A guideline to study the feasibility domain of multi-trophic and changing ecological communities

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.

Parameter Sensitivity of Transient Community Dynamics

AbstractTransient dynamics have always intrigued ecologists, but current rapid environmental change (inducing transients even in previously undisturbed systems) has highlighted their importance more than ever. Here, I introduce a method for analyzing the sensitivity of transient ecological dynamics to parameter perturbations. The question the method answers is: how would the community dynamics have unfolded for some time horizon had the parameters been slightly different? I apply the method to three empirically parameterized models: competition between native forbs and exotic grasses in California, a host-parasitoid system, and an experimental chemostat predator-prey model. These applications showcase the ecological insights one can gain from models using transient sensitivity analysis. First, one can find parameters and their combinations whose perturbations disproportionately affect a system. Second, one can identify particular windows of time during which the predicted deviation from the unperturbed trajectories is especially large and utilize this information for management purposes. Third, there is an inverse relationship between transient and long-term sensitivities whenever the interacting populations are ecologically similar; paradoxically, the smaller the immediate response of the system, the more extreme its long-term response will be.

Multitrophic Higher-Order Interactions Modulate Species Persistence

AbstractEcologists increasingly recognize that interactions between two species can be affected by the density of a third species. How these higher-order interactions (HOIs) affect species persistence remains poorly understood. To explore the effect of HOIs stemming from multiple trophic layers on a plant community composition, we experimentally built a mesocosm with three plants and three pollinator species arranged in a fully nested and modified network structure. We estimated pairwise interactions among plants and between plants and pollinators, as well as HOIs initiated by a plant or a pollinator affecting plant species pairs. Using a structuralist approach, we evaluated the consequences of the statistically supported HOIs on the persistence probability of each of the three competing plant species and their combinations. HOIs substantially redistribute the strength and sign of pairwise interactions between plant species, promoting the opportunities for multispecies communities to persist compared with a non-HOI scenario. However, the physical elimination of a plant-pollinator link in the modified network structure promotes changes in per capita pairwise interactions and HOIs, resulting in a single-species community. Our study provides empirical evidence of the joint importance of HOIs and network structure in determining species persistence within diverse communities.

Assessing mutualistic metacommunity capacity by integrating spatial and interaction networks

We develop a spatially realistic model of mutualistic metacommunities that exploits the joint structure of spatial and interaction networks. Assuming that all species have the same colonisation and extinction parameters, this model exhibits a sharp transition between stable non-null equilibrium states and a global extinction state. This behaviour allows defining a threshold on colonisation/extinction parameters for the long-term metacommunity persistence. This threshold, the 'metacommunity capacity', extends the metapopulation capacity concept and can be calculated from the spatial and interaction networks without needing to simulate the whole dynamics. In several applications we illustrate how the joint structure of the spatial and the interaction networks affects metacommunity capacity. It results that a weakly modular spatial network and a power-law degree distribution of the interaction network provide the most favourable configuration for the long-term persistence of a mutualistic metacommunity. Our model that encodes several explicit ecological assumptions should pave the way for a larger exploration of spatially realistic metacommunity models involving multiple interaction types.

Quantifying soil microbial effects on plant species coexistence: A conceptual synthesis

Soil microorganisms play a critical role in shaping the biodiversity dynamics of plant communities. These microbial effects can arise through direct mediation of plant fitness by pathogens and mutualists, and over the past two decades, numerous studies have shined a spotlight on the role of dynamic feedbacks between plants and soil microorganisms as key determinants of plant species coexistence. Such feedbacks occur when plants modify the composition of the soil community, which in turn affects plant performance. Stimulated by a theoretical model developed in the 1990s, a bulk of the empirical evidence for microbial controls over plant coexistence comes from experiments that quantify plant growth in soil communities that were previously conditioned by conspecific or heterospecific plants. These studies have revealed that soil microbes can generate strong negative to positive frequency-dependent dynamics among plants. Even as soil microbes have become recognized as a key player in determining plant coexistence outcomes, the past few years have seen a renewed interest in expanding the conceptual foundations of this field. New results include re-interpretations of key metrics from classic two-species models, extensions of plant-soil feedback theory to multispecies communities, and frameworks to integrate plant-soil feedbacks with processes like intra- and interspecific competition. Here, I review the implications of theoretical developments for interpreting existing empirical results and highlight proposed analyses and designs for future experiments that can enable a more complete understanding of microbial regulation of plant community dynamics.

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

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

Interaction network structure explains species' temporal persistence in empirical plant-pollinator communities

Despite clear evidence that some pollinator populations are declining, our ability to predict pollinator communities prone to collapse or species at risk of local extinction is remarkably poor. Here, we develop a model grounded in the structuralist approach that allows us to draw sound predictions regarding the temporal persistence of species in mutualistic networks. Using high-resolution data from a six-year study following 12 independent plant-pollinator communities, we confirm that pollinator species with more persistent populations in the field are theoretically predicted to tolerate a larger range of environmental changes. Persistent communities are not necessarily more diverse, but are generally located in larger habitat patches, and present a distinctive combination of generalist and specialist species resulting in a more nested structure, as predicted by previous theoretical work. Hence, pollinator interactions directly inform about their ability to persist, opening the door to use theoretically informed models to predict species' fate within the ongoing global change.

Plant interaction networks reveal the limits of our understanding of diversity maintenance

Species interactions are key drivers of biodiversity and ecosystem stability. Current theoretical frameworks for understanding the role of interactions make many assumptions which unfortunately, do not always hold in natural, diverse communities. This mismatch extends to annual plants, a common model system for studying coexistence, where interactions are typically averaged across environmental conditions and transitive competitive hierarchies are assumed to dominate. We quantify interaction networks for a community of annual wildflowers in Western Australia across a natural shade gradient at local scales. Whilst competition dominated, intraspecific and interspecific facilitation were widespread in all shade categories. Interaction strengths and directions varied substantially despite close spatial proximity and similar levels of local species richness, with most species interacting in different ways under different environmental conditions. Contrary to expectations, all networks were predominantly intransitive. These findings encourage us to rethink how we conceive of and categorize the mechanisms driving biodiversity in plant systems.

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.

Robustness versus productivity during evolutionary community assembly: short-term synergies and long-term trade-offs

The realization that evolutionary feedbacks need to be considered to fully grasp ecological dynamics has sparked interest in the effect of evolution on community properties like coexistence and productivity. However, little is known about the evolution of community robustness and productivity along diversification processes in species-rich systems. We leverage the recent structural approach to coexistence together with adaptive dynamics to study such properties and their relationships in a general trait-based model of competition on a niche axis. We show that the effects of coevolution on coexistence are two-fold and contrasting depending on the time scale considered. In the short term, evolution of niche differentiation strengthens coexistence, while long-term diversification leads to niche packing and decreased robustness. Moreover, we find that coevolved communities tend to be on average more robust and more productive than non-evolutionary assemblages. We illustrate how our theoretical predictions echo in observed empirical patterns and the implications of our results for empiricists and applied ecologists. We suggest that some of our results such as the improved robustness of Evolutionarily Stable Communities could be tested experimentally in suitable model systems.

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.

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.

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.

Nitrogen enrichment and foliar fungal pathogens affect the mechanisms of multispecies plant coexistence

Changes in resources (e.g. nitrogen) and enemies (e.g. foliar pathogens) are key drivers of plant diversity and composition. However, their effects have not been connected to the niche and fitness differences that determine multispecies coexistence. Here, we combined a structuralist theoretical approach with a detailed grassland experiment factorially applying nitrogen addition and foliar fungal pathogen suppression to evaluate the joint effect of nitrogen and pathogens on niche and fitness differences, across a gradient from two to six interacting species. Nitrogen addition and pathogen suppression modified species interaction strengths and intrinsic growth rates, leading to reduced multispecies fitness differences. However, contrary to expected, we also observed that they promote stabilising niche differences. Although these modifications did not substantially alter species richness, they predicted major changes in community composition. Indirect interactions between species explained these community changes in smaller assemblages (three and four species) but lost importance in favour of direct pairwise interactions when more species were involved (five and six). Altogether, our work shows that explicitly considering the number of interacting species is critical for better understanding the direct and indirect processes by which nitrogen enrichment and pathogen communities shape coexistence in grasslands.

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.

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.

Understanding the impact of third-party species on pairwise coexistence

The persistence of virtually every single species depends on both the presence of other species and the specific environmental conditions in a given location. Because in natural settings many of these conditions are unknown, research has been centered on finding the fraction of possible conditions (probability) leading to species coexistence. The focus has been on the persistence probability of an entire multispecies community (formed of either two or more species). However, the methodological and philosophical question has always been whether we can observe the entire community and, if not, what the conditions are under which an observed subset of the community can persist as part of a larger multispecies system. Here, we derive long-term (using analytical calculations) and short-term (using simulations and experimental data) system-level indicators of the effect of third-party species on the coexistence probability of a pair (or subset) of species under unknown environmental conditions. We demonstrate that the fraction of conditions incompatible with the possible coexistence of a pair of species tends to become vanishingly small within systems of increasing numbers of species. Yet, the probability of pairwise coexistence in isolation remains approximately the expected probability of pairwise coexistence in more diverse assemblages. In addition, we found that when third-party species tend to reduce (resp. increase) the coexistence probability of a pair, they tend to exhibit slower (resp. faster) rates of competitive exclusion. Long-term and short-term effects of the remaining third-party species on all possible specific pairs in a system are not equally distributed, but these differences can be mapped and anticipated under environmental uncertainty.

It is debated whether the frequency with which two species coexist in isolation or within a single environmental context is representative of their coexistence expectation within larger multispecies systems and across different environmental conditions. Here, using analytical calculations, simulations, and experimental data, we show why and how third-party species can provide the opportunity for pairwise coexistence regardless of whether a pair of species can coexist in isolation across different environmental conditions. However, we show that this opportunity is not homogeneously granted across all pairs within the same system. We provide a framework to understand and map the long-term and short-term effects that third-party species have on the coexistence of each possible subset in a multispecies system.

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.

A keystone gene underlies the persistence of an experimental food web

Genes encode information that determines an organism's fitness. Yet we know little about whether genes of one species influence the persistence of interacting species in an ecological community. Here, we experimentally tested the effect of three plant defense genes on the persistence of an insect food web and found that a single allele at a single gene promoted coexistence by increasing plant growth rate, which in turn increased the intrinsic growth rates of species across multiple trophic levels. Our discovery of a "keystone gene" illustrates the need to bridge between biological scales, from genes to ecosystems, to understand community persistence.

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.

On the sensitivity of food webs to multiple stressors

Evaluating the effects of multiple stressors on ecosystems is becoming increasingly vital with global changes. The role of species interactions in propagating the effects of stressors, although widely acknowledged, has yet to be formally explored. Here, we conceptualise how stressors propagate through food webs and explore how they affect simulated three-species motifs and food webs of the Canadian St. Lawrence System. We find that overlooking species interactions invariably underestimate the effects of stressors, and that synergistic and antagonistic effects through food webs are prevalent. We also find that interaction type influences a species' susceptibility to stressors; species in omnivory and tri-trophic food chain interactions in particular are sensitive and prone to synergistic and antagonistic effects. Finally, we find that apex predators were negatively affected and mesopredators benefited from the effects of stressors due to their trophic position in the St. Lawrence System, but that species sensitivity is dependent on food web structure. In conceptualising the effects of multiple stressors on food webs, we bring theory closer to practice and show that considering the intricacies of ecological communities is key to assess the net effects of stressors on species.

Generalizing game-changing species across microbial communities

Microbes form multispecies communities that play essential roles in our environment and health. Not surprisingly, there is an increasing need for understanding if certain invader species will modify a given microbial community, producing either a desired or undesired change in the observed collection of resident species. However, the complex interactions that species can establish between each other and the diverse external factors underlying their dynamics have made constructing such understanding context-specific. Here we integrate tractable theoretical systems with tractable experimental systems to find general conditions under which non-resident species can change the collection of resident communities-game-changing species. We show that non-resident colonizers are more likely to be game-changers than transients, whereas game-changers are more likely to suppress than to promote resident species. Importantly, we find general heuristic rules for game-changers under controlled environments by integrating mutual invasibility theory with in vitro experimental systems, and general heuristic rules under changing environments by integrating structuralist theory with in vivo experimental systems. Despite the strong context-dependency of microbial communities, our work shows that under an appropriate integration of tractable theoretical and experimental systems, it is possible to unveil regularities that can then be potentially extended to understand the behavior of complex natural communities.

Experimental evidence of the importance of multitrophic structure for species persistence

Ecological theory predicts that species interactions embedded in multitrophic networks shape the opportunities for species to persist. However, the lack of experimental support of this prediction has limited our understanding of how species interactions occurring within and across trophic levels simultaneously regulate the maintenance of biodiversity. Here, we integrate a mathematical approach and detailed experiments in plant-pollinator communities to demonstrate the need to jointly account for species interactions within and across trophic levels when estimating the ability of species to persist. Within the plant trophic level, we show that the persistence probability of plant species increases when introducing the effects of plant-pollinator interactions. Across trophic levels, we show that the persistence probabilities of both plants and pollinators exhibit idiosyncratic changes when experimentally manipulating the multitrophic structure. Importantly, these idiosyncratic effects are not recovered by traditional simulations. Our work provides tractable experimental and theoretical platforms upon which it is possible to investigate the multitrophic factors affecting species persistence in ecological communities.

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.

Observed Ecological Communities Are Formed by Species Combinations That Are among the Most Likely to Persist under Changing Environments

AbstractDespite the rich biodiversity found in nature, it is unclear to what extent some combinations of interacting species, while conceivable in a given place and time, may never be realized. Yet solving this problem is important for understanding the role of randomness and predictability in the assembly of ecological communities. Here we show that the specific combinations of interacting species that emerge from the ecological dynamics within regional species pools are not all equally likely to be seen; rather, they are among the most likely to persist under changing environments. First, we use niche-based competition matrices and Lotka-Volterra models to demonstrate that realized combinations of interacting species are more likely to persist under random parameter perturbations than the majority of potential combinations with the same number of species that could have been formed from the regional pool. We then corroborate our theoretical results using a 10-year observational study, recording 88 plant-herbivore communities across three different forest successional stages. By inferring and validating plant-mediated communities of competing herbivore species, we find that observed combinations of herbivores have an expected probability of species persistence higher than half of all potential combinations. Our findings open up the opportunity to establish a formal probabilistic and predictive understanding of the composition of ecological communities.

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.

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.

Structural forecasting of species persistence under changing environments

The persistence of a species in a given place not only depends on its intrinsic capacity to consume and transform resources into offspring, but also on how changing environmental conditions affect its growth rate. However, the complexity of factors has typically taken us to choose between understanding and predicting the persistence of species. To tackle this limitation, we propose a probabilistic approach rooted on the statistical concepts of ensemble theory applied to statistical mechanics and on the mathematical concepts of structural stability applied to population dynamics models - what we call structural forecasting. We show how this new approach allows us to estimate a probability of persistence for single species in local communities; to understand and interpret this probability conditional on the information we have concerning a system; and to provide out-of-sample predictions of species persistence as good as the best experimental approaches without the need of extensive amounts of data.

Species multidimensional effects explain idiosyncratic responses of communities to environmental change

Environmental change can alter species' abundances within communities consistently; for example, increasing all abundances by the same percentage, or more idiosyncratically. Here, we show how comparing effects of temperature on species grown in isolation and when grown together helps our understanding of how ecological communities more generally respond to environmental change. In particular, we find that the shape of the feasibility domain (the parameter space of carrying capacities compatible with positive species' abundances) helps to explain the composition of experimental microbial communities under changing environmental conditions. First, we introduce a measure to quantify the asymmetry of a community's feasibility domain using the column vectors of the corresponding interaction matrix. These column vectors describe the effects each species has on all other species in the community (hereafter referred to as species' multidimensional effects). We show that as the asymmetry of the feasibility domain increases the relationship between species' abundance when grown together and when grown in isolation weakens. We then show that microbial communities experiencing different temperature environments exhibit patterns consistent with this theory. Specifically, communities at warmer temperatures show relatively more asymmetry; thus, the idiosyncrasy of responses is higher compared with that in communities at cooler temperatures. These results suggest that while species' interactions are typically defined at the pairwise level, multispecies dynamics can be better understood by focusing on the effects of these interactions at the community level.

Telling ecological networks apart by their structure: An environment-dependent approach

The network architecture of an ecological community describes the structure of species interactions established in a given place and time. It has been suggested that this architecture presents unique features for each type of ecological interaction: e.g., nested and modular architectures would correspond to mutualistic and antagonistic interactions, respectively. Recently, Michalska-Smith and Allesina (2019) proposed a computational challenge to test whether it is indeed possible to differentiate ecological interactions based on network architecture. Contrary to the expectation, they found that this differentiation is practically impossible, moving the question to why it is not possible to differentiate ecological interactions based on their network architecture alone. Here, we show that this differentiation becomes possible by adding the local environmental information where the networks were sampled. We show that this can be explained by the fact that environmental conditions are a confounder of ecological interactions and network architecture. That is, the lack of association between network architecture and type of ecological interactions changes by conditioning on the local environmental conditions. Additionally, we find that environmental conditions are linked to the stability of ecological networks, but the direction of this effect depends on the type of interaction network. This suggests that the association between ecological interactions and network architectures exists, but cannot be fully understood without attention to the environmental conditions acting upon them.

It has been suggested that different types of species interactions lead to ecological networks with different architectures. For example, mutualistic and antagonistic interaction networks have been shown to have nested and modular architectures, respectively. Importantly, this differentiation can provide clues about the link between the dynamics and structures shaping ecological communities. Recently, Michalska-Smith and Allesina (2019) turned this assumption into a serious computational challenge for the scientific community. Here, we embrace this challenge. We confirm that network architecture alone is not enough to differentiate interaction networks. However, we show that network architectures can differentiate between mutualistic and antagonistic interaction networks by using information about their local environmental conditions. In other words, ignoring environmental information throws out the predictable patterns of network architectures along environmental gradients. Thus, this response is also a reminder that ecological networks may only make sense in the light of environmental information.

Towards a Probabilistic Understanding About the Context-Dependency of Species Interactions

Observational and experimental studies have shown that an interaction class between two species (be it mutualistic, competitive, antagonistic, or neutral) may switch to a different class, depending on the biotic and abiotic factors within which species are observed. This complexity arising from the evidence of context-dependencies has underscored a difficulty in establishing a systematic analysis about the extent to which species interactions are expected to switch in nature and experiments. Here, we propose an overarching theoretical framework, by integrating probabilistic and structural approaches, to establish null expectations about switches of interaction classes across environmental contexts. This integration provides a systematic platform upon which it is possible to establish new hypotheses, clear predictions, and quantifiable expectations about the context-dependency of species interactions.

On the Consequences of the Interdependence of Stabilizing and Equalizing Mechanisms

We present an overlooked but important property of modern coexistence theory (MCT), along with two key new results and their consequences. The overlooked property is that stabilizing mechanisms (increasing species' niche differences) and equalizing mechanisms (reducing species' fitness differences) have two distinct sets of meanings within MCT: one in a two-species context and another in a general multispecies context. We demonstrate that the two-species framework is not a special case of the multispecies one, and therefore these two parallel frameworks must be studied independently. Our first result is that, using the two-species framework and mechanistic consumer-resource models, stabilizing and equalizing mechanisms exhibit complex interdependence, such that changing one will simultaneously change the other. Furthermore, the nature and direction of this simultaneous change sensitively depend on model parameters. The second result states that while MCT is often seen as bridging niche and neutral modes of coexistence by building a niche-neutrality continuum, the interdependence between stabilizing and equalizing mechanisms acts to break this continuum under almost any biologically relevant circumstance. We conclude that the complex entanglement of stabilizing and equalizing terms makes their impact on coexistence difficult to understand, but by seeing them as aggregated effects (rather than underlying causes) of coexistence, we may increase our understanding of ecological dynamics.

Is the addition of higher-order interactions in ecological models increasing the understanding of ecological dynamics?

Recent work has shown that higher-order terms in population dynamics models can increase the stability, promote the diversity, and better explain the dynamics of ecological systems. While it is known that these perceived benefits come from an increasing number of alternative solutions given by the nature of multivariate polynomials, this mathematical advantage has not been formally quantified. Here, we develop a general method to quantify the mathematical advantage of adding higher-order interactions in ecological models based on the number of free-equilibrium points that can emerge in a system (i.e., equilibria that can be feasible or unfeasible as a function of model parameters). We apply this method to calculate the number of free-equilibrium points in Lotka-Volterra dynamics. While it is known that Lotka-Volterra models without higher-order interactions only have one free-equilibrium point regardless of the number of parameters, we find that by adding higher-order terms this number increases exponentially with the dimension of the system. Hence, the number of free-equilibrium points can be used to compare more fairly between ecological models. Our results suggest that while adding higher-order interactions in ecological models may be good for prediction purposes, they cannot provide additional explanatory power of ecological dynamics if model parameters are not ecologically restricted.

Non-parametric estimation of the structural stability of non-equilibrium community dynamics

Environmental factors are important drivers of community dynamics. Yet, despite extensive research, it is still extremely challenging to predict the effect of environmental changes on the dynamics of ecological communities. Equilibrium- and model-based approaches have provided a theoretical framework with which to investigate this problem systematically. However, the applicability of this framework to empirical data has been limited because equilibrium dynamics of populations within communities are seldom observed in nature and exact equations for community dynamics are rarely known. To overcome these limitations, here we develop a data-driven non-parametric framework to estimate the tolerance of non-equilibrium community dynamics to environmental perturbations (that is, their structural stability). Following our approach, we show that in non-equilibrium systems, structural stability can vary significantly across time. As a case study, we investigate the structural stability of a rocky intertidal community with dynamics at the edge of chaos. The structural stability of the community as a whole exhibited a clear seasonal pattern, despite the persistent chaotic dynamics of individual populations. Importantly, we show that this seasonal pattern of structural stability is causally driven by sea temperature. Overall, our approach provides novel opportunities for estimating the tolerance of ecological communities to environmental changes within a non-parametric framework.

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.