Does deterministic coexistence theory matter in a finite world?

Contemporary studies of species coexistence are underpinned by deterministic models that assume that competing species have continuous (i.e., noninteger) densities, live in infinitely large landscapes, and coexist over infinite time horizons. By contrast, in nature, species are composed of discrete individuals subject to demographic stochasticity and occur in habitats of finite size where extinctions occur in finite time. One consequence of these discrepancies is that metrics of species' coexistence derived from deterministic theory may be unreliable predictors of the duration of species coexistence in nature. These coexistence metrics include invasion growth rates and niche and fitness differences, which are now commonly applied in theoretical and empirical studies of species coexistence. In this study, we tested the efficacy of deterministic coexistence metrics on the duration of species coexistence in a finite world. We introduce new theoretical and computational methods to estimate coexistence times in stochastic counterparts of classic deterministic models of competition. Importantly, we parameterized this model using experimental field data for 90 pairwise combinations of 18 species of annual plants, allowing us to derive biologically informed estimates of coexistence times for a natural system. Strikingly, we found that for species expected to deterministically coexist, community sizes containing only 10 individuals had predicted coexistence times of more than 1000 years. We also found that invasion growth rates explained 60% of the variation in intrinsic coexistence times, reinforcing their general usefulness in studies of coexistence. However, only by integrating information on both invasion growth rates and species' equilibrium population sizes could most (>99%) of the variation in species coexistence times be explained. This integration was achieved with demographically uncoupled single-species models solely determined by the invasion growth rates and equilibrium population sizes. Moreover, because of a complex relationship between niche overlap/fitness differences and equilibrium population sizes, increasing niche overlap and increasing fitness differences did not always result in decreasing coexistence times, as deterministic theory would predict. Nevertheless, our results tend to support the informed use of deterministic theory for understanding the duration of species' coexistence while highlighting the need to incorporate information on species' equilibrium population sizes in addition to invasion growth rates.

An experimental test of the area-heterogeneity tradeoff

A fundamental property of ecosystems is a tradeoff between the number and size of habitats: as the number of habitats within a fixed area increases, the average area per habitat must decrease. This tradeoff is termed the "area-heterogeneity tradeoff." Theoretical models suggest that the reduction in habitat sizes under high levels of heterogeneity may cause a decline in species richness because it reduces the amount of effective area available for individual species under high levels of heterogeneity, thereby increasing the likelihood of stochastic extinctions. Here, we test this prediction using an experiment that allows us to separate the effect of the area-heterogeneity tradeoff from the total effect of habitat heterogeneity. Surprisingly, despite considerable extinctions, reduction in the amount of effective area available per species facilitated rather than reduced richness in the study communities. Our data suggest that the mechanism behind this positive effect was a decrease in the probability of deterministic competitive exclusion. We conclude that the area-heterogeneity tradeoff may have both negative and positive implications for biodiversity and that its net effect depends on the relative importance of stochastic vs. deterministic drivers of extinction in the relevant system. Our finding that the area-heterogeneity tradeoff may contribute to biodiversity adds a dimension to existing ecological theory and is highly relevant for understanding and predicting biodiversity responses to natural and anthropogenic variations in the environment.

CHAOTIC COEXISTENCE IN A RESOURCE–CONSUMER MODEL

We study the interspecies competition in a simple resource–consumer model which includes the resource supply as a dynamic variable. In the model, an organism of each species needs to consume a certain minimum rate of resource (food) to survive; a higher rate of consumption is required for reproduction. We analyze the orbit diagrams and Lyapunov exponents in order to characterize the system dynamical behavior. We observe that under a periodic food supply, the system can exhibit coexistence in the form of limit cycle oscillations. For a certain parameter range, we observe chaotic behavior emerging from successive period doublings and quasi-periodicity.

The advantage of being slow: The quasi-neutral contact process

According to the competitive exclusion principle, in a finite ecosystem, extinction occurs naturally when two or more species compete for the same resources. An important question that arises is: when coexistence is not possible, which mechanisms confer an advantage to a given species against the other(s)? In general, it is expected that the species with the higher reproductive/death ratio will win the competition, but other mechanisms, such as asymmetry in interspecific competition or unequal diffusion rates, have been found to change this scenario dramatically. In this work, we examine competitive advantage in the context of quasi-neutral population models, including stochastic models with spatial structure as well as macroscopic (mean-field) descriptions. We employ a two-species contact process in which the “biological clock” of one species is a factor of αslower than that of the other species. Our results provide new insights into how stochasticity and competition interact to determine extinction in finite spatial systems. We find that a species with a slower biological clock has an advantage if resources are limited, winning the competition against a species with a faster clock, in relatively small systems. Periodic or stochastic environmental variations also favor the slower species, even in much larger systems.

The effects of demographic stochasticity and parameter uncertainty on predicting the establishment of introduced species

Invasive species are a serious threat to biodiversity worldwide and predicting whether an introduced species will first establish and then become invasive can be useful to preserve ecosystem services. Establishment is influenced by multiple factors, such as the interactions between the introduced individuals and the resident community, and demographic and environmental stochasticity. Field observations are often incomplete or biased. This, together with an imperfect knowledge of the ecological traits of the introduced species, makes the prediction of establishment challenging. Methods that consider the combined effects of these factors on our ability to predict the establishment of an introduced species are currently lacking. We develop an inference framework to assess the combined effects of demographic stochasticity and parameter uncertainty on our ability to predict the probability of establishment following the introduction of a small number of individuals. We find that even moderate levels of demographic stochasticity influence both the probability of establishment, and, crucially, our ability to correctly predict that probability. We also find that estimation of the demographic parameters of an introduced species is fundamental to obtain precise estimates of the interaction parameters. For typical values of demographic stochasticity, the drop in our ability to predict an establishment can be 30% when having priors on the demographic parameters compared to having their accurate values. The results from our study illustrate how demographic stochasticity may bias the prediction of the probability of establishment. Our method can be applied to estimate probability of establishment of introduced species in field scenarios, where time series data and prior information on the demographic traits of the introduced species are available.

Modelling the multidimensional niche by linking functional traits to competitive performance

Linking competitive outcomes to environmental conditions is necessary for understanding species' distributions and responses to environmental change. Despite this importance, generalizable approaches for predicting competitive outcomes across abiotic gradients are lacking, driven largely by the highly complex and context-dependent nature of biotic interactions. Here, we present and empirically test a novel niche model that uses functional traits to model the niche space of organisms and predict competitive outcomes of co-occurring populations across multiple resource gradients. The model makes no assumptions about the underlying mode of competition and instead applies to those settings where relative competitive ability across environments correlates with a quantifiable performance metric. To test the model, a series of controlled microcosm experiments were conducted using genetically related strains of a widespread microbe. The model identified trait microevolution and performance differences among strains, with the predicted competitive ability of each organism mapped across a two-dimensional carbon and nitrogen resource space. Areas of coexistence and competitive dominance between strains were identified,and the predicted competitive outcomes were validated in approximately 95% of the pairings. By linking trait variation to competitive ability, our work demonstrates a generalizable approach for predicting and modelling competitive outcomes across changing environmental contexts.