Will Observation Error and Biases Ruin the Use of Simple Extinction Models?

Abstract: Estimating the risk of extinction for populations of endangered species is an important component of conservation biology. These estimates must be made from data that contain both environmental noise in the year-to-year transitions in population size (so-called "process error"), random errors in sampling, and possible biases in sampling ( both forms of observation errors). To determine how much faith to place in estimated extinction rates, it is important to know how sensitive they are to observation error. We used three simple, commonly employed models of population dynamics to generate simulated population time series. We then combined random observation error or systematic biases with those data, fit models to the time series data, and observed how close the extinction dynamics of the fitted models compared with the dynamics of the underlying models. We found that systematic biases in sampling rarely affected estimates of extinction risk. We also found that even moderate levels of random observation error do not significantly affect extinction estimates except over a small range of process errors, corresponding to the region where extinction risk is most uncertain. With more substantial sampling error, estimates of extinction risk degraded rapidly. Field census techniques for a variety of taxa often involve observation errors within ±32% of actual population sizes. For typical time series used in conservation, therefore, we often may not need to be overly concerned about observation errors as an extra source of imperfection in our estimated extinction rates.     

Effects of sampling error and temporal correlations in population growth on process variance estimators

Estimates of a population’s growth rate and process variance from time-series data are often used to calculate risk metrics such as the probability of quasi-extinction, but temporal correlations in the data from sampling error, intrinsic population factors, or environmental conditions can bias process variance estimators and detrimentally affect risk predictions. It has been claimed (McNamara and Harding, Ecol Lett 7:16–20, 2004) that estimates of the long-term variance that incorporate observed temporal correlations in population growth are unaffected by sampling error; however, no estimation procedures were proposed for time-series data. We develop a suite of such long-term variance estimators, and use simulated data with temporally autocorrelated population growth and sampling error to evaluate their performance. In some cases, we get nearly unbiased long-term variance estimates despite ignoring sampling error, but the utility of these estimators is questionable because of large estimation uncertainty and difficulties in estimating correlation structure in practice. Process variance estimators that ignored temporal correlations generally gave more precise estimates of the variability in population growth and of the probability of quasi-extinction. We also found that the estimation of probability of quasi-extinction was greatly improved when quasi-extinction thresholds were set relatively close to population levels. Because of precision concerns, we recommend using simple models for risk estimates despite potential biases, and limiting inference to quantifying relative risk; e.g., changes in risk over time for a single population or comparative risk among populations.     

Estimating the functional form for the density dependence from life history data

Two contrasting approaches to the analysis of population dynamics are currently popular: demographic approaches where the associations between demographic rates and statistics summarizing the population dynamics are identified; and time series approaches where the associations between population dynamics, population density, and environmental covariates are investigated. In this paper, we develop an approach to combine these methods and apply it to detailed data from Soay sheep (Ovis aries). We examine how density dependence and climate contribute to fluctuations in population size via age- and sex-specific demographic rates, and how fluctuations in demographic structure influence population dynamics. Density dependence contributes most, followed by climatic variation, age structure fluctuations and interactions between density and climate. We then simplify the density-dependent, stochastic, age-structured demographic model and derive a new phenomenological time series which captures the dynamics better than previously selected functions. The simple method we develop has potential to provide substantial insight into the relative contributions of population and individual-level processes to the dynamics of populations in stochastic environments.     

Incorporating known sources of uncertainty to determine precautionary harvests of saltwater crocodiles.

It has been demonstrated repeatedly that the degree to which regulation operates and the magnitude of environmental variation in an exploited population will together dictate the type of sustainable harvest achievable. Yet typically, harvest models fail to incorporate uncertainty in the underlying dynamics of the target population by assuming a particular (unknown) form of endogenous control. We use a novel approach to estimate the sustainable yield of saltwater crocodile (Crocodylus porosus) populations from major river systems in the Northern Territory, Australia, as an example of a system with high uncertainty. We used multimodel inference to incorporate three levels of uncertainty in yield estimation: (1) uncertainty in the choice of the underlying model(s) used to describe population dynamics, (2) the error associated with the precision and bias of model parameter estimation, and (3) environmental fluctuation (process error). We demonstrate varying strength of evidence for density regulation (1.3-96.7%) for crocodiles among 19 river systems by applying a continuum of five dynamical models (density-independent with and without drift and three alternative density-dependent models) to time series of density estimates. Evidence for density dependence increased with the number of yearly transitions over which each river system was monitored. Deterministic proportional maximum sustainable yield (PMSY) models varied widely among river systems (0.042-0.611), and there was strong evidence for an increasing PMSY as support for density dependence rose. However, there was also a large discrepancy between PMSY values and those produced by the full stochastic simulation projection incorporating all forms of uncertainty, which can be explained by the contribution of process error to estimates of sustainable harvest. We also determined that a fixed-quota harvest strategy (up to 0.2K, where K is the carrying capacity) reduces population size much more rapidly than proportional harvest (the latter strategy requiring temporal monitoring of population size to adjust harvest quotas) and greatly inflates the risk of resource depletion. Using an iconic species recovering from recent extreme overexploitation to examine the potential for renewed sustainable harvest, we have demonstrated that incorporating major forms of uncertainty into a single quantitative framework provides a robust approach to modeling the dynamics of exploited populations.     

Spatial and temporal variability modify density dependence in populations of large herbivores

A central challenge in ecology is to understand the interplay of internal and external controls on the growth of populations. We examined the effects of temporal variation in weather and spatial variation in vegetation on the strength of density dependence in populations of large herbivores. We fit three subsets of the model ln(Nt) = a + (1 + b) x ln(N(t-1)) + c x ln(N(t-2)) to five time series of estimates (Nt) of abundance of ungulates in the Rocky Mountains, USA. The strength of density dependence was estimated by the magnitude of the coefficient b. We regressed the estimates of b on indices of temporal heterogeneity in weather and spatial heterogeneity in resources. The 95% posterior intervals of the slopes of these regressions showed that temporal heterogeneity strengthened density-dependent feedbacks to population growth, whereas spatial heterogeneity weakened them. This finding offers the first empirical evidence that density dependence responds in different ways to spatial heterogeneity and temporal heterogeneity.     

Population Viability Analysis with Species Occurrence Data from Museum Collections

Abstract: The most comprehensive data on many species come from scientific collections. Thus, we developed a method of population viability analysis (PVA) in which this type of occurrence data can be used. In contrast to classical PVA, our approach accounts for the inherent observation error in occurrence data and allows the estimation of the population parameters needed for viability analysis. We tested the sensitivity of the approach to spatial resolution of the data, length of the time series, sampling effort, and detection probability with simulated data and conducted PVAs for common, rare, and threatened species. We compared the results of these PVAs with results of standard method PVAs in which observation error is ignored. Our method provided realistic estimates of population growth terms and quasi‐extinction risk in cases in which the standard method without observation error could not. For low values of any of the sampling variables we tested, precision decreased, and in some cases biased estimates resulted. The results of our PVAs with the example species were consistent with information in the literature on these species. Our approach may facilitate PVA for a wide range of species of conservation concern for which demographic data are lacking but occurrence data are readily available.     

Predicting [corrected] extinction risk in spite of predator-prey oscillations.

Most population viability analyses (PVA) assume that the effects of species interactions are subsumed by population-level parameters. We examine how robust five commonly used PVA models are to violations of this assumption. We develop a stochastic, stage-structured predator-prey model and simulate prey population vital rates and abundance. We then use simulated data to parameterize and estimate risk for three demographic models (static projection matrix, stochastic projection matrix, stochastic vital rate matrix) and two time series models (diffusion approximation [DA], corrupted diffusion approximation [CDA]). Model bias is measured as the absolute deviation between estimated and observed quasi-extinction risk. Our results highlight three generalities about the application of single-species models to multi-species conservation problems. First, our collective model results suggest that most single-species PVA models overestimate extinction risk when species interactions cause periodic variation in abundance. Second, the DA model produces the most (conservatively) biased risk forecasts. Finally, the CDA model is the most robust PVA to population cycles caused by species interactions. CDA models produce virtually unbiased and relatively precise risk estimates even when populations cycle strongly. High performance of simple time series models like the CDA owes to their ability to effectively partition stochastic and deterministic sources of variation in population abundance.     

Determining Optimal Population Monitoring for Rare Butterflies

Abstract: Determining population viability of rare insects depends on precise, unbiased estimates of population size and other demographic parameters. We used data on the endangered St. Francis' satyr butterfly (Neonympha mitchellii francisci) to evaluate 2 approaches (mark–recapture and transect counts) for population analysis of rare butterflies. Mark–recapture analysis provided by far the greatest amount of demographic information, including estimates (and standard errors) of population size, detection, survival, and recruitment probabilities. Mark–recapture analysis can also be used to estimate dispersal and temporal variation in rates, although we did not do this here. Models of seasonal flight phenologies derived from transect counts (Insect Count Analyzer) provided an index of population size and estimates of survival and statistical uncertainty. Pollard–Yates population indices derived from transect counts did not provide estimates of demographic parameters. This index may be highly biased if detection and survival probabilities vary spatially and temporally. In terms of statistical performance, mark–recapture and Pollard–Yates indices were least variable. Mark–recapture estimates were less likely to fail than Insect Count Analyzer, but mark–recapture estimates became less precise as sampling intensity decreased. In general, count‐based approaches are less costly and less likely to cause harm to rare insects than mark–recapture. The optimal monitoring approach must reconcile these trade‐offs. Thus, mark–recapture should be favored when demographic estimates are needed, when financial resources enable frequent sampling, and when marking does not harm the insect populations. The optimal sampling strategy may use 2 sampling methods together in 1 overall sampling plan: limited mark–recapture sampling to estimate survival and detection probabilities and frequent but less expensive transect counts.     

The use of replicated plot,line and point sampling for estimating species abundance and ecological diversity

The paper deals with the problem of estimating diversity indexes for an ecological community. First the species abundances are unbiasedly and consistently estimated using designs based on n random and independent selections of plots, points or lines over the study area. The problem of sampling elusive populations is also considered. Finally, the diversity index estimates are obtained as functions of the abundance estimates. The resulting estimators turn out to be asymptotically (n large) unbiased, even if a considerable bias may occur for a small n. Accordingly, the method of jackknifing is made use of in order to reduce bias.     

Optimization of capture–recapture monitoring of elusive species illustrated with a threatened grasshopper

Information on population sizes and trends of threatened species is essential for their conservation, but obtaining reliable estimates can be challenging. We devised a method to improve the precision of estimates of population size obtained from capture–recapture studies for species with low capture and recapture probabilities and short seasonal activity, illustrated with population data of an elusive grasshopper (Prionotropis rhodanica). We used data from 5 capture–recapture studies to identify methodological and environmental factors affecting capture and recapture probabilities and estimates of population size. In a simulation, we used the population size and capture and recapture probability estimates obtained from the field studies to identify the minimum number of sampling occasions needed to obtain unbiased and robust estimates of population size. Based on these results we optimized the capture–recapture design, implemented it in 2 additional studies, and compared their precision with those of the nonoptimized studies. Additionally, we simulated scenarios based on thresholds of population size in criteria C and D of the International Union for Conservation of Nature (IUCN) Red List to investigate whether estimates of population size for elusive species can reliably inform red-list assessments. Identifying parameters that affect capture and recapture probabilities (for the grasshopper time since emergence of first adults) and optimizing field protocols based on this information reduced study effort (−6% to −27% sampling occasions) and provided more precise estimates of population size (reduced coefficient of variation) compared with nonoptimized studies. Estimates of population size from the scenarios based on the IUCN thresholds were mostly unbiased and robust (only the combination of very small populations and little study effort produced unreliable estimates), suggesting capture–recapture can be considered reliable for informing red-list assessments. Although capture–recapture remains difficult and costly for elusive species, our optimization procedure can help determine efficient protocols to increase data quality and minimize monitoring effort.     

A hierarchical model for spatial capture-recapture data

Estimating density is a fundamental objective of many animal population studies. Application of methods for estimating population size from ostensibly closed populations is widespread, but ineffective for estimating absolute density because most populations are subject to short-term movements or so-called temporary emigration. This phenomenon invalidates the resulting estimates because the effective sample area is unknown. A number of methods involving the adjustment of estimates based on heuristic considerations are in widespread use. In this paper, a hierarchical model of spatially indexed capture-recapture data is proposed for sampling based on area searches of spatial sample units subject to uniform sampling intensity. The hierarchical model contains explicit models for the distribution of individuals and their movements, in addition to an observation model that is conditional on the location of individuals during sampling. Bayesian analysis of the hierarchical model is achieved by the use of data augmentation, which allows for a straightforward implementation in the freely available software WinBUGS. We present results of a simulation study that was carried out to evaluate the operating characteristics of the Bayesian estimator under variable densities and movement patterns of individuals. An application of the model is presented for survey data on the flat-tailed horned lizard (Phrynosoma mcallii) in Arizona, USA.     

Synergistic influences of phase, density, and climatic variation on the dynamics of fluctuating populations

Although ecologists have long recognized that certain mammalian species exhibit high-amplitude, often multiannual, fluctuations in abundance, their causes have remained poorly understood and the subject of intense debate. A key contention has been the relative role of density-dependent and density-independent processes in governing population dynamics. We applied capture-mark-recapture analysis to 25 years of monthly trapping data from a fluctuating prairie vole Microtus ochrogaster population in Illinois, USA, to estimate realized population growth rates and associated vital rates (survival and recruitment) and modeled them as a function of vole density and density-independent climatic variation. We also tested for phase dependence and seasonality in the effects of the above processes. Variation in the realized population growth rate was best explained by phase-specific changes in vole density lagged by one month and mean monthly temperatures with no time lags. The underlying vital rates, survival and recruitment, were influenced by the additive and interactive effects of phase, vole density, and mean monthly temperatures. Our results are consistent with the observation that large-scale population fluctuations are characterized by phase-specific changes in demographic and physiological characteristics. Our findings also support the growing realization that the interaction between climatic variables and density-dependent factors may be a widespread phenomenon, and they suggest that the direction and magnitude of such interactive effects may be phase specific. We conclude that density-dependent and density-independent climatic variables work in tandem during each phase of density fluctuations to drive the dynamics of fluctuating populations.     

Sampling variability and estimates of density dependence: a composite-likelihood approach

It is well known that sampling variability, if not properly taken into account, affects various ecologically important analyses. Statistical inference for stochastic population dynamics models is difficult when, in addition to the process error, there is also sampling error. The standard maximum-likelihood approach suffers from large computational burden. In this paper, I discuss an application of the composite-likelihood method for estimation of the parameters of the Gompertz model in the presence of sampling variability. The main advantage of the method of composite likelihood is that it reduces the computational burden substantially with little loss of statistical efficiency. Missing observations are a common problem with many ecological time series. The method of composite likelihood can accommodate missing observations in a straightforward fashion. Environmental conditions also affect the parameters of stochastic population dynamics models. This method is shown to handle such nonstationary population dynamics processes as well. Many ecological time series are short, and statistical inferences based on such short time series tend to be less precise. However, spatial replications of short time series provide an opportunity to increase the effective sample size. Application of likelihood-based methods for spatial time-series data for population dynamics models is computationally prohibitive. The method of composite likelihood is shown to have significantly less computational burden, making it possible to analyze large spatial time-series data. After discussing the methodology in general terms, I illustrate its use by analyzing a time series of counts of American Redstart (Setophaga ruticilla) from the Breeding Bird Survey data, San Joaquin kit fox (Vulpes macrotis mutica) population abundance data, and spatial time series of Bull trout (Salvelinus confluentus) redds count data.     

Nonlinear effects of climate and density in the dynamics of a fluctuating population of reindeer

Nonlinear and irregular population dynamics may arise as a result of phase dependence and coexistence of multiple attractors. Here we explore effects of climate and density in the dynamics of a highly fluctuating population of wild reindeer (Rangifer tarandus platyrhynchus) on Svalbard observed over a period of 29 years. Time series analyses revealed that density dependence and the effects of local climate (measured as the degree of ablation [melting] of snow during winter) on numbers were both highly nonlinear: direct negative density dependence was found when the population was growing (Rt > 0) and during phases of the North Atlantic Oscillation (NAO) characterized by winters with generally high (1979-1995) and low (1996-2007) indices, respectively. A growth-phase-dependent model explained the dynamics of the population best and revealed the influence of density-independent processes on numbers that a linear autoregressive model missed altogether. In particular, the abundance of reindeer was enhanced by ablation during phases of growth (Rt > 0), an observation that contrasts with the view that periods of mild weather in winter are normally deleterious for reindeer owing to icing of the snowpack. Analyses of vital rates corroborated the nonlinearity described in the population time series and showed that both starvation mortality in winter and fecundity were nonlinearly related to fluctuations in density and the level of ablation. The erratic pattern of growth of the population of reindeer in Adventdalen seems, therefore, to result from a combination of the effects of nonlinear density dependence, strong density-dependent mortality, and variable density independence related to ablation in winter.     

Reliability of Absolute and Relative Predictions of Population Persistence Based on Time Series

Abstract:  Conventional population viability analysis (PVA) is often impractical because data are scarce for many threatened species. For this reason, simple count-based models are being advocated. The simplest of these models requires nothing more than a time series of abundance estimates, from which variance and autocorrelation in growth rate are estimated and predictions of population persistence are generated. What remains unclear, however, is how many years of data are needed to generate reliable estimates of these parameters and hence reliable predictions of persistence. By analyzing published and simulated time series, we show that several decades of data are needed. Predictions based on short time series were very unreliable mainly because limited data yielded biased, unreliable estimates of variance in growth rate, especially when growth rate was strongly autocorrelated. More optimistically, our results suggest that count-based PVA is sometimes useful for relative risk assessment (i.e., for ranking populations by extinction risk), even when time series are only a decade long. However, some conditions consistently lead to backward rankings. We explored the limited conditions under which simple count-based PVA may be useful for relative risk assessment.     

Spatially Explicit Power Analyses for Occupancy‐Based Monitoring of Wolverine in the U.S. Rocky Mountains

Conservation scientists and resource managers often have to design monitoring programs for species that are rare or patchily distributed across large landscapes. Such programs are frequently expensive and seldom can be conducted by one entity. It is essential that a prospective power analysis be undertaken to ensure stated monitoring goals are feasible. We developed a spatially based simulation program that accounts for natural history, habitat use, and sampling scheme to investigate the power of monitoring protocols to detect trends in population abundance over time with occupancy‐based methods. We analyzed monitoring schemes with different sampling efforts for wolverine (Gulo gulo) populations in 2 areas of the U.S. Rocky Mountains. The relation between occupancy and abundance was nonlinear and depended on landscape, population size, and movement parameters. With current estimates for population size and detection probability in the northern U.S. Rockies, most sampling schemes were only able to detect large declines in abundance in the simulations (i.e., 50% decline over 10 years). For small populations reestablishing in the Southern Rockies, occupancy‐based methods had enough power to detect population trends only when populations were increasing dramatically (e.g., doubling or tripling in 10 years), regardless of sampling effort. In general, increasing the number of cells sampled or the per‐visit detection probability had a much greater effect on power than the number of visits conducted during a survey. Although our results are specific to wolverines, this approach could easily be adapted to other territorial species. Poder de Análisis Espacialmente Explícito para el Monitoreo Basado en Ocupación del Glotón (Gulo gulo) en las Montañas Rocallosas de Estados Unidos     

Allee Effect and the Uncertainty of Population Recovery

Recovery of depleted populations is fundamentally important for conservation biology and sustainable resource harvesting. At low abundance, population growth rate, a primary determinant of population recovery, is generally assumed to be relatively fast because competition is low (i.e., negative density dependence). But population growth can be limited in small populations by an Allee effect. This is particularly relevant for collapsed populations or species that have not recovered despite large reductions in, or elimination of, threats. We investigated how an Allee effect can influence the dynamics of recovery. We used Atlantic cod (Gadus morhua) as the study organism and an empirically quantified Allee effect for the species to parameterize our simulations. We simulated recovery through an individual‐based mechanistic simulation model and then compared recovery among scenarios incorporating an Allee effect, negative density dependence, and an intermediate scenario. Although an Allee effect significantly slowed recovery, such that population increase could be negligible even after 100 years or more, it also made the time required for biomass rebuilding much less predictable. Our finding that an Allee effect greatly increased the uncertainty in recovery time frames provides an empirically based explanation for why the removal of threat does not always result in the recovery of depleted populations or species. El Efecto Allee y la Incertidumbre de la Recuperación de Poblaciones     

Multinomial mixture model with heterogeneous classification probabilities

Royle and Link (Ecology 86(9):2505?C2512, 2005) proposed an analytical method that allowed estimation of multinomial distribution parameters and classification probabilities from categorical data measured with error. While useful, we demonstrate algebraically and by simulations that this method yields biased multinomial parameter estimates when the probabilities of correct category classifications vary among sampling units. We address this shortcoming by treating these probabilities as logit-normal random variables within a Bayesian framework. We use Markov chain Monte Carlo to compute Bayes estimates from a simulated sample from the posterior distribution. Based on simulations, this elaborated Royle-Link model yields nearly unbiased estimates of multinomial and correct classification probability estimates when classification probabilities are allowed to vary according to the normal distribution on the logit scale or according to the Beta distribution. The method is illustrated using categorical submersed aquatic vegetation data.     

Using the bootstrap and fast Fourier transform to estimate confidence intervals of 2D kernel densities

Kernel density estimators are often used to estimate the utilization distributions (UDs) of animals. Kernel UD estimates have a strong theoretical basis and perform well, but are usually reported without estimates of error or uncertainty. It is intuitively and theoretically appealing to estimate the sampling error in kernel UD estimates using bootstrapping. However, standard equations for kernel density estimates are complicated and computationally expensive. Bootstrapping requires computing hundreds or thousands of probability densities and is impractical when the number of observations, or the area of interest is large. We used the fast Fourier transform (FFT) and discrete convolution theorem to create a bootstrapping algorithm fast enough to run on commonly available desktop or laptop computers. Application of the FFT method to a large (n>20,000) set of radio telemetry data would provide a 99.6% reduction in computation time (i.e., 1.6 as opposed to 444 hours) for 1000 bootstrap UD estimates. Bootstrap error contours were computed using data from a radio-collared polar bear (Ursus maritimus) in the Beaufort Sea north of Alaska.     

Temporal Changes in Allele Frequencies and a Population's History of Severe Bottlenecks

Monitoring temporal changes in genetic variation has been suggested as a means of determining if a population has experienced a demographic bottleneck. Simulations have shown that the variance in allele frequencies over time ( F ) can provide reasonable estimates of effective population size ( N_e ). This relationship between F and N_e suggests that changes in allele frequencies may provide a way to determine the severity of recent demographic bottlenecks experienced by a population. We examined allozyme variation in experimental populations of the eastern mosquitofish ( Gambusia holbrooki ) to evaluate the relationship between the severity of demographic bottlenecks and temporal variation in allele frequencies. Estimates of F from both the fish populations and computer simulations were compared to expected rates of drift. We found that different methods for estimating F had little effect on the analysis. The variance in estimates of F was large among both experimental and simulated populations experiencing similar demographic bottlenecks. Temporal changes in allele frequencies suggested that the experimental populations had experienced bottlenecks, but there was no relationship between observed and expected values of F . Furthermore, genetic drift was likely to be underestimated in populations experiencing the most severe bottlenecks. The weak relationship between F and bottleneck severity is probably due to both sampling error associated with the number of polymorphic loci examined and the loss of alleles during the bottlenecks. For populations that may have experienced severe bottlenecks, caution should be used in making evolutionary interpretations or management recommendations based on temporal changes in allele frequencies.