Predicting the process of extinction in experimental microcosms and accounting for interspecific interactions in single‐species time series

Predicting population extinction risk is a fundamental application of ecological theory to the practice of conservation biology. Here, we compared the prediction performance of a wide array of stochastic, population dynamics models against direct observations of the extinction process from an extensive experimental data set. By varying a series of biological and statistical assumptions in the proposed models, we were able to identify the assumptions that affected predictions about population extinction. We also show how certain autocorrelation structures can emerge due to interspecific interactions, and that accounting for the stochastic effect of these interactions can improve predictions of the extinction process. We conclude that it is possible to account for the stochastic effects of community interactions on extinction when using single‐species time series.     

Value of long‐term ecological studies

Long‐term ecological studies are critical for providing key insights in ecology, environmental change, natural resource management and biodiversity conservation. In this paper, we briefly discuss five key values of such studies. These are: (1) quantifying ecological responses to drivers of ecosystem change; (2) understanding complex ecosystem processes that occur over prolonged periods; (3) providing core ecological data that may be used to develop theoretical ecological models and to parameterize and validate simulation models; (4) acting as platforms for collaborative studies, thus promoting multidisciplinary research; and (5) providing data and understanding at scales relevant to management, and hence critically supporting evidence‐based policy, decision making and the management of ecosystems. We suggest that the ecological research community needs to put higher priority on communicating the benefits of long‐term ecological studies to resource managers, policy makers and the general public. Long‐term research will be especially important for tackling large‐scale emerging problems confronting humanity such as resource management for a rapidly increasing human population, mass species extinction, and climate change detection, mitigation and adaptation. While some ecologically relevant, long‐term data sets are now becoming more generally available, these are exceptions. This deficiency occurs because ecological studies can be difficult to maintain for long periods as they exceed the length of government administrations and funding cycles. We argue that the ecological research community will need to coordinate ongoing efforts in an open and collaborative way, to ensure that discoverable long‐term ecological studies do not become a long‐term deficiency. It is important to maintain publishing outlets for empirical field‐based ecology, while simultaneously developing new systems of recognition that reward ecologists for the use and collaborative sharing of their long‐term data sets. Funding schemes must be re‐crafted to emphasize collaborative partnerships between field‐based ecologists, theoreticians and modellers, and to provide financial support that is committed over commensurate time frames.     

Stochastic ecological network occupancy (SENO) models: a new tool for modeling ecological networks across spatial scales

Stochastic ecological network occupancy (SENO) models predict the probability that species will occur in a sample of an ecological network. In this review, we introduce SENO models as a means to fill a gap in the theoretical toolkit of ecologists. As input, SENO models use a topological interaction network and rates of colonization and extinction (including consumer effects) for each species. A SENO model then simulates the ecological network over time, resulting in a series of sub-networks that can be used to identify commonly encountered community modules. The proportion of time a species is present in a patch gives its expected probability of occurrence, whose sum across species gives expected species richness. To illustrate their utility, we provide simple examples of how SENO models can be used to investigate how topological complexity, species interactions, species traits, and spatial scale affect communities in space and time. They can categorize species as biodiversity facilitators, contributors, or inhibitors, making this approach promising for ecosystem-based management of invasive, threatened, or exploited species.     

How is extinction risk related to population-size variability over time? A family of models for species with repeated extinction and immigration

It is well known that for an isolated population, the probability of extinction is positively related to population size variation: more variation is associated with more extinction. What, then, is the relation of extinction to population size variation for a population embedded in a metapopulation and subjected to repeated extinction and recolonization? In this case, the extinction risk can be measured by the extinction rate, the frequency at which local extinction occurs. Using several population dynamics models with immigration, we find, in general, a negative correlation between extinction and variation. More precisely, with increasing length of the time series, an initially negative regression coefficient first becomes more negative, then becomes less negative, and eventually attains positive values before decreasing again to 0. This pattern holds under substantial variation in values of parameters representing species and environmental properties. It is also rather robust to census interval length and the fraction of missed individuals but fails to hold for high thresholds (population size values below which extinction is deemed to occur) when quasi extinction rather than true extinction is represented. The few departures from the initial negative correlation correspond to populations at risk: low growth rate or frequent catastrophes.     

Distinguishing intrinsic limit cycles from forced oscillations in ecological time series

Ecological cycles are ubiquitous in nature and have triggered ecologists’ interests for decades. Deciding whether a cyclic ecological variable, such as population density, is part of an intrinsically emerging limit cycle or simply driven by a varying environment is still an unresolved issue, particularly when the only available information is in the form of a recorded time series. We investigate the possibility of discerning intrinsic limit cycles from oscillations forced by a cyclic environment based on a single time series. We argue that such a distinction is possible because of the fundamentally different effects that perturbations have on the focal system in these two cases. Using a set of generic mathematical models, we show that random perturbations leave characteristic signatures on the power spectrum and autocovariance that differ between limit cycles and forced oscillations. We quantify these differences through two summary variables and demonstrate their predictive power using numerical simulations. Our work demonstrates that random perturbations of ecological cycles can give valuable insight into the underlying deterministic dynamics.     

From local interactions to population dynamics in site-based models of ecology

A central problem in ecology is relating the interactions of individuals-described in terms of competition, predation, interference, etc.-to the dynamics of the populations of these individuals-in terms of change in numbers of individuals over time. Here, we address this problem for a class of site-based ecological models, where local interactions between individuals take place at a finite number of discrete resource sites over non-overlapping generations and, between generations, individuals move randomly between sites over the entire system. Such site-based models have previously been applied to a wide range of ecological systems: from those involving contest or scramble competition for resources to host-parasite interactions and meta-populations. We show how the population dynamics of site-based models can be accurately approximated by and understood through deterministic and stochastic difference equations. Conversely, we use the inverse of this approximation to show what implicit assumptions are made about individual interactions by modelling of population dynamics in terms of difference equations. To this end, we prove a useful and general theorem: that any model in our class of site-based models has a corresponding stochastic difference equation population model, by which it can be approximated. This theorem allows us to calculate long-term population dynamics, evolutionary stable strategies and, by extending our theory to account for large deviations, extinction probabilities for a wide range of site-based systems. Our methodology is then illustrated to various examples of between species competition, predator-prey interactions and co-operation.     

Statistical indicators and state–space population models predict extinction in a population of bobwhite quail

Early warning systems of extinction thresholds have been developed for and tested in microcosm experiments, but have not been applied to populations of wild animals. We used state–space population models and a statistical indicator to detect a transcritical bifurcation extinction threshold in a population of bobwhite quail (Colinus virginianus) located in an agricultural region experiencing habitat deterioration and loss. The extinction threshold was detectible using two independent data sets. We compared predictions from state–space population models to predictions from a statistical indicator and found that predictions were corroborated. Using state–space population models, we estimated that our study population crossed the extinction threshold in 2010 (2002–2036; 95 % confidence intervals [CI]) using the whistle count (WC) data set and in 2008 (1999–2064; 95 % CI) using the Breeding Bird Survey (BBS) data. With the statistical indicator, we estimated that the extinction threshold will be crossed in 2018 (2004–2031; 95 % CI) using the WC data and will be crossed in 2012 (2006–2018; 95 % CI) using the BBS data. We expect extinction in our study population soon after crossing the extinction threshold, but the time to extinction and potential reversibility of the threshold are unknown. Our results suggest that neither small nor decreasing population size will warn of the transcritical bifurcation extinction threshold. We suggest that managers of wildlife populations in regions experiencing land use change should try to predict extinction thresholds and make management decisions to ensure the persistence of the species.     

On the mechanistic underpinning of discrete-time population models with complex dynamics

We present a mechanistic underpinning for various discrete-time population models that can produce limit cycles and chaotic dynamics. Specific examples include the discrete-time logistic model and the Hassell model, which for a long time eluded convincing mechanistic interpretations, and also the Ricker- and Beverton-Holt models. We first formulate a continuous-time resource consumption model for the dynamics within a year, and from that we derive a discrete-time model for the between-year dynamics. Without influx of resources from the outside into the system, the resulting between-year dynamics is always overcompensating and hence may produce complex dynamics as well as extinction in finite time. We recover a connection between various standard types of continuous-time models for the resource dynamics within a year on the one hand and various standard types of discrete-time models for the population dynamics between years on the other. The model readily generalizes to several resource and consumer species as well as to more than two trophic levels for the within-year dynamics.     

Occupancy versus colonization–extinction models for projecting population trends at different spatial scales

Understanding spatiotemporal population trends and their drivers is a key aim in population ecology. We further need to be able to predict how the dynamics and sizes of populations are affected in the long term by changing landscapes and climate. However, predictions of future population trends are sensitive to a range of modeling assumptions. Deadwood‐dependent fungi are an excellent system for testing the performance of different predictive models of sessile species as these species have different rarity and spatial population dynamics, the populations are structured at different spatial scales, and they utilize distinct substrates. We tested how the projected large‐scale occupancies of species with differing landscape‐scale occupancies are affected over the coming century by different modeling assumptions. We compared projections based on occupancy models against colonization–extinction models, conducting the modeling at alternative spatial scales and using fine‐ or coarse‐resolution deadwood data. We also tested effects of key explanatory variables on species occurrence and colonization–extinction dynamics. The hierarchical Bayesian models applied were fitted to an extensive repeated survey of deadwood and fungi at 174 patches. We projected higher occurrence probabilities and more positive trends using the occupancy models compared to the colonization–extinction models, with greater difference for the species with lower occupancy, colonization rate, and colonization:extinction ratio than for the species with higher estimates of these statistics. The magnitude of future increase in occupancy depended strongly on the spatial modeling scale and resource resolution. We encourage using colonization–extinction models over occupancy models, modeling the process at the finest resource‐unit resolution that is utilizable by the species, and conducting projections for the same spatial scale and resource resolution at which the model fitting is conducted. Further, the models applied should include key variables driving the metapopulation dynamics, such as the availability of suitable resource units, habitat quality, and spatial connectivity.     

Signal extraction from long-term ecological data using Bayesian and non-Bayesian state-space models

Emerging ecological time series from long-term ecological studies and remote sensing provide excellent opportunities for ecologists to study the dynamic patterns and governing processes of ecological systems. However, signal extraction from long-term time series often requires system learning (e.g., estimation of true system state) to process the large amount of information, to reconstruct system state, to account for measurement error, and to handle missing data. State-space models (SSMs) are a natural choice for these tasks and thus have received increasing attention in ecological and environmental studies. Data-based learning using SSMs that connect ecological processes to the measurement of system state becomes a useful technique in the ecological informatics toolkit. The present study illustrates the use of the Kalman filter (KF), an estimator of SSMs, with case studies of population dynamics. The examples of the SSM applications include the reconstruction of system state using the KF method and Markov chain Monte Carlo methods, estimation of measurement-error variances in the estimates of animal population abundance using basic structural models (BSMs), and estimation of missing values using the KF and Kalman smoother. Estimation of measurement-error variances by BSMs does not require knowledge of the functional form that generates the time series data. Instead, BSMs approximate the trajectory or deterministic skeleton of a system dynamics in a semi-parametric fashion, and provide a robust estimator of measurement-error variances. The present study also compares Bayesian SSMs with non-Bayesian SSMs. The joint use of the KF method or its extensions and Markov chain Monte Carlo (MCMC) methods is a promising approach to the parameter estimation of SSMs.     

Circumventing structural uncertainty: A Bayesian perspective on nonlinear forecasting for ecology

As a consequence of the complexity of ecosystems and context-dependence of species interactions, structural uncertainty is pervasive in ecological modeling. This is particularly problematic when ecological models are used to make conservation and management plans whose outcomes may depend strongly on model formulation. Nonlinear time series approaches allow us to circumvent this issue by using the observed dynamics of the system to guide policy development. However, these methods typically require long time series from stationary systems, which are rarely available in ecological settings. Here we present a Bayesian approach to nonlinear forecasting based on Gaussian processes that readily integrates information from several short time series and allows for nonstationary dynamics. We demonstrate the utility of our modeling methods on simulated from a wide range of ecological scenarios. We expect that these models will extend the range of ecological systems to which nonlinear forecasting methods can be usefully applied.     

Reconstructing Local Population Dynamics in Noisy Metapopulations—The Role of Random Catastrophes and Allee Effects

Reconstructing the dynamics of populations is complicated by the different types of stochasticity experienced by populations, in particular if some forms of stochasticity introduce bias in parameter estimation in addition to error. Identification of systematic biases is critical when determining whether the intrinsic dynamics of populations are stable or unstable and whether or not populations exhibit an Allee effect, i.e., a minimum size below which deterministic extinction should follow. Using a simulation model that allows for Allee effects and a range of intrinsic dynamics, we investigated how three types of stochasticity—demographic, environmental, and random catastrophes— affect our ability to reconstruct the intrinsic dynamics of populations. Demographic stochasticity aside, which is only problematic in small populations, we find that environmental stochasticity—positive and negative environmental fluctuations—caused increased error in parameter estimation, but bias was rarely problematic, except at the highest levels of noise. Random catastrophes, events causing large-scale mortality and likely to be more common than usually recognized, caused immediate bias in parameter estimates, in particular when Allee effects were large. In the latter case, population stability was predicted when endogenous dynamics were actually unstable and the minimum viable population size was overestimated in populations with small or non-existent Allee effects. Catastrophes also generally increased extinction risk, in particular when endogenous Allee effects were large. We propose a method for identifying data points likely resulting from catastrophic events when such events have not been recorded. Using social spider colonies (Anelosimus spp.) as models for populations, we show that after known or suspected catastrophes are accounted for, reconstructed growth parameters are consistent with intrinsic dynamical instability and substantial Allee effects. Our results are applicable to metapopulation or time series data and are relevant for predicting extinction in conservation applications or the management of invasive species.     

Metapopulation extinction risk is increased by environmental stochasticity and assemblage complexity

Extinction risk is a key area of investigation for contemporary ecologists and conservation biologists. Practical conservation efforts for vulnerable species can be considerably enhanced by thoroughly understanding the ecological processes that interact to determine species persistence or extinction. Theory has highlighted the importance of both extrinsic environmental factors and intrinsic demographic processes. In laboratory microcosms, single-species single-habitat patch experimental designs have been widely used to validate the theoretical prediction that environmental heterogeneity can increase extinction risk. Here, we develop on this theme by testing the effects of fluctuating resource levels in experimental multispecies metapopulations. We compare a three-species host-parasitoid assemblage that exhibits apparent competition to the individual pairwise, host-parasitoid interactions. Existing theory is broadly supported for two-species assemblages: environmental stochasticity reduces trophic interaction persistence time, while metapopulation structure increases persistence time. However, with increasing assemblage complexity, the effects of trophic interactions mask environmental impacts and persistence time is further reduced, regardless of resource renewal regime. We relate our findings to recent theory, highlighting the importance of taking into account both intrinsic and extrinsic factors, over a range of spatial scales, in order to understand resource-consumer dynamics.     

Stochastic predation exposes prey to predator pits and local extinction

Understanding how predators affect prey populations is a fundamental goal for ecologists and wildlife managers. A well-known example of regulation by predators is the predator pit, where two alternative stable states exist and prey can be held at a low density equilibrium by predation if they are unable to pass the threshold needed to attain a high density equilibrium. While empirical evidence for predator pits exists, deterministic models of predator–prey dynamics with realistic parameters suggest they should not occur in these systems. Because stochasticity can fundamentally change the dynamics of deterministic models, we investigated if incorporating stochasticity in predation rates would change the dynamics of deterministic models and allow predator pits to emerge. Based on realistic parameters from an elk–wolf system, we found predator pits were predicted only when stochasticity was included in the model. Predator pits emerged in systems with highly stochastic predation and high carrying capacities, but as carrying capacity decreased, low density equilibria with a high likelihood of extinction became more prevalent. We found that incorporating stochasticity is essential to fully understand alternative stable states in ecological systems, and due to the interaction between top–down and bottom–up effects on prey populations, habitat management and predator control could help prey to be resilient to predation stochasticity.     

Plant population dynamics, pollinator foraging, and the selection of self-fertilization

Many flowering plants rely on pollinators, self-fertilization, or both for reproduction. We model the consequences of these features for plant population dynamics and mating system evolution. Our mating systems-based population dynamics model includes an Allee effect. This often leads to an extinction threshold, defined as a density below which population densities decrease. Reliance on generalist pollinators who primarily visit higher density plant species increases the extinction threshold, whereas autonomous modes of selfing decrease and can eliminate the threshold. Generalist pollinators visiting higher density plant species coupled with autonomous selfing may introduce an effect where populations decreasing in density below the extinction threshold may nonetheless persist through selfing. The extinction threshold and selfing at low density result in populations where individuals adopting a single reproductive strategy exhibit mating systems that depend on population density. The ecological and evolutionary analyses provide a mechanism where prior selfing evolves even though inbreeding depression is greater than one-half. Simultaneous consideration of ecological and evolutionary dynamics confirms unusual features (e.g., evolution into extinction or abrupt increases in population density) implicit in our separate consideration of ecological and evolutionary scenarios. Our analysis has consequences for understanding pollen limitation, reproductive assurance, and the evolution of mating systems.     

Assessing variation in life‐history tactics within a population using mixture regression models: a practical guide for evolutionary ecologists

Mixed models are now well‐established methods in ecology and evolution because they allow accounting for and quantifying within‐ and between‐individual variation. However, the required normal distribution of the random effects can often be violated by the presence of clusters among subjects, which leads to multi‐modal distributions. In such cases, using what is known as mixture regression models might offer a more appropriate approach. These models are widely used in psychology, sociology, and medicine to describe the diversity of trajectories occurring within a population over time (e.g. psychological development, growth). In ecology and evolution, however, these models are seldom used even though understanding changes in individual trajectories is an active area of research in life‐history studies. Our aim is to demonstrate the value of using mixture models to describe variation in individual life‐history tactics within a population, and hence to promote the use of these models by ecologists and evolutionary ecologists. We first ran a set of simulations to determine whether and when a mixture model allows teasing apart latent clustering, and to contrast the precision and accuracy of estimates obtained from mixture models versus mixed models under a wide range of ecological contexts. We then used empirical data from long‐term studies of large mammals to illustrate the potential of using mixture models for assessing within‐population variation in life‐history tactics. Mixture models performed well in most cases, except for variables following a Bernoulli distribution and when sample size was small. The four selection criteria we evaluated [Akaike information criterion (AIC), Bayesian information criterion (BIC), and two bootstrap methods] performed similarly well, selecting the right number of clusters in most ecological situations. We then showed that the normality of random effects implicitly assumed by evolutionary ecologists when using mixed models was often violated in life‐history data. Mixed models were quite robust to this violation in the sense that fixed effects were unbiased at the population level. However, fixed effects at the cluster level and random effects were better estimated using mixture models. Our empirical analyses demonstrated that using mixture models facilitates the identification of the diversity of growth and reproductive tactics occurring within a population. Therefore, using this modelling framework allows testing for the presence of clusters and, when clusters occur, provides reliable estimates of fixed and random effects for each cluster of the population. In the presence or expectation of clusters, using mixture models offers a suitable extension of mixed models, particularly when evolutionary ecologists aim at identifying how ecological and evolutionary processes change within a population. Mixture regression models therefore provide a valuable addition to the statistical toolbox of evolutionary ecologists. As these models are complex and have their own limitations, we provide recommendations to guide future users.     

Preventing extinction and outbreaks in chaotic populations

Interactions in ecological communities are inherently nonlinear and can lead to complex population dynamics including irregular fluctuations induced by chaos. Chaotic population dynamics can exhibit violent oscillations with extremely small or large population abundances that might cause extinction and recurrent outbreaks, respectively. We present a simple method that can guide management efforts to prevent crashes, peaks, or any other undesirable state. At the same time, the irregularity of the dynamics can be preserved when chaos is desirable for the population. The control scheme is easy to implement because it relies on time series information only. The method is illustrated by two examples: control of crashes in the Ricker map and control of outbreaks in a stage-structured model of the flour beetle Tribolium. It turns out to be effective even with few available data and in the presence of noise, as is typical for ecological settings.     

Species coexistence in a variable world

The contribution of deterministic and stochastic processes to species coexistence is widely debated. With the introduction of powerful statistical techniques, we can now better characterise different sources of uncertainty when quantifying niche differentiation. The theoretical literature on the effect of stochasticity on coexistence, however, is often ignored by field ecologists because of its technical nature and difficulties in its application. In this review, we examine how different sources of variability in population dynamics contribute to coexistence. Unfortunately, few general rules emerge among the different models that have been studied to date. Nonetheless, we believe that a greater understanding is possible, based on the integration of coexistence and population extinction risk theories. There are two conditions for coexistence in the presence of environmental and demographic variability: (1) the average per capita growth rates of all coexisting species must be positive when at low densities, and (2) these growth rates must be strong enough to overcome negative random events potentially pushing densities to extinction. We propose that critical tests for species coexistence must account for niche differentiation arising from this variability and should be based explicitly on notions of stability and ecological drift.     

Forest cover, extinction thresholds and time lags in woody plants (Myrtaceae) in the Brazilian Atlantic Forest: resources for conservation

Efforts to conserve tropical forests could be strengthened based on ecological knowledge, such as extinction thresholds in ecological processes. Many studies of extinction thresholds associated with habitat reduction have focused on animals, generally at the patch scale. However, certain plant groups are very interesting models with which to study this type of relationship, such as Myrtaceae in Neotropical forests. Because trees are long-lived organisms, local extinctions in response to habitat loss may occur in different ways due to a time lag. In this study, our objective was to assess the occurrence of extinction thresholds at the landscape scale for Myrtaceae in a large biome and the pattern of species reduction in different tree size classes. We studied nine landscapes with different amounts of available habitat (between 5 and 55 % forest cover) in different parts of the Atlantic Forest in Bahia, Brazil, and in each landscape, we evaluated four plant classes based on tree circumference: saplings (CBH between 8 and 15 cm), young (CBH between 15 and 30 cm) adults (CBH ≥30 cm) and total (all individuals with CBH ≥8 cm). Landscapes with forest cover less than 25 % presented an approximately sixfold reduction in Myrtaceae total species richness compared with landscapes with forest cover greater than 40 %. We identified a relationship with a threshold between the amount of available habitat at the landscape level and Myrtaceae richness, with a reduction in total, sapling and young species below a threshold of 40 % forest cover, whereas adults had an extinction threshold at 30 % forest cover. We discuss the differences among the categories of plants associated with a time lag and the possibilities and limitations in applying these results in environmental management.