Understanding long‐term coexistence of numerous competing species is a longstanding challenge in ecology. Progress requires determining which processes and species differences are most important for coexistence when multiple processes operate and species differ in many ways. Modern coexistence theory (MCT), formalised by Chesson, holds out the promise of doing that, but empirical applications remain scarce. We argue that MCT's mathematical complexity and subtlety have obscured the simplicity and power of its underlying ideas and hindered applications. We present a general computational approach that extends our previous solution for the storage effect to all of standard MCT's spatial and temporal coexistence mechanisms, and also process‐defined mechanisms amenable to direct study such as resource partitioning, indirect competition, and life history trade‐offs. The main components are a method to partition population growth rates into contributions from different mechanisms and their interactions, and numerical calculations in which some mechanisms are removed and others retained. We illustrate how our approach handles features that have not been analysed in the standard framework through several case studies: competing diatom species under fluctuating temperature, plant–soil feedbacks in grasslands, facilitation in a beach grass community, and niche differences with independent effects on recruitment, survival and growth in sagebrush steppe.

In the presence of permanent spatial heterogeneity, local dispersal, especially short-range dispersal, can facilitate coexistence by concentrating low-density species in the areas where their rates of increase are higher. We present a framework for predicting the effects of local dispersal on coexistence for arbitrary forms of dispersal and arbitrary spatial patterns of environmental variation. Using the lottery model as an example, we find that local dispersal contributes to coexistence by enhancing the effects of environmental variation on scales longer than typical dispersal distances, which can be characterized solely by the variance of the dispersal kernel. Higher moments of the dispersal kernel are not important.

Spatial coexistence depends on a variety of biological and physical processes, and the relative scales of these processes may promote or suppress coexistence. We model plant competition in a spatially varying environment to show how shifting scales of dispersal, competition, and environmental heterogeneity affect coexistence. Spatial coexistence mechanisms are partitioned into three types: the storage effect, nonlinear competitive variance, and growth-density covariance. We first describe how the strength of each of these mechanisms depends on covariances between population densities and between population densities and the environment, and we then explain how changes in the scales of dispersal, competition, and environmental heterogeneity should affect these covariances. Our quantitative approach allows us to show how changes in the scales of biological and physical processes can shift the relative importance of different classes of spatial coexistence mechanisms and gives us a more complete understanding of how environmental heterogeneity can enable coexistence. For example, we show how environmental heterogeneity can promote coexistence even when competing species have identical responses to the environment.

The storage effect has become a core concept in community ecology, explaining how environmental fluctuations can promote coexistence and maintain biodiversity. However, limitations of existing theory have hindered empirical applications: the need for detailed mathematical analysis whenever the study system requires a new model, and restricted theory for structured populations. We present a new approach that overcomes both these limitations. We show how temporal storage effect can be quantified by Monte Carlo simulations in a wide range of models for competing species. We use the lottery model and a generic integral projection model (IPM) to introduce ideas, and present two empirical applications: (1) algal species in a chemostat with variable temperature, showing that the storage effect can operate without a long-lived life stage and (2) a sagebrush steppe community IPM. Our results highlight the need for careful modelling of nonlinearities so that conclusions are not driven by unrecognised model constraints.

In a spatiotemporally variable environment, plants use seed dispersal and dormancy to reduce risk. Intuition suggests that dormancy should be able to substitute for dispersal, so that dormancy will reduce the optimal mean dispersal distance, and previous theoretical studies using temporally uncorrelated environments have found this to be true. I show that in the presence of positive temporal correlations, dormancy instead increases dispersal: dormancy and dispersal are not interchangeable risk reduction mechanisms. Dispersal has both costs (seeds landing in unfavourable habitat) and benefits (seeds being in place to exploit newly favourable habitat). I discuss how the costs and benefits balance to determine optimal dispersal and how dormancy shifts this balance, causing dispersal to increase. I also find that an interaction between spatial and temporal correlations determines whether an evolutionarily stable dispersal distance exists at all and confirm the expectation that increasing the scale of spatial correlations causes dispersal to increase.

Ecologists have traditionally neglected demographic stochasticity in describing the spread of an invading species. However, the region most critical in determining wave speed is often the leading edge, precisely the point where demographic stochasticity is most pronounced. In this paper, I analyze a common class of one-dimensional, singlespecies invasion models and find that, for very general conditions, demographic stochasticity slows biological invasions. Nonetheless, the slowing is not large enough to be noticeable in most ecological time series. I also briefly discuss the role of transient dynamics and rare, long-distance dispersal.

While there has been extensive interest in how intraspecific trait variation affects ecological processes, outcomes are highly variable even when individuals are identical: some are lucky while others are not. Trait variation is therefore only important if it adds substantially to the variability produced by luck. We ask when trait variation has a substantial effect on variability in lifetime reproductive success (LRS), using two approaches: 1) we partition the variation in LRS into contributions from luck and trait variation; 2) we ask what can be inferred about an individual's traits, and with what certainty, given their observed LRS. In theoretical stage-and size-structured models, and two empirical case studies, we find that luck usually dominates the variance of LRS. Even when individuals differ substantially in ways that affect expected LRS, unless the effects of luck are substantially reduced (e.g. low variability in reproductive lifespan or in annual fecundity), most variance in lifetime outcomes is due to luck, implying that departures from "null" models omitting trait variation will be hard to detect. Luck also obscures the relationship between realized LRS and individual traits. While trait variation may influence the fate of populations, luck often governs the lives of individuals.

The ability of environmental variation to affect species coexistence is much studied, yet environmental variation is not always important. I present an approximate calculation for the long-run growth rate of a species in the presence of spatially and temporally correlated environmental variation. I then perform a factorial numerical experiment, varying the mean seed dispersal distances, competition radii, and overwinter seed survival probabilities for two competing species for an array of variational regimes, noting the effects on their long-run growth rates. I find, first, that purely spatial variation has a greater capacity for influence than variation with a temporal component. Second, spatiotemporal variation can promote coexistence as strongly as purely temporal variation or more so, given the right species traits. Third, if the environmental variation has a spatial component, traits which enable species to become spatially segregated promote coexistence most strongly. That is, it is the possibility of spatial segregation which gives spatial variation its large potential to promote coexistence.