three approximations in addition to the assumption that the network has a negligible degree correlation. In the present study, we analyze the accuracy of the one-dimensional reduction assuming networks without degree correlation. We do so mainly through examining the validity of the individual assumptions underlying the method. Across five dynamical system models, we find that the accuracy of the one-dimensional reduction hinges on the spread of the equilibrium value of the state variable across the nodes in most cases. Specifically, the one-dimensional reduction tends to be accurate when the dispersion of the node's state is small. We also find that the correlation between the node's state and the node's degree, which is common for various dynamical systems on networks, is unrelated to the accuracy of the one-dimensional reduction.

Eigenvalue sensitivity-based analysis for evaluation of biological network stability versus disturbances

Network modeling is an effective tool for understanding the properties of complex systems. Networks are widely used to help us gain insight into biological systems. In this way, the cell, gene, and protein are denoted as nodes, and the connection elements are regarded as links or edges. In this paper, a novel stochastic strategy is developed for identifying the most influential edges on the stability of biological networks. Regarding the principles of networks and control-theory basics like Jacobian and eigenvalue sensitivity-based analysis, a new criterion is proposed, called "random sensitivity index matrix" (RSIM). RSIM evaluates the eigenvalue sensitivity of all edges in a network in the presents of stochastic disturbances based on the Monte Carlo algorithm. Through the values of RSIM elements, the sensitive edges are identifiable. In addition, the contribution of each edge in network instability has been compared through different percentages of disturbances. Different percentages of disturbances did not change the results. The performance of the proposed method was verified by simulation results for Lac (lactose) operon and MAPK (Mitogen-activated protein kinases) as two sample biological networks.

Stability and selective extinction in complex mutualistic networks

We study species abundance in the empirical plant-pollinator mutualistic networks exhibiting broad degree distributions, with uniform intragroup competition assumed, by the Lotka-Volterra equation. The stability of a fixed point is found to be identified by the signs of its nonzero components and those of its neighboring fixed points. Taking the annealed approximation, we derive the nonzero components to be formulated in terms of degrees and the rescaled interaction strengths, which lead us to find different stable fixed points depending on parameters, and we obtain the phase diagram. The selective extinction phase finds small-degree species extinct and effective interaction reduced, maintaining stability and hindering the onset of instability. The nonzero minimum species abundances from different empirical networks show data collapse when rescaled as predicted theoretically.

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.

Spatial heterogeneity enhance robustness of large multi-species ecosystems

Understanding ecosystem stability and functioning is a long-standing goal in theoretical ecology, with one of the main tools being dynamical modelling of species abundances. With the help of spatially unresolved (well-mixed) population models and equilibrium dynamics, limits to stability and regions of various ecosystem robustness have been extensively mapped in terms of diversity (number of species), types of interactions, interaction strengths, varying interaction networks (for example plant-pollinator, food-web) and varying structures of these networks. Although many insights have been gained, the impact of spatial extension is not included in this body of knowledge. Recent studies of spatially explicit modelling on the other hand have shown that stability limits can be crossed and diversity increased for systems with spatial heterogeneity in species interactions and/or chaotic dynamics. Here we show that such crossing and diversity increase can appear under less strict conditions. We find that the mere possibility of varying species abundances at different spatial locations make possible the preservation or increase in diversity across previous boundaries thought to mark catastrophic transitions. In addition, we introduce and make explicit a multitude of different dynamics a spatially extended complex system can use to stabilise. This expanded stabilising repertoire of dynamics is largest at intermediate levels of dispersal. Thus we find that spatially extended systems with intermediate dispersal are more robust, in general have higher diversity and can stabilise beyond previous stability boundaries, in contrast to well-mixed systems.

One of the major challenges facing humanity is the fragmentation of wildlife habitats and decline in biodiversity due to human land-use practices and need for resources. We need to find ways to combine human prosperity with biodiversity conservation. To achieve this a solid understanding of ecosystem stability and functioning is paramount. One way to gain such insight is to find limits when we expect species to go extinct or ecosystems to collapse by simulations of interacting species populations. Many such stability limits have been found theoretically the last decades, but for simplification of modelling, studies often exclude that ecosystems are spread out in space. Here, we explicitly include space and thus allow for dispersal and spatial heterogeneity (local differences) in species abundances. We find that for an ecosystem with the possibility of local spatial heterogeneity, the repertoire of the system’s dynamical behaviour increases dramatically. This increase in possibilities increases system robustness, enables limits previously marking extinction or collapse to be crossed without any remarkable change in global species abundances, and increases biodiversity. Thus we elucidate an additional mechanism pointing to spatial heterogeneity as crucial for ecosystem stability. We find intermediate dispersal as the most favourable for robustness and diversity of ecosystems since they display the largest repertoire of dynamical behaviour.