A user-friendly guide to using distance measures to compare time series in ecology.

Time series are a critical component of ecological analysis, used to track changes in biotic and abiotic variables. Information can be extracted from the properties of time series for tasks such as classification (e.g., assigning species to individual bird calls); clustering (e.g., clustering similar responses in population dynamics to abrupt changes in the environment or management interventions); prediction (e.g., accuracy of model predictions to original time series data); and anomaly detection (e.g., detecting possible catastrophic events from population time series). These common tasks in ecological research all rely on the notion of (dis-) similarity, which can be determined using distance measures. A plethora of distance measures have been described, predominantly in the computer and information sciences, but many have not been introduced to ecologists. Furthermore, little is known about how to select appropriate distance measures for time-series-related tasks. Therefore, many potential applications remain unexplored. Here, we describe 16 properties of distance measures that are likely to be of importance to a variety of ecological questions involving time series. We then test 42 distance measures for each property and use the results to develop an objective method to select appropriate distance measures for any task and ecological dataset. We demonstrate our selection method by applying it to a set of real-world data on breeding bird populations in the UK and discuss other potential applications for distance measures, along with associated technical issues common in ecology. Our real-world population trends exhibit a common challenge for time series comparisons: a high level of stochasticity. We demonstrate two different ways of overcoming this challenge, first by selecting distance measures with properties that make them well suited to comparing noisy time series and second by applying a smoothing algorithm before selecting appropriate distance measures. In both cases, the distance measures chosen through our selection method are not only fit-for-purpose but are consistent in their rankings of the population trends. The results of our study should lead to an improved understanding of, and greater scope for, the use of distance measures for comparing ecological time series and help us answer new ecological questions.

Quantifying reliability and data deficiency in global vertebrate population trends using the Living Planet Index.

Global biodiversity is facing a crisis, which must be solved through effective policies and on-the-ground conservation. But governments, NGOs, and scientists need reliable indicators to guide research, conservation actions, and policy decisions. Developing reliable indicators is challenging because the data underlying those tools is incomplete and biased. For example, the Living Planet Index tracks the changing status of global vertebrate biodiversity, but taxonomic, geographic and temporal gaps and biases are present in the aggregated data used to calculate trends. However, without a basis for real-world comparison, there is no way to directly assess an indicator's accuracy or reliability. Instead, a modelling approach can be used. We developed a model of trend reliability, using simulated datasets as stand-ins for the "real world", degraded samples as stand-ins for indicator datasets (e.g., the Living Planet Database), and a distance measure to quantify reliability by comparing partially sampled to fully sampled trends. The model revealed that the proportion of species represented in the database is not always indicative of trend reliability. Important factors are the number and length of time series, as well as their mean growth rates and variance in their growth rates, both within and between time series. We found that many trends in the Living Planet Index need more data to be considered reliable, particularly trends across the global south. In general, bird trends are the most reliable, while reptile and amphibian trends are most in need of additional data. We simulated three different solutions for reducing data deficiency, and found that collating existing data (where available) is the most efficient way to improve trend reliability, whereas revisiting previously studied populations is a quick and efficient way to improve trend reliability until new long-term studies can be completed and made available.

Are experiment sample sizes adequate to detect biologically important interactions between multiple stressors?

As most ecosystems are being challenged by multiple, co-occurring stressors, an important challenge is to understand and predict how stressors interact to affect biological responses. A popular approach is to design factorial experiments that measure biological responses to pairs of stressors and compare the observed response to a null model expectation. Unfortunately, we believe experiment sample sizes are inadequate to detect most non-null stressor interaction responses, greatly hindering progress. Using both real and simulated data, we show sample sizes typical of many experiments (<6) can (i) only detect very large deviations from the additive null model, implying many important non-null stressor-pair interactions are being missed, and (ii) potentially lead to mostly statistical outliers being reported. Computer code that simulates data under either additive or multiplicative null models is provided to estimate statistical power for user-defined responses and sample sizes, and we recommend this is used to aid experimental design and interpretation of results. We suspect that most experiments may require 20 or more replicates per treatment to have adequate power to detect nonadditive. However, estimates of power need to be made while considering the smallest interaction of interest, i.e., the lower limit for a biologically important interaction, which is likely to be system-specific, meaning a general guide is unavailable. We discuss ways in which the smallest interaction of interest can be chosen, and how sample sizes can be increased. Our main analyses relate to the additive null model, but we show similar problems occur for the multiplicative null model, and we encourage similar investigations into the statistical power of other null models and inference methods. Without knowledge of the detection abilities of the statistical tools at hand or the definition of the smallest meaningful interaction, we will undoubtedly continue to miss important ecosystem stressor interactions.

Priority effects and the macroevolutionary dynamics of biodiversity.

Priority effects can play a fundamental role in the assembly of ecological communities, but how they shape the dynamics of biodiversity over macroevolutionary timescales remains unclear. Here we develop and analyse a metacommunity model combining local priority effects with niche evolution, speciation and extinction. We show that by promoting the persistence of rare species, local priority effects cause the evolution of higher metacommunity diversity as well as major disparities in richness among evolutionary lineages. However, we also show how classic macroevolutionary patterns of niche incumbency-whereby rates of regional diversification and invasion slow down as ecological niches are filled-do not depend on local priority effects, arising even when invading species continuously displace residents. Together, these results clarify the connection between local priority effects and the filling of ecological niche space, and reveal how the impact of species arrival order on competition fundamentally shapes the generation and maintenance of biodiversity.

Classifying ecosystem stressor interactions: Theory highlights the data limitations of the additive null model and the difficulty in revealing ecological surprises.

Understanding how multiple co-occurring environmental stressors combine to affect biodiversity and ecosystem services is an on-going grand challenge for ecology. Currently, progress has been made through accumulating large numbers of smaller-scale empirical studies that are then investigated by meta-analyses to detect general patterns. There is particular interest in detecting, understanding and predicting 'ecological surprises' where stressors interact in a non-additive (e.g. antagonistic or synergistic) manner, but so far few general results have emerged. However, the ability of the statistical tools to recover non-additive interactions in the face of data uncertainty is unstudied, so crucially, we do not know how well the empirical results reflect the true stressor interactions. Here, we investigate the performance of the commonly implemented additive null model. A meta-analysis of a large (545 interactions) empirical dataset for the effects of pairs of stressors on freshwater communities reveals additive interactions dominate individual studies, whereas pooling the data leads to an antagonistic summary interaction class. However, analyses of simulated data from food chain models, where the underlying interactions are known, suggest both sets of results may be due to observation error within the data. Specifically, we show that the additive null model is highly sensitive to observation error, with non-additive interactions being reliably detected at only unrealistically low levels of data uncertainty. Similarly, plausible levels of observation error lead to meta-analyses reporting antagonistic summary interaction classifications even when synergies co-dominate. Therefore, while our empirical results broadly agree with those of previous freshwater meta-analyses, we conclude these patterns may be driven by statistical sampling rather than any ecological mechanisms. Further investigation of candidate null models used to define stressor-pair interactions is essential, and once any artefacts are accounted for, the so-called 'ecological surprises' may be more frequent than was previously assumed.

Relative importance of evolutionary dynamics depends on the composition of microbial predator-prey community.

Community dynamics are often studied in subsets of pairwise interactions. Scaling pairwise interactions back to the community level is, however, problematic because one given interaction might not reflect ecological and evolutionary outcomes of other functionally similar species interactions or capture the emergent eco-evolutionary dynamics arising only in more complex communities. Here we studied this experimentally by exposing Pseudomonas fluorescens SBW25 prey bacterium to four different protist predators (Tetrahymena pyriformis, Tetrahymena vorax, Chilomonas paramecium and Acanthamoeba polyphaga) in all possible single-predator, two-predator and four-predator communities for hundreds of prey generations covering both ecological and evolutionary timescales. We found that only T. pyriformis selected for prey defence in single-predator communities. Although T. pyriformis selection was constrained in the presence of the intraguild predator, T. vorax, T. pyriformis selection led to evolution of specialised prey defence strategies in the presence of C. paramecium or A. polyphaga. At the ecological level, adapted prey populations were phenotypically more diverse, less stable and less productive compared with non-adapted prey populations. These results suggest that predator community composition affects the relative importance of ecological and evolutionary processes and can crucially determine when rapid evolution has the potential to change ecological properties of microbial communities.

The memory of spatial patterns: changes in local abundance and aggregation in a tropical forest.

The current spatial pattern of a population is the result of previous individual birth, death, and dispersal events. We present a simple model followed by a comparative analysis for a species-rich plant community to show how the current spatial aggregation of a population may hold information about recent population dynamics. Previous research has shown how locally restricted seed dispersal often leads to stronger aggregation in less abundant populations than it does in more abundant populations. In contrast, little is known about how changes in the local abundance of a species may affect the spatial distribution of individuals. If the level of aggregation within a species depends to some extent on the abundance of the species, then changes in abundance should lead to subsequent changes in aggregation. However, an overall change of spatial pattern relies on many individual birth and death events, and a surplus of deaths or births may have short-term effects on aggregation that are opposite to the long-term change predicted by the change in abundance. The change in aggregation may therefore lag behind the change in abundance, and consequently, the current aggregation may hold information about recent population dynamics. Using an individual-based simulation model with local dispersal and density-dependent competition, we show that, on average, recently growing populations should be more aggregated than shrinking populations of the same current local abundance. We tested this hypothesis using spatial data on individuals from a long-term tropical rain forest plot, and find support for this relationship in canopy trees, but not in understory and shrub species. On this basis we argue that current spatial aggregation is an important characteristic that contains information on recent changes in local abundance, and may be applied to taxonomic groups where dispersal is limited and within-species aggregation is observed.

Quantifying the likelihood of co-existence for communities with asymmetric competition.

Trade-offs in performance of different ecological functions within a species are commonly offered as an explanation for co-existence in natural communities. Single trade-offs between competitive ability and other life history traits have been shown to support a large number of species, as a result of strong competitive asymmetry. We consider a single competition-fecundity trade-off in a homogeneous environment, and examine the effect of the form of asymmetry on the likelihood of species co-existing. We find conditions that allow co-existence of two species for a general competition function, and show that (1) two species can only co-exist if the competition function is sufficiently steep when the species are similar; (2) when competition is determined by a linear function, no more than two species can co-exist; (3) when the competition between two individuals is determined by a discontinuous step function, this single trade-off can support an arbitrarily large number of species. Further, we show analytically that as the degree of asymmetry in competition increases, the probability of a given number of species co-existing also increases, but note that even in the most favourable conditions, large numbers of species co-existing along a single trade-off is highly unlikely. On this basis, we suggest it is unlikely that single trade-offs are able to support high levels of bio-diversity without interacting other processes.

Intense or spatially heterogeneous predation can select against prey dispersal.

Dispersal theory generally predicts kin competition, inbreeding, and temporal variation in habitat quality should select for dispersal, whereas spatial variation in habitat quality should select against dispersal. The effect of predation on the evolution of dispersal is currently not well-known: because predation can be variable in both space and time, it is not clear whether or when predation will promote dispersal within prey. Moreover, the evolution of prey dispersal affects strongly the encounter rate of predator and prey individuals, which greatly determines the ecological dynamics, and in turn changes the selection pressures for prey dispersal, in an eco-evolutionary feedback loop. When taken all together the effect of predation on prey dispersal is rather difficult to predict. We analyze a spatially explicit, individual-based predator-prey model and its mathematical approximation to investigate the evolution of prey dispersal. Competition and predation depend on local, rather than landscape-scale densities, and the spatial pattern of predation corresponds well to that of predators using restricted home ranges (e.g. central-place foragers). Analyses show the balance between the level of competition and predation pressure an individual is expected to experience determines whether prey should disperse or stay close to their parents and siblings, and more predation selects for less prey dispersal. Predators with smaller home ranges also select for less prey dispersal; more prey dispersal is favoured if predators have large home ranges, are very mobile, and/or are evenly distributed across the landscape.

When does local spatial structure hinder competitive coexistence and reverse competitive hierarchies?

Classical theory states that if conspecifics have a greater competitive effect on individuals than heterospecifics then coexistence should occur, and ecologists have spent much effort exploring ways to generate coexistence when this condition is not met. One process that has received particular attention in the last two decades is the effect of within-species aggregation and between-species segregation caused by limited dispersal. A number of theories have emerged as to how this common spatial pattern may help maintain biodiversity, and the general conclusion that has emerged is that spatial structure should almost always help competitors to coexist. But does spatial structure really always aid biodiversity? An individual-based model based on a spatial extension to the Lotka-Volterra competition equations and its mathematical approximation are presented to determine how local spatial structure may affect communities in which there is strong niche differentiation. Two main results emerge from analyses of the models. First, intraspecific competition being greater than interspecific competition coexistence may no longer be sufficient to generate coexistence when spatial structure is strong; and the species with the highest intraspecific competition coefficient is likely to be excluded. Second, dominance hierarchies may be reversed so that a competitor may become the subordinate species when dispersal and competitive interactions occur over short spatial scales. Both results emerge because, even though a species may be globally rare, intense clumping means most interactions occur between conspecifics, and if this is very intense it may be sufficient to stop a species from invading. However, long-range dispersal may ameliorate these effects by reducing the frequency of conspecific interactions, and this is especially important when a species is rare since it is very likely to land in an area dominated by heterospecifics. These results are most relevant to sessile organisms that produce relatively few viable offspring that survive to adulthood and that have relatively weak dispersal. The conclusion is that within-species aggregation may hinder coexistence when the toughest competitor an individual is likely to face is a member of its own species.

Testing spatial theories of plant coexistence: no consistent differences in intra- and interspecific interaction distances.

Plants stand still and interact with their immediate neighbors. Theory has shown that the distances over which these interactions occur may have important consequences for population and community dynamics. In particular, if intraspecific competition occurs over longer distances than interspecific competition (heteromyopia), coexistence can be promoted. We examined how intraspecific and interspecific competition scales with neighbor distance in a target-neighbor greenhouse competition experiment. Individuals from co-occurring forbs from calcareous grasslands were grown in isolation and with single conspecific or heterospecific neighbors at distances of 5, 10, or 15 cm (Plantago lanceolata vs. Plantago media and Hieracium pilosella vs. Prunella grandiflora). Neighbor effects were strong and declined with distance. Interaction distances varied greatly within and between species, but we found no evidence for heteromyopia. Instead, neighbor identity effects were mostly explained by relative size differences between target and neighbor. We found a complex interaction between final neighbor size and identity such that neighbor identity may become important only as the neighbor becomes very large compared with the target individual. Our results suggest that species-specific size differences between neighboring individuals determine both the strength of competitive interactions and the distance over which these interactions occur.

Local spatial structure and predator-prey dynamics: counterintuitive effects of prey enrichment.

The Lotka-Volterra predator-prey model with prey density dependence shows the final prey density to be independent of its vital rates. This result assumes the community to be well mixed so that encounters between predators and prey occur as a product of the landscape densities, yet empirical evidence suggests that over small spatial scales this may not be the normal pattern. Starting from an individual-based model with neighborhood interactions and movements, a deterministic approximation is derived, and the effect of local spatial structure on equilibrium densities is investigated. Incorporating local movements and local interactions has important consequences for the community dynamics. Now the final prey density is very much dependent on its birth, death, and movement rates and in ways that seem counterintuitive. Increasing prey fecundity or mobility and decreasing the coefficient of competition can all lead to decreases in the final density of prey if the predator is also relatively immobile. However, analysis of the deterministic approximation makes the mechanism for these results clear; each of these changes subtly alters the emergent spatial structure, leading to an increase in the predator-prey spatial covariance at short distances and hence to a higher predation pressure on the prey.

On moment closures for population dynamics in continuous space.

A first-order moment closure, the mean-field assumption that organisms encounter one another in proportion to their spatial average densities, lies at the heart of much theoretical ecology. This assumption ignores all spatial information and, at the very least, needs to be replaced by a second-order closure to gain understanding of ecological dynamics in spatially structured populations. We describe a number of conditions that a second-order closure should satisfy and use these conditions to evaluate some closures currently available in the literature. Two conditions are particularly helpful in discriminating among the alternatives: that the closure should be positive, and that the dynamics should be unaltered when identical individuals are given different labels. On this basis, a class of closures we refer to as 'power-2' turns out to provide a good compromise between positivity and dynamical invariance under relabelling.