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.

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.

Stability criteria for the consumption and exchange of essential resources.

Models of consumer effects on a shared resource environment have helped clarify how the interplay of consumer traits and resource supply impact stable coexistence. Recent models generalize this picture to include the exchange of resources alongside resource competition. These models exemplify the fact that although consumers shape the resource environment, the outcome of consumer interactions is context-dependent: such models can have either stable or unstable equilibria, depending on the resource supply. However, these recent models focus on a simplified version of microbial metabolism where the depletion of resources always leads to consumer growth. Here, we model an arbitrarily large system of consumers governed by Liebig's law, where species require and deplete multiple resources, but each consumer's growth rate is only limited by a single one of these resources. Resources that are taken up but not incorporated into new biomass are leaked back into the environment, possibly transformed by intracellular reactions, thereby tying the mismatch between depletion and growth to cross-feeding. For this set of dynamics, we show that feasible equilibria can be either stable or unstable, again depending on the resource environment. We identify special consumption and production networks which protect the community from instability when resources are scarce. Using simulations, we demonstrate that the qualitative stability patterns derived analytically apply to a broader class of network structures and resource inflow profiles, including cases where multiple species coexist on only one externally supplied resource. Our stability criteria bear some resemblance to classic stability results for pairwise interactions, but also demonstrate how environmental context can shape coexistence patterns when resource limitation and exchange are modeled directly.

Integrating eco-evolutionary dynamics and modern coexistence theory.

Community ecology typically assumes that competitive exclusion and species coexistence are unaffected by evolution on the time scale of ecological dynamics. However, recent studies suggest that rapid evolution operating concurrently with competition may enable species coexistence. Such findings necessitate general theory that incorporates the coexistence contributions of eco-evolutionary processes in parallel with purely ecological mechanisms and provides metrics for quantifying the role of evolution in shaping competitive outcomes in both modelling and empirical contexts. To foster the development of such theory, here we extend the interpretation of the two principal metrics of modern coexistence theory-niche and competitive ability differences-to systems where competitors evolve. We define eco-evolutionary versions of these metrics by considering how invading and resident species adapt to conspecific and heterospecific competitors. We show that the eco-evolutionary niche and competitive ability differences are sums of ecological and evolutionary processes, and that they accurately predict the potential for stable coexistence in previous theoretical studies of eco-evolutionary dynamics. Finally, we show how this theory frames recent empirical assessments of rapid evolution effects on species coexistence, and how empirical work and theory on species coexistence and eco-evolutionary dynamics can be further integrated.

Competition for water and species coexistence in phenologically structured annual plant communities.

Both competition for water and phenological variation are important determinants of plant community structure, but ecologists lack a synthetic theory for how they affect coexistence outcomes. We developed an analytically tractable model of water competition for Mediterranean annual communities and demonstrated that variation in phenology alone can maintain high diversity in spatially homogenous assemblages of water-limited plants. We modelled a system where all water arrives early in the season and species vary in their ability to grow under drying conditions. As a consequence, species differ in growing season length and compete by shortening the growing season of their competitors. This model replicates and offers mechanistic explanations for patterns observed in empirical studies of how phenology influences coexistence among Mediterranean annuals. Additionally, we found that a decreasing, concave-up trade-off between growth rate and access to water can maintain high diversity under simple but realistic assumptions. High diversity is possible because: (1) later plants escape competition after their earlier season competitors have gone to seed and (2) early-season species are more than compensated for their shortened growing season by a growth rate advantage. Together, these mechanisms provide an explanation for how phenologically variable annual plant species might coexist when competing only for water.

Emergent neutrality in consumer-resource dynamics.

Neutral theory assumes all species and individuals in a community are ecologically equivalent. This controversial hypothesis has been tested across many taxonomic groups and environmental contexts, and successfully predicts species abundance distributions across multiple high-diversity communities. However, it has been critiqued for its failure to predict a broader range of community properties, particularly regarding community dynamics from generational to geological timescales. Moreover, it is unclear whether neutrality can ever be a true description of a community given the ubiquity of interspecific differences, which presumably lead to ecological inequivalences. Here we derive analytical predictions for when and why non-neutral communities of consumers and resources may present neutral-like outcomes, which we verify using numerical simulations. Our results, which span both static and dynamical community properties, demonstrate the limitations of summarizing distributions to detect non-neutrality, and provide a potential explanation for the successes of neutral theory as a description of macroecological pattern.

Effect of population abundances on the stability of large random ecosystems.

Random matrix theory successfully connects the structure of interactions of large ecological communities to their ability to respond to perturbations. One of the most debated aspects of this approach is that so far studies have neglected the role of population abundances on stability. While species abundances are well studied and empirically accessible, studies on stability have so far failed to incorporate this information. Here we tackle this question by explicitly including population abundances in a random matrix framework. We derive an analytical formula that describes the spectrum of a large community matrix for arbitrary feasible species abundance distributions. The emerging picture is remarkably simple: while population abundances affect the rate to return to equilibrium after a perturbation, the stability of large ecosystems is uniquely determined by the interaction matrix. We confirm this result by showing that the likelihood of having a feasible and unstable solution in the Lotka-Volterra system of equations decreases exponentially with the number of species for stable interaction matrices.