Coexistence is simultaneously one of the most fundamental concepts of ecology, and one of the most difficult to define. A particular challenge is that, despite a well-developed body of research, several different schools of thought have developed over the past century, leading to multiple independent, and largely isolated, branches of literature with distinct methodologies. Here, we provide a broad overview of the most common concepts and metrics currently used to detect and characterise ecological coexistence. We first introduce four classes of behaviour, which jointly describe the ways in which community dynamics can unfold: (i) the existence of a feasible steady state (or invariant set), i.e. where all coexisting species retain positive abundances in the long-term in the absence of interference by external forces; (ii) the existence of a local attractor that draws the community towards a feasible steady state from within a restricted set of starting conditions; (iii) the existence of a global attractor that draws the community towards feasible steady states from any non-zero starting condition; and (o) a null transient state, where species abundances vary over time irrespective of steady states and attractors. Next, we explain how these classes of behaviour relate to commonly used metrics for identifying and characterising coexistence, including analyses of parameter sensitivity, asymptotic return rates, invasion growth rates, and time to extinction. We then discuss the scope and limitations of each of these behavioural classes and corresponding metrics, with a particular focus on applications in empirical systems. Finally, we provide a potential workflow for matching empirical questions to theoretical tools, and present a brief prospectus looking forward to opportunities for advancing and integrating research on coexistence.

Transient 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.

Urban vegetable gardens provide an opportunity to recycle nutrients from food waste back into the human food system through the application of compost. However, a reliance on compost for soil fertility can lead to excess phosphorus (P) inputs that can build up in garden soil and potentially be exported via leachate or runoff. We report the results of a 7-year experiment in a campus research garden in which replicated raised-bed garden plots received manure-based compost or municipal compost that was applied at a higher rate targeted to meet crop nitrogen demand or a lower rate targeted to meet crop P demand. Control plots received either no soil inputs or targeted synthetic fertilizer. Higher input treatments for both types of composts showed steadily increasing concentrations of soil plant-available P, with a corresponding increase in leachate phosphate concentration. For both higher input compost treatments, approximately 30% of P added as compost was recovered in harvested crops over the 7-year period, compared to >88% in the lower input compost treatments. In both high- and low-input manure compost treatments, export of P as leachate accounted for approximately 10% of total P input, compared to 4% for the municipal compost. Over the 7-year study period, P exported as leachate ranged from 0.8 g P/m2 in the no-input treatments to 6.5 g P/m2 in the higher input manure compost treatments. These results show that tradeoffs are not inevitable as targeted compost applications can lead to high yield and low leachate export.

Individual-level trait diversity has been identified as an essential component of trait diversity (TD), influencing community assembly and structure. Traditionally, one employs trait diversity indices to measure facets of individual-level trait diversity (divergence, richness and evenness). However, the application of species-level trait diversity indices to individual-level traits data and their implications have not been adequately studied. Thus, we examined the possible challenges of using four commonly used multi-trait TD indices: Raos quadratic entropy (Rao), functional dispersion (FDis), functional evenness (FEve) and functional richness (FRic); two indices primarily developed to measure individual-level trait diversity: trait evenness distribution (TED-for evenness) and trait onion peeling (TOP-for richnness); and a modified version of TED (TEDM-for evenness). Additionally, we considered an index that integrates both evenness and richness by generalizing ordinary Hill indices for traits (coined HIT). We measured individual-level trait diversity with these indices using simulated traits data and experimental data from a growth experiment with cyanobacteria. Comparing the observed trends from the indices with the expected trends, we observed that only the trait divergence indices (FDis and Rao) produced the expected trends in the simulation scenarios and experimental data. TED and TEDM are not robust against the number of individuals used, and FEve is not sensitive to some changes in the location of individuals in the trait space. Also, TOP proved to be a discontinuous function dependent on the number of individuals, and FRic did not produce the anticipated trend when changes in the trait space did not affect the edges of the trait space. HIT did produce the anticipated changes, but it was only reliable when many individuals were sampled. In summary, applying these individual-level trait diversity indices to quantify anything except trait divergence may lead to misinterpretation of the original situation of trait distribution in the trait space if their specific properties are not adequately considered.

We analyze a frequent but undertheorized form of structural injustice, one that arises due to the difficulty of reaching numerically equitable representation of underrepresented subgroups within a larger group. This form of structural injustice is significant because it could occur even if it were possible to completely eliminate bias and overt discrimination from hiring and recruitment practices. The conceptual toolkit we develop can be used to analyze such situations and propose remedies. Specifically, based on a simple mathematical model, we offer a new argument in favour of quotas, explore implications for policy-making, and consider the wider philosophical significance of the problem. We show that in order to reach more equitable representations, quota-based recruitment may often be practically unavoidable. Assuming that members of groups in statistical minority are more likely to quit due to their marginalization, their proportions can stabilize at a low level, preventing a shift towards more equal representation and conserving the minority status of the subgroup. We show that this argument has important implications for addressing, preventing, and remediating the structural injustice of unfair representation.

Restoration ecology commonly seeks to re-establish species of interest in degraded habitats. Despite a rich understanding of how succession influences re-establishment, there are several outstanding questions that remain unaddressed: are short-term abundances sufficient to determine long-term re-establishment success, and what factors contribute to unpredictable restorations outcomes? In other words, when restoration fails, is it because the restored habitat is substandard, because of strong competition with invasive species, or alternatively due to changing environmental conditions that would equally impact established populations? Here, we re-purpose tools developed from modern coexistence theory to address these questions, and apply them to an effort to restore the endangered Contra Costa goldfields (Lasthenia conjugens) in constructed ("restored") California vernal pools. Using 16 years of data, we construct a population model of L. conjugens, a species of conservation concern due primarily to habitat loss and invasion of exotic grasses. We show that initial, short-term appearances of restoration success from population abundances is misleading, as year-to-year fluctuations cause long-term population growth rates to fall below zero. The failure of constructed pools is driven by lower maximum growth rates compared with reference ("natural") pools, coupled with a stronger negative sensitivity to annual fluctuations in abiotic conditions that yield decreased maximum growth rates. Nonetheless, our modeling shows that fluctuations in competition (mainly with exotic grasses) benefit L. conjugens through periods of competitive release, especially in constructed pools of intermediate pool depth. We therefore show how reductions in invasives and seed addition in pools of particular depths could change the outcome of restoration for L. conjugens. By applying a largely theoretical framework to the urgent goal of ecological restoration, our study provides a blueprint for predicting restoration success, and identifies future actions to reverse species loss.

Disturbance has long been recognized as a critical driver of species diversity in community ecology. Recently, it has been found that the well-known intermediate disturbance hypothesis, which predicts a unimodal diversity-disturbance relationship (DDR), fails to describe numerous experimental observations, as empirical DDRs are diverse. Consequently, the precise form of the DDR remains a topic of debate. Here we develop a simple yet comprehensive metacommunity framework that can account for complex competition patterns. Using both numerical simulations and analytical arguments, we show that strongly multimodal DDRs arise naturally, and this multimodality is quite robust to changing parameters or relaxing the assumption of a strict competitive hierarchy. Having multimodality as a robust property of DDRs in competition models suggests that much of the noise observed in empirical DDRs could be a critical signature of the underlying competitive dynamics.

Modelling species interactions in diverse communities traditionally requires a prohibitively large number of species-interaction coefficients, especially when considering environmental dependence of parameters. We implemented Bayesian variable selection via sparsity-inducing priors on non-linear species abundance models to determine which species interactions should be retained and which can be represented as an average heterospecific interaction term, reducing the number of model parameters. We evaluated model performance using simulated communities, computing out-of-sample predictive accuracy and parameter recovery across different input sample sizes. We applied our method to a diverse empirical community, allowing us to disentangle the direct role of environmental gradients on species intrinsic growth rates from indirect effects via competitive interactions. We also identified a few neighbouring species from the diverse community that had non-generic interactions with our focal species. This sparse modelling approach facilitates exploration of species interactions in diverse communities while maintaining a manageable number of parameters.

Species abundance distributions must reflect the dynamic processes involved in community assembly, but whether and when specific processes lead to distinguishable signals is not well understood. Biodiversity and species abundances may be shaped by a variety of influences, but particular attention has been paid to competition, which can involve neutral dynamics, where competitor abundances are governed only by demographic stochasticity and immigration, and dynamics driven by trait differences that enable stable coexistence through the formation of niches. Key recent studies of the species abundance patterns of communities with niches employ simple models with pre-imposed niche structure. These studies suggest that species abundance distributions are insensitive to the relative contributions of niche and neutral processes, especially when diversity is much higher than the number of niches. Here we analyze results from a stochastic population model with competition driven by trait differences. With this model, niche structure emerges as clumps of species that persist along the trait axis, and leads to more substantial differences from neutral species abundance distributions than have been previously shown. We show that heterogeneity in between-niche interaction strength (i.e., in the strength of competition between species in different niches) plays the dominant role in shaping the species abundances along the trait axis, acting as a biotic filter favoring species at the centers of niches. Furthermore, we show that heterogeneity in within-niche interactions (i.e., in the competition between species in the same niche) counteracts the influence of heterogeneity in between-niche interactions on the SAD to some degree. Our results suggest that competitive interactions that produce niches can also influence the shapes of SADs.