On the latent state estimation of nonlinear population dynamics using Bayesian and non-Bayesian state-space models

Nonlinear state-space models have been increasingly applied to study population dynamics and data assimilation in environmental sciences. State-space models can account for process error and measurement error simultaneously to correct for the bias in the estimates of system state and model parameters. However, few studies have compared the performance of different nonlinear state-space models for reconstructing the state of population dynamics from noisy time series. This study compared the performance of the extended Kalman filter (EKF), unscented Kalman filter (UKF) and Bayesian nonlinear state-space models (BNSSM) through simulations. Synthetic population time series were generated using the theta logistic model with known parameters, and normally distributed process and measurement errors were introduced using the Monte Carlo simulations. At higher levels of nonlinearity, the UKF and BNSSM had lower root mean square error (RMSE) than the EKF. The BNSSM performed reliably across all levels of nonlinearity, whereas increased levels of nonlinearity resulted in higher RMSE of the EKF. The Metropolis–Hastings algorithm within the Gibbs algorithm was used to fit the theta logistic model to synthetic time series to estimate model parameters. The estimated posterior distribution of the parameter θ indicated that the 95% credible intervals included the true values of θ (=0.5 and 1.5), but did not include 1.0 and 0.0. Future studies need to incorporate the adaptive Metropolis algorithm to estimate unknown model parameters for broad applications of Bayesian nonlinear state-space models in ecological studies.     

Simulated Responses of a Forest-Interior Bird Population to Forest Management Options in Central Hardwood Forests of the United States

I used estimates of carrying capacity, survival, fecundity, and edge effects to simulate the responses of a forest-interior bird population to selection cutting clearcutting, and no timber harvest. I also modeled population sensitivity to changes in fecundity, survival, K , and edge relationships. Because model parameters are based on scant data, results should he regarded as hypotheses to be further investigated or measures of the relative impact or sensitivity (given model assumptions). Simulated population size was greater with no timber harvest than with clearcutting and greater with clearcutting than with group selection when edge effects were included in the model. Without edge effects, population levels were only slightly lower under group selection than under no timber harvest, and greater than clearcutting. Edge effects had only a small impact on population levels under clearcutting. Clearcut size did not have much effect on population levels, but longer and shorter rotation ages resulted in higher and lower population levels, respectively. The model was very sensitive to declines in mean fecundity and survival, suggesting that factors affecting mean demographic rates could be more important than local edge effects. Some methods of timber harvest may be compatible with the conservation of forest-interior birds, but better demographic data and information on habitat suitability of selectively cut forests and young even-aged stands is needed to adequately evaluate management options.     

Use of Integrated Modeling to Enhance Estimates of Population Dynamics Obtained from Limited Data

Abstract:  Demographic data of rare and endangered species are often too sparse to estimate vital rates and population size with sufficient precision for understanding population growth and decline. Yet, the combination of different sources of demographic data into one statistical model holds promise. We applied Bayesian integrated population modeling to demographic data from a colony of the endangered greater horseshoe bats (Rhinolophus ferrumequinum) . Available data were the number of subadults and adults emerging from the colony roost at dusk, the number of newborns from 1991 to 2005, and recapture data of subadults and adults from 2004 and 2005. Survival rates did not differ between sexes, and demographic rates remained constant across time. The greater horseshoe bat is a long-lived species with high survival rates (first year: 0.49 [SD 0.06]; adults: 0.91 [SD 0.02]) and low fecundity (0.74 [SD 0.12]). The yearly average population growth was 4.4% (SD 0.1%) and there were 92 (SD 10) adults in the colony in year 2005. Had we analyzed each data set separately, we would not have been able to estimate fecundity, the estimates of survival would have been less precise, and the estimate of population growth biased. Our results demonstrate that integrated models are suitable for obtaining crucial demographic information from limited data.     

A combination line transect and capture-recapture sampling model for multiple observers in aerial surveys

We present a robust sampling methodology to estimate population size using line transect and capture-recapture procedures for aerial surveys. Aerial surveys usually underestimate population density due to animals being missed. A combination of capture-recapture and line transect sampling methods with multiple observers allows violation of the assumption that all animals on the centreline are sighted from the air. We illustrate our method with an example of inanimate objects which shows evidence of failure of the assumption that all objects on the centreline have probability 1 of being detected. A simulation study is implemented to evaluate the performance of three variations of the Lincoln-Petersen estimator: the overall estimator, the stratified estimator, and the general stratified estimator based on the combined likelihood proposed in this paper. The stratified Lincoln-Petersen estimator based on the combined likelihood is found to be generally superior to the other estimators.     

Fitting probability models to population dynamics data

Methods for modeling population dynamics in probability using the generalized point process approach are developed. The life history of these populations is such that seasonal reproduction occurs during a short time. Several models are developed and analyzed. Data about two species: colonial spiders (Stegodyphus dumicola) and a migratory bird (wood thrush, Hylocichla mustelina) are used to estimate model parameters with appropriate log maximum likelihood functions. For the spiders, the model is fitted to provide evolutionary feasible colony size based on maximum likelihood estimates of fecundity and survival data. For the migratory bird species, a maximum likelihood estimates are derived for the fecundity and survival rates of young and adult birds and immigration rate. The presented approach allows computation of quantities of interest such as probability of extinction and average time to extinction.     

Extinction Rate Estimates for Plant Populations in Revisitation Studies: Importance of Detectability

Abstract:  Many researchers have obtained extinction-rate estimates for plant populations by comparing historical and current records of occurrence. A population that is no longer found is assumed to have gone extinct. Extinction can then be related to characteristics of these populations, such as habitat type, size, or species, to test ideas about what factors may affect extinction. Such studies neglect the fact that a population may be overlooked, however, which may bias estimates of extinction rates upward. In addition, if populations are unequally detectable across groups to be compared, such as habitat type or population size, comparisons become distorted to an unknown degree. To illustrate the problem, I simulated two data sets, assuming a constant extinction rate, in which populations occurred in different habitats or habitats of different size and these factors affected their detectability. The conventional analysis implicitly assumed that detectability equalled 1 and used logistic regression to estimate extinction rates. It wrongly identified habitat and population size as factors affecting extinction risk. In contrast, with capture-recapture methods, unbiased estimates of extinction rates were recovered. I argue that capture-recapture methods should be considered more often in estimations of demographic parameters in plant populations and communities.     

Using Age Structure to Detect Impacts on Threatened Populations: a Case Study with Steller Sea Lions

Abstract:   A delayed response to change is often a characteristic of long-lived species and presents a major challenge to monitoring their status. However, rapid shifts in age structure can occur even while population size remains relatively static. We used time-varying matrix models to study age-structure information as a tool for improving detection of survivorship and fecundity change and status. We applied the methods to Steller sea lions (  Eumetopias jubatus ), a long-lived endangered marine mammal found throughout the North Pacific Rim. Population and newborn counts were supplemented with information on the fraction of the population that was juvenile, obtained by measuring animals in aerial photographs taken during range-wide censuses. By fitting the model to 1976–1998 data, we obtained maximum-likelihood estimates and 95% confidence intervals for juvenile survivorship, adult survivorship, and adult fecundity in the mid-1980s, late 1980s, and 1990s. We used a series of nested models to test whether the data were best fit by a model with one, two, or three temporal changes in demographic rates, and we fit the models to different lengths of data to test the number of years of data needed to detect a demographic change. The declines in the early 1980s were associated with severely low juvenile survivorship, whereas declines in the 1990s were associated with disproportionately low fecundity. We repeated these analyses, fitting only to the count data without the juvenile-fraction information, to determine whether the age-structure information changed the conclusions and/or changed the certainty and speed with which demographic-rate changes could be detected. The juvenile-fraction data substantially improved the degree to which estimates from the model were consistent with field data and significantly improved the speed and certainty with which changes in demographic rates were detected.     

Use of the superpopulation approach to estimate breeding population size: an example in asynchronously breeding birds

Many populations of animals are fluid in both space and time, making estimation of numbers difficult. Much attention has been devoted to estimation of bias in detection of animals that are present at the time of survey. However, an equally important problem is estimation of population size when all animals are not present on all survey occasions. Here, we showcase use of the superpopulation approach to capture-recapture modeling for estimating populations where group membership is asynchronous, and where considerable overlap in group membership among sampling occasions may occur. We estimate total population size of long-legged wading bird (Great Egret and White Ibis) breeding colonies from aerial observations of individually identifiable nests at various times in the nesting season. Initiation and termination of nests were analogous to entry and departure from a population. Estimates using the superpopulation approach were 47-382% larger than peak aerial counts of the same colonies. Our results indicate that the use of the superpopulation approach to model nesting asynchrony provides a considerably less biased and more efficient estimate of nesting activity than traditional methods. We suggest that this approach may also be used to derive population estimates in a variety of situations where group membership is fluid.     

An Age-Structured Demographic Model for the Endangered Stephens' Kangaroo Rat

Effective conservation of endangered species often is hampered by inadequate knowledge of demography. We extracted information on survival and fecundity from an 18-month, live-trapping study of Dipodomys stephensi , and from this we developed an age-structured demographic model to assess population viability. Adult Stephens' kangaroo rats persisted longer than juveniles, and adult females persisted longer than adult males. Disappearance rates were high in the first months after initial capture. Thereafter, the fraction of animals persisting decreased slowly and in an approximately linear fashion on a semilogarithmic scale, suggesting age-independent mortality factors such as predation. Juvenile persistence did not differ substantially between two years of strikingly different rainfall. Onset of breeding followed the start of winter rains. Length of the breeding season, average number of litters per female, and the fraction of first-year females breeding were much greater in the year of higher rainfall. We propose a birth-pulse demographic model for D. stephensi that distinguishes juvenile and adult age classes. Temporal environmental variation can be modeled adequately with a constant survivorship schedule and variable fecundity determined by yearly precipitation. Several issues should be resolved, however, before conservation decisions are based on the model. Better estimates of juvenile survivorship are critical, the quantitative relationship between precipitation and fecundity must be determined, and the potential for density dependence and source-sink population dynamics must be evaluated.     

A robust design capture-recapture model with multiple age classes augmented with population assignment data

The relative contribution of in situ reproduction versus immigration to the recruitment process is important to ecologists. Here we consider a robust design superpopulation capture-recapture model for a population with two age classes augmented with population assignment data. We first use age information to estimate the entry probabilities of new animals originating via in situ reproduction and immigration separately for all except the first period. Then we combine age and population assignment information with the capture-recapture model, which enables us to estimate the entry probability of in situ births and the entry probability of immigrants separately for all sampling periods. Further, this augmentation of age specific capture-recapture data with population assignment data greatly improves the estimators’ precision. We apply our new model to a capture-recapture data set with genetic information for banner-tailed kangaroo rats in Southern Arizona. We find that many more individuals are born in situ than are immigrants for all time periods. Young animals have lower survival probabilities than adults born in situ. Adult animals born in situ have higher survival probabilities than adults that were immigrants.     

Measuring and modeling the seasonal changes of an urban Green Treefrog (Hyla cinerea) population

Green Treefrogs (Hyla cinerea) were captured, marked, measured and released at an urban study site in Lafayette, LA, during the 2004 and 2005 breeding seasons. A statistical method based on a generalization of the hypergeometric distribution was used to derive weekly time-series estimates of the population sizes. To describe the population dynamics, a stage structured mathematical model was developed and compared to time-series obtained from the weekly population estimates study using a least-squares approach. Two fitting experiments were done: (1) Using uniform distribution for the birth rate during the breeding season; (2) Using a birth rate distributed according to weekly data on frog calling intensity. Although both model-to-data fits look very promising during the years 2004 and 2005 and result in similar inherent survivorship rates for the tadpoles, juvenile and adult frogs, the fit that uses the calling data predicts a lower number of tadpoles and frogs in the long term than the one that uses uniform birth distribution. The parameter estimates resulting from these fitting experiments are used in the context of stochastic simulations to derive extinction and persistence probabilities for this population. Due to the oscillatory dynamics (with high amplitude) evidenced by the capture-recapture data and corroborated by the model, it is suggested that anuran monitoring efforts should take into account the natural intra-annual variation in population size.     

Population viability analysis for several populations using multivariate state-space models

The International Union for the Conservation of Nature and Natural Resources (IUCN), the world's largest and most important global conservation network, has listed approximately 16,000 species worldwide as threatened. The most important tool for recognizing and listing species as threatened is population viability analysis (PVA), which estimates the probability of extinction of a population or species over a specified time horizon. The most common PVA approach is to apply it to single time series of population abundance. This approach to population viability analysis ignores covariability of local populations. Covariability can be important because high synchrony of local populations reduces the effective number of local populations and leads to greater extinction risk. Needed is a way of extending PVA to model correlation structure among multiple local populations. Multivariate state-space modeling is applied to this problem and alternative estimation methods are compared. The multivariate state-space technique is applied to endangered populations of pacific salmon, USA. Simulations demonstrated that the correlation structure can strongly influence population viability and is best estimated using restricted maximum likelihood instead of maximum likelihood.     

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.     

Population level effects of reduced fecundity in the fish species perch (Perca fluviatilis) and the implications for environmental monitoring

A 40% reduction in relative gonad size in perch (Perca fluviatilis) has been observed over that past two decades at the Swedish national reference site Kvädöfjärden. This biomarker response could be interpreted as a reduction in fecundity and increased risk of local extinction. However, abundance estimates from the same area has not provided any evidence of a reduction in population size. In the present study, a matrix population model was developed to investigate if a reduction in fecundity can be expected to have long term effects on population viability for perch and to evaluate the probability to detect such effects through abundance estimates. The model was parameterized from 17 years of population data from Kvädöfjärden as well as from other studies on perch. The model included density dependence and environmental stochasticity. The results indicated that a reduction in fecundity that is in level with the observed reduction in relative gonad size in Kvädöfjärden will cause a substantial risk for local extinction. The risk that the population will fall below 20% of the carrying capacity within 50 years is 44% when the fecundity is reduced by 40%. However, due to variability in abundance measurements it will take some time before a reduction in gonad size leads to statistically significant effects on the population. If the fecundity is reduced by 40% successively over a 10-year period, the probability to detect this through abundance estimates within 10 years is less than 50%. The results of the present study clearly show that relevant biomarkers have an important role in environmental monitoring as early warning signals, preferably in combination with measurements at higher levels of biological organization.     

An estimating function approach to the inference of catch-effort models

A class of catch-effort models, which allows for heterogeneous removal probabilities, is proposed for closed populations. The model includes three types of removal probabilities: multiplicative, Poisson and logistic. The usual removal and generalized removal models then become special cases. The equivalence of the proposed model and a special type of capture-recapture model is discussed. A unified estimating function approach is used to estimate the initial population size. For the homogeneous model, the resulting population size estimator based on optimal estimating functions is asymptotically equivalent to the maximum likelihood estimator. One advantage for our approach is that it can be extended to handle the heterogeneous populations in which the maximum likelihood estimators do not exist. The bootstrap method is applied to construct variance estimators and confidence intervals. We illustrate the method by two real data examples. Results of a simulation study investigating the performance of the proposed estimation procedure are presented.     

Estimating the complexity of bird song by using capture-recapture approaches from community ecology

Repertoire size, the number of unique song or syllable types in the repertoire, is a widely used measure of song complexity in birds, but it is difficult to calculate this exactly in species with large repertoires. A new method of repertoire size estimation applies species richness estimation procedures from community ecology, but such capture-recapture approaches have not been much tested. Here, we establish standardized sampling schemes and estimation procedures using capture-recapture models for syllable repertoires from 18 bird species, and suggest how these may be used to tackle problems of repertoire estimation. Different models, with different assumptions regarding the heterogeneity of the use of syllable types, performed best for different species with different song organizations. For most species, models assuming heterogeneous probability of occurrence of syllables (so-called detection probability) were selected due to the presence of both rare and frequent syllables. Capture-recapture estimates of syllable repertoire size from our small sample did not differ significantly from previous estimates using larger samples of count data. However, the enumeration of syllables in 15 songs yielded significantly lower estimates than previous reports. Hence, heterogeneity in detection probability of syllables should be addressed when estimating repertoire size. This is neglected using simple enumeration procedures, but is taken into account when repertoire size is estimated by appropriate capture-recapture models adjusted for species-specific song organization characteristics. We suggest that such approaches, in combination with standardized sampling, should be applied in species with potentially large repertoire size. On the other hand, in species with small repertoire size and homogenous syllable usage, enumerations may be satisfactory. Although researchers often use repertoire size as a measure of song complexity, listeners to songs are unlikely to count entire repertoires and they may rely on other cues, such as syllable detection probability.Communicated by A. Cockburn     

A generalized estimating equations approach for capture–recapture closed population models

The estimation of population density animal population parameters, such as capture probability, population size, or population density, is an important issue in many ecological applications. Capture–recapture data may be considered as repeated observations that are often correlated over time. If these correlations are not taken into account then parameter estimates may be biased, possibly producing misleading results. We propose a generalized estimating equations (GEE) approach to account for correlation over time instead of assuming independence as in the traditional closed population capture–recapture studies. We also account for heterogeneity among observed individuals and over-dispersion, modelling capture probabilities as a function of covariates. The GEE versions of all closed population capture–recapture models and their corresponding estimating equations are proposed. We evaluate the effect of accounting for correlation structures on capture–recapture model selection based on the quasi-likelihood information criterion (QIC). An example is used for an illustrative application and for comparison to currently used methodology. A Horvitz–Thompson-like estimator is used to obtain estimates of population size based on conditional arguments. A simulation study is conducted to evaluate the performance of the GEE approach in capture-recapture studies. The GEE approach performs well for estimating population parameters, particularly when capture probabilities are high. The simulation results also reveal that estimated population size varies on the nature of the existing correlation among capture occasions.     

Climate change threatens polar bear populations: a stochastic demographic analysis

The polar bear (Ursus maritimus) depends on sea ice for feeding, breeding, and movement. Significant reductions in Arctic sea ice are forecast to continue because of climate warming. We evaluated the impacts of climate change on polar bears in the southern Beaufort Sea by means of a demographic analysis, combining deterministic, stochastic, environment-dependent matrix population models with forecasts of future sea ice conditions from IPCC general circulation models (GCMs). The matrix population models classified individuals by age and breeding status; mothers and dependent cubs were treated as units. Parameter estimates were obtained from a capture-recapture study conducted from 2001 to 2006. Candidate statistical models allowed vital rates to vary with time and as functions of a sea ice covariate. Model averaging was used to produce the vital rate estimates, and a parametric bootstrap procedure was used to quantify model selection and parameter estimation uncertainty. Deterministic models projected population growth in years with more extensive ice coverage (2001-2003) and population decline in years with less ice coverage (2004-2005). LTRE (life table response experiment) analysis showed that the reduction in lambda in years with low sea ice was due primarily to reduced adult female survival, and secondarily to reduced breeding. A stochastic model with two environmental states, good and poor sea ice conditions, projected a declining stochastic growth rate, log lambdas, as the frequency of poor ice years increased. The observed frequency of poor ice years since 1979 would imply log lambdas approximately - 0.01, which agrees with available (albeit crude) observations of population size. The stochastic model was linked to a set of 10 GCMs compiled by the IPCC; the models were chosen for their ability to reproduce historical observations of sea ice and were forced with "business as usual" (A1B) greenhouse gas emissions. The resulting stochastic population projections showed drastic declines in the polar bear population by the end of the 21st century. These projections were instrumental in the decision to list the polar bear as a threatened species under the U.S. Endangered Species Act.     

Modeling Dynamics of Habitat-Interior Bird Populations in Fragmented Landscapes

Abstract: A stochastic computer model was used to examine the effects of varying degrees of habitat fragmentation on the dynamics of a hypothetical population of forest-interior bid. The primary demographic parameter that influenced the population's dynamics was fecundity, which varied as a function of how far a birds territory was from an ecological edge. As our model landscape became more fragmented the proportion of forest habitat that was near edges increased geometrically, and the population's overall fecundity dropped as a result. The model demonstrates that impaired reproduction in a fragmented landscape is, by itself a sufficient disruption of the population's dynamics to generate population declines and shifts in distribution similar to those observed in the fragmented forests of southern Wisconsin. Without immigration of recruits from other regions where reproduction is better, habitat-interior populations in a severely fragmented landscape can become locally extinct.     

Analysis of neighborhood dynamics of forest ecosystems using likelihood methods and modeling.

Advances in computing power in the past 20 years have led to a proliferation of spatially explicit, individual-based models of population and ecosystem dynamics. In forest ecosystems, the individual-based models encapsulate an emerging theory of "neighborhood" dynamics, in which fine-scale spatial interactions regulate the demography of component tree species. The spatial distribution of component species, in turn, regulates spatial variation in a whole host of community and ecosystem properties, with subsequent feedbacks on component species. The development of these models has been facilitated by development of new methods of analysis of field data, in which critical demographic rates and ecosystem processes are analyzed in terms of the spatial distributions of neighboring trees and physical environmental factors. The analyses are based on likelihood methods and information theory, and they allow a tight linkage between the models and explicit parameterization of the models from field data. Maximum likelihood methods have a long history of use for point and interval estimation in statistics. In contrast, likelihood principles have only more gradually emerged in ecology as the foundation for an alternative to traditional hypothesis testing. The alternative framework stresses the process of identifying and selecting among competing models, or in the simplest case, among competing point estimates of a parameter of a model. There are four general steps involved in a likelihood analysis: (1) model specification, (2) parameter estimation using maximum likelihood methods, (3) model comparison, and (4) model evaluation. Our goal in this paper is to review recent developments in the use of likelihood methods and modeling for the analysis of neighborhood processes in forest ecosystems. We will focus on a single class of processes, seed dispersal and seedling dispersion, because recent papers provide compelling evidence of the potential power of the approach, and illustrate some of the statistical challenges in applying the methods.