Kernel-based home range method for data with irregular sampling intervals

Studies of habitat selection and movements often use radio-tracking data for defining animal home ranges. Home ranges (HR) can be approximated by a utilization density distribution (UD) that instead of assuming uniform use of areas within HR boundary provides a probabilistic measure of animal space use. In reality, radio-tracking data contain periods of frequent autocorrelated observations interspersed with temporally more independent observations. Using such temporally irregular data directly may result in biased UD estimates, because areas that have been sampled intensively receive too much weight. The problem of autocorrelation has been tackled by resampling data with an appropriate time interval. However, resampling may cause a large reduction in the data set size along with a loss of information. Evidently, biased UD estimates or reduction in data may prejudice the results on animal habitat selection and movement. We introduce a new method for estimating UDs with temporally irregular data. The proposed method, called the time kernel, accounts for temporal aggregation of observations and gives less weight to temporally autocorrelated observations. A further extension of the method accounts also for spatially aggregated observations with relatively low weights given to observations that are both temporally and spatially aggregated. We test the behaviour of the time kernel method and its spatiotemporal version using simulated data. In addition, the method is applied to a data set of brown bear locations.     

Kernel density estimators of home range: smoothing and the autocorrelation red herring

Two oft-cited drawbacks of kernel density estimators (KDEs) of home range are their sensitivity to the choice of smoothing parameter(s) and their need for independent data. Several simulation studies have been conducted to compare the performance of objective, data-based methods of choosing optimal smoothing parameters in the context of home range and utilization distribution (UD) estimation. Lost in this discussion of choice of smoothing parameters is the general role of smoothing in data analysis, namely, that smoothing serves to increase precision at the cost of increased bias. A primary goal of this paper is to illustrate this bias-variance trade-off by applying KDEs to sampled locations from simulated movement paths. These simulations will also be used to explore the role of autocorrelation in estimating UDs. Autocorrelation can be reduced (1) by increasing study duration (for a fixed sample size) or (2) by decreasing the sampling rate. While the first option will often be reasonable, for a fixed study duration higher sampling rates should always result in improved estimates of space use. Further, KDEs with typical data-based methods of choosing smoothing parameters should provide competitive estimates of space use for fixed study periods unless autocorrelation substantially alters the optimal level of smoothing.     

Estimating animal densities and home range in regions with irregular boundaries and holes: A lattice-based alternative to the kernel density estimator

Density estimates based on point processes are often restrained to regions with irregular boundaries or holes. We propose a density estimator, the lattice-based density estimator, which produces reasonable density estimates under these circumstances. The estimation process starts with overlaying the region with nodes, linking these together in a lattice and then computing the density of random walks of length k on the lattice. We use an approximation to the unbiased crossvalidation criterion to find the optimal walk length k. The technique is illustrated using walleye (Sander vitreus) radiotelemetry relocations in Lake Monroe, Indiana. We also use simulation to compare the technique to the traditional kernel density estimate in the situation where there are no significant boundary effects.     

Density estimation in tiger populations: combining information for strong inference

A productive way forward in studies of animal populations is to efficiently make use of all the information available, either as raw data or as published sources, on critical parameters of interest. In this study, we demonstrate two approaches to the use of multiple sources of information on a parameter of fundamental interest to ecologists: animal density. The first approach produces estimates simultaneously from two different sources of data. The second approach was developed for situations in which initial data collection and analysis are followed up by subsequent data collection and prior knowledge is updated with new data using a stepwise process. Both approaches are used to estimate density of a rare and elusive predator, the tiger, by combining photographic and fecal DNA spatial capture-recapture data. The model, which combined information, provided the most precise estimate of density (8.5 +/- 1.95 tigers/100 km2 [posterior mean +/- SD]) relative to a model that utilized only one data source (photographic, 12.02 +/- 3.02 tigers/100 km2 and fecal DNA, 6.65 +/- 2.37 tigers/100 km2). Our study demonstrates that, by accounting for multiple sources of available information, estimates of animal density can be significantly improved.     

Hierarchical Bayes estimation of hunting success rates

Although most post-season harvest surveys are conducted at the state level, the effective management of wildlife populations often requires estimates of hunting success rate, hunting pressure and harvest at the sub-area (such as management unit, regional, or county) level.Sample sizes for some sub-areas are often very small or even zero. Because of small sample sizes, estimates for small sub-areas often yield unacceptably large standard errors. In this article, a hierarchical Bayes model is used to estimate hunting success rates at the sub-area level from post-season harvest surveys. The computation is done by Gibbs sampling and adaptive rejection sampling techniques. The method is illustrated using data from the Missouri Turkey Hunting Survey 1994 Spring Season. The Bayesian estimates are close to the frequency estimates for the sub-areas with large sample sizes and more stable than the frequency estimates for those with small sample sizes. The Bayesian estimates will be more useful to wildlife biologists in estab-lishing hunting regulation on small sub-areas at no additional survey cost.     

Estimation of population parameters from aerial counts of North American mallards: a cautionary tale.

The accuracy of population estimates strongly interferes with our ability to obtain unbiased estimates of population parameters based on analyses of time series of population fluctuations. Here we use long-term data on fluctuations in the size of Mallard populations collected as part of the May Breeding Waterfowl Survey covering a large section of North America. We assume a log-linear model of density dependence and use a hierarchical Bayesian state-space approach in which all parameters are assumed to be realizations from a common underlying distribution. Thus, parameters for different populations are not allowed to vary independently of each other. We then simulated independent time series of aerial counts, using the estimated parameters and adding various levels of observation error. These simulations showed that the estimates of stochastic population growth rate and strength of density dependence were biased even when moderate sampling errors were present. In contrast, the estimates of the environmental stochasticity and the carrying capacity were unbiased even for short time series and large observation error. Our results underline the importance of reducing the magnitude of sampling error in the design of large-scale monitoring programs of population fluctuations.     

Evaluating orangutan census techniques using nest decay rates: implications for population estimates.

An accurate estimate for orangutan nest decay time is a crucial factor in commonly used methods for estimating orangutan population size. Decay rates are known to vary, but the decay process and, thus, the temporal and spatial variation in decay time are poorly understood. We used established line-transect methodology to survey orangutan nests in a lowland forest in East Kalimantan, Indonesia, and monitored the decay of 663 nests over 20 months. Using Markov chain analysis we calculated a decay time of 602 days, which is significantly longer than times found in other studies. Based on this, we recalculated the orangutan density estimate for a site in East Kalimantan; the resulting density is much lower than previous estimates (previous estimates were 3-8 times higher than our recalculated density). Our data suggest that short-term studies where decay times are determined using matrix mathematics may produce unreliable decay times. Our findings have implications for other parts of the orangutan range where population estimates are based on potentially unreliable nest decay rate estimates, and we recommend that for various parts of the orangutan range census estimates be reexamined. Considering the high variation in decay rates there is a need to move away from using single-number decay time estimates and, preferably, to test methods that do not rely on nest decay times as alternatives for rapid assessments of orangutan habitat for conservation in Borneo.     

Reconstructibility of Density Dependence and the Conservative Assessment of Extinction Risks

Abstract: The probability of extinction is sensitive to the presence and character of density dependence controlling the dynamics of a population. This means that our capacity to estimate a population's risks of extinction under varying environmental conditions or competing management regimes is linked to our ability to reconstruct from data the density-dependence relationships governing the natural dynamics, especially when data do not reveal a trend of population growth or decline. In an example using Gadus morhua , we show that even 10- or 20-year data sets are too short to make precise estimates of these risks. We also observe, however, that under moderate or weak density dependence, the computed risks are lower than when density dependence is not included in the model. We propose, therefore, that when available data sets are insufficient for reconstructing reliable measurements of density dependence, conservative estimates of extinction probabilities can be made from models that simply omit density dependence.     

Using the bootstrap with two-phase adaptive stratified samples from multiple populations at multiple locations

Recently the two-phase adaptive stratified sampling design proposed by Francis (1984) has been extended by Manly et al. (2002) for situations where several biological populations are sampled simultaneously, and where this is done at several different geographical locations in order to estimate population totals or means. The method uses the results from a first phase sample to decide how best to allocate a second phase sample to locations and strata, in order to maximise a criterion (based on estimated coefficients of variation) that measures the accuracy of estimation for population totals, for all variables at all locations. One potential problem with this method is bias in the estimators of the population totals and means. In this paper bootstrapping is considered as a means of overcoming these biases. It is shown using model populations of Pacific walrus and shellfish, based on real data, that bootstrapping is a useful tool for removing about half of the bias. This is also confirmed from some simulations using artificial data.     

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.     

Toward accurate and precise estimates of lion density

Reliable estimates of animal density are fundamental to understanding ecological processes and population dynamics. Furthermore, their accuracy is vital to conservation because wildlife authorities rely on estimates to make decisions. However, it is notoriously difficult to accurately estimate density for wide‐ranging carnivores that occur at low densities. In recent years, significant progress has been made in density estimation of Asian carnivores, but the methods have not been widely adapted to African carnivores, such as lions (Panthera leo). Although abundance indices for lions may produce poor inferences, they continue to be used to estimate density and inform management and policy. We used sighting data from a 3‐month survey and adapted a Bayesian spatially explicit capture‐recapture (SECR) model to estimate spatial lion density in the Maasai Mara National Reserve and surrounding conservancies in Kenya. Our unstructured spatial capture‐recapture sampling design incorporated search effort to explicitly estimate detection probability and density on a fine spatial scale, making our approach robust in the context of varying detection probabilities. Overall posterior mean lion density was estimated to be 17.08 (posterior SD 1.310) lions >1 year old/100 km², and the sex ratio was estimated at 2.2 females to 1 male. Our modeling framework and narrow posterior SD demonstrate that SECR methods can produce statistically rigorous and precise estimates of population parameters, and we argue that they should be favored over less reliable abundance indices. Furthermore, our approach is flexible enough to incorporate different data types, which enables robust population estimates over relatively short survey periods in a variety of systems. Trend analyses are essential to guide conservation decisions but are frequently based on surveys of differing reliability. We therefore call for a unified framework to assess lion numbers in key populations to improve management and policy decisions.     

Density surface modelling with line transect sampling as a tool for abundance estimation of marine benthic species: the <Emphasis Type="Italic">Pinna nobilis</Emphasis> example in a marine lake

Density estimation of marine benthic fauna is most often conducted with fishery surveys using dredges or trawls. These estimates are often unreliable due to low and variable efficiency and are inappropriate when dealing with rare or endangered species. In the marine Lake Vouliagmeni, a density surface modelling (DSM) approach using survey data from line transects, integrated with a Geographic Information System (GIS), was used to estimate the population density of the endangered fan mussel Pinna nobilis. This is the first time that such an approach has been applied for a marine benthic species. DSM was beneficial in relation to traditional distance sampling. Apart from providing a more precise total abundance estimate, it related the density of the species to spatial covariates of interest, gave a depiction of the species dispersion in the study area, and provided abundance estimates in any sub-region of the study area. In Lake Vouliagmeni, a marked zonation of P. nobilis distribution was revealed, with the species being restricted in the shallow peripheral zone at depths <22 m. Two density peaks were observed, a major peak at depths between 12 and 13 m and a secondary peak at ∼4 m. A main hotspot of high density was also observed in the northeastern part of the lake. Total abundance of the species was estimated to be 6,770 individuals with a 95% confidence interval of 5,460–8,393 individuals.     

The effects of plant distribution and frugivore density on the scale and shape of dispersal kernels

For many plant species, seed dispersal is one of the most important spatial demographic processes. We used a diffusion approximation and a spatially explicit simulation model to explore the mechanisms generating seed dispersal kernels for plants dispersed by frugivores. The simulation model combined simple movement and foraging rules with seed gut passage time, plant distribution, and fruit production. A simulation experiment using plant spatial aggregation and frugivore density as factors showed that seed dispersal scale was largely determined by the degree of plant aggregation, whereas kernel shape was mostly dominated by frugivore density. Kernel shapes ranged from fat tailed to thin tailed, but most shapes were between an exponential and that of the solution of a diffusion equation. The proportion of dispersal kernels with fat tails was highest for landscapes with clumped plant distributions and increased with increasing number of dispersers. The diffusion model provides a basis for models including more behavioral details but can also be used to approximate dispersal kernels once a diffusion rate is estimated from animal movement data. Our results suggest that important characteristics of dispersal kernels will depend on the spatial pattern of plant distribution and on disperser density when frugivores mediate seed dispersal.     

Local spatial modeling of white-tailed deer distribution

Complex spatial heterogeneity of ecological systems is difficult to capture and interpret using global models alone. For this reason, recent attention has been paid to local spatial modeling techniques. We used one local modeling approach, geographically weighted regression (GWR), to investigate the effects of local spatial heterogeneity on multivariate relationships of white-tailed deer distribution using land cover patch metrics and climate factors. The results of these analyses quantify differences in the contributions of model parameters to estimates of deer density over space. A GWR model with local kernel bandwidth was compared to a GWR model with global kernel bandwidth and an ordinary least-squares regression (OLS) model with the same parameters to evaluate their relative abilities in modeling deer distributions. The results indicated that the GWR models predicted deer density better than the traditional ordinary least-squares model and also provided useful information regarding local environmental processes affecting deer distribution. GWR model comparisons showed that the local kernel bandwidth GWR model was more realistic than the global kernel bandwidth GWR model, as the latter exaggerated local spatial variation. The parameter estimates and model statistics (e.g., model R²) of the GWR models were mapped using geographic information systems (GIS) to illustrate local spatial variation in the regression relationship and to identify causes of large-scale model misspecifications and low estimation efficiencies.     

Density derived estimates of standing crop and net primary production in the giant kelp <Emphasis Type="Italic">Macrocystis pyrifera</Emphasis>

Assemblages of macroalgae are believe to be among the most productive ecosystems in the world, yet difficulties in obtaining direct estimates of biomass and primary production have led to few macroalgal data sets from which the consequences of long-term change can be assessed. We evaluated the validity of using two easily measured population variables (frond density and plant density) to estimate the more difficult to measure variables of standing crop and net primary production (NPP) in the giant kelp Macrocystis pyrifera off southern California. Standing crop was much more strongly correlated to frond density than to plant density. Frond density data collected in summer were particularly useful for estimating annual NPP, explaining nearly 80% of the variation in the NPP from year to year. Data on frond densities also provided a relatively good estimate of seasonal NPP for the season that the data were collected. In contrast, estimates of seasonal and annual NPP derived from plant density data were less reliable. These results indicate that data on frond density collected at the proper time of year can make assessments of NPP by giant kelp more tractable. They also suggest that other easily measured variables that are strongly correlated with standing crop, such as surface canopy area, might serve as similarly useful proxies of NPP.     

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.     

Estimating Sonoran pronghorn abundance and survival with fecal DNA and capture–recapture methods

Population abundance estimates are important for management but can be challenging to determine in low‐density, wide‐ranging, and endangered species, such as Sonoran pronghorn (Antilocapra americana sonoriensis). The Sonoran pronghorn population has been increasing; however, population estimates are currently derived from a biennial aerial count that does not provide survival or recruitment estimates. We identified individuals through noninvasively collected fecal DNA and used robust‐design capture–recapture to estimate abundance and survival for Sonoran pronghorn in the United States from 2013 to 2014. In 2014 we generated separate population estimates for pronghorn gathered near 13 different artificial water holes and for pronghorn not near water holes. The population using artificial water holes had 116 (95% CI 102–131) and 121 individuals (95% CI 112–132) in 2013 and 2014, respectively. For all locations, we estimated there were 144 individuals (95% CI 132–157). Adults had higher annual survival probabilities (0.83, 95% CI 0.69–0.92) than fawns (0.41, 95% CI 0.21–0.65). Our use of targeted noninvasive genetic sampling and capture–recapture with Sonoran pronghorn fecal DNA was an effective method for monitoring a large proportion of the population. Our results provided the first survival estimates for this population in over 2 decades and precise estimates of the population using artificial water holes. Our method could be used for targeted sampling of broadly distributed species in other systems, such as in African savanna ecosystems, where many species congregate at watering sites.     

Bayes Estimation of a Distribution Function Using Ranked Set Samples

Aranked set sample (RSS), if not balanced, is simply a sample of independent order statistics gener- ated from the same underlying distribution F. Kvam and Samaniego (1994) derived maximum likelihood estimates of F for a general RSS. In many applications, including some in the environ- mental sciences, prior information about F is available to supplement the data-based inference. In such cases, Bayes estimators should be considered for improved estimation. Bayes estimation (using the squared error loss function) of the unknown distribution function F is investigated with such samples. Additionally, the Bayes generalized maximum likelihood estimator (GMLE) is derived. An iterative scheme based on the EM Algorithm is used to produce the GMLE of F. For the case of squared error loss, simple solutions are uncommon, and a procedure to find the solution to the Bayes estimate using the Gibbs sampler is illustrated. The methods are illustrated with data from the Natural Environmental Research Council of Great Britain (1975), representing water discharge of floods on the Nidd River in Yorkshire, England     

Modeling resource selection using polytomous logistic regression and kernel density estimates

Wildlife resource selection studies typically compare used to available resources; selection or avoidance occurs when use is disproportionately greater or less than availability. Comparing used to available resources is problematic because results are often greatly influenced by what is considered available to the animal. Moreover, placing relocation points within resource units is often difficult due to radiotelemetry and mapping errors. Given these problems, we suggest that an animal’s resource use be summarized at the scale of the home range (i.e., the spatial distribution of all point locations of an animal) rather than by individual points that are considered used or available. To account for differences in use-intensity throughout an animal’s home range, we model resource selection using kernel density estimates and polytomous logistic regression. We present a case study of elk (Cervus elaphus) resource selection in South Dakota to illustrate the procedure. There are several advantages of our proposed approach. First, resource availability goes undefined by the investigator, which is a difficult and often arbitrary decision. Instead, the technique compares the intensity of animal use throughout the home range. This technique also avoids problems with classifying locations rigidly as used or unused. Second, location coordinates do not need to be placed within mapped resource units, which is problematic given mapping and telemetry error. Finally, resource use is considered at an appropriate scale for management because most wildlife resource decisions are made at the level of the patch. Despite the advantages of this use-intensity procedure, future research should address spatial autocorrelation and develop spatial models for ordered categorical variables.     

Multimethod,multistate Bayesian hierarchical modeling approach for use in regional monitoring of wolves

In many cases, the first step in large‐carnivore management is to obtain objective, reliable, and cost‐effective estimates of population parameters through procedures that are reproducible over time. However, monitoring predators over large areas is difficult, and the data have a high level of uncertainty. We devised a practical multimethod and multistate modeling approach based on Bayesian hierarchical‐site‐occupancy models that combined multiple survey methods to estimate different population states for use in monitoring large predators at a regional scale. We used wolves (Canis lupus) as our model species and generated reliable estimates of the number of sites with wolf reproduction (presence of pups). We used 2 wolf data sets from Spain (Western Galicia in 2013 and Asturias in 2004) to test the approach. Based on howling surveys, the naïve estimation (i.e., estimate based only on observations) of the number of sites with reproduction was 9 and 25 sites in Western Galicia and Asturias, respectively. Our model showed 33.4 (SD 9.6) and 34.4 (3.9) sites with wolf reproduction, respectively. The number of occupied sites with wolf reproduction was 0.67 (SD 0.19) and 0.76 (0.11), respectively. This approach can be used to design more cost‐effective monitoring programs (i.e., to define the sampling effort needed per site). Our approach should inspire well‐coordinated surveys across multiple administrative borders and populations and lead to improved decision making for management of large carnivores on a landscape level. The use of this Bayesian framework provides a simple way to visualize the degree of uncertainty around population‐parameter estimates and thus provides managers and stakeholders an intuitive approach to interpreting monitoring results. Our approach can be widely applied to large spatial scales in wildlife monitoring where detection probabilities differ between population states and where several methods are being used to estimate different population parameters.