Continuous-time correlated random walk model for animal telemetry data

We propose a continuous-time version of the correlated random walk model for animal telemetry data. The continuous-time formulation allows data that have been nonuniformly collected over time to be modeled without subsampling, interpolation, or aggregation to obtain a set of locations uniformly spaced in time. The model is derived from a continuous-time Ornstein-Uhlenbeck velocity process that is integrated to form a location process. The continuous-time model was placed into a state-space framework to allow parameter estimation and location predictions from observed animal locations. Two previously unpublished marine mammal telemetry data sets were analyzed to illustrate use of the model, by-products available from the analysis, and different modifications which are possible. A harbor seal data set was analyzed with a model that incorporates the proportion of each hour spent on land. Also, a northern fur seal pup data set was analyzed with a random drift component to account for directed travel and ocean currents.     

Measurement error causes scale-dependent threshold erosion of biological signals in animal movement data.

Recent advances in telemetry technology have created a wealth of tracking data available for many animal species moving over spatial scales from tens of meters to tens of thousands of kilometers. Increasingly, such data sets are being used for quantitative movement analyses aimed at extracting fundamental biological signals such as optimal searching behavior and scale-dependent foraging decisions. We show here that the location error inherent in various tracking technologies reduces the ability to detect patterns of behavior within movements. Our analyses endeavored to set out a series of initial ground rules for ecologists to help ensure that sampling noise is not misinterpreted as a real biological signal. We simulated animal movement tracks using specialized random walks known as Lévy flights at three spatial scales of investigation: 100-km, 10-km, and 1-km maximum daily step lengths. The locations generated in the simulations were then blurred using known error distributions associated with commonly applied tracking methods: the Global Positioning System (GPS), Argos polar-orbiting satellites, and light-level geolocation. Deviations from the idealized Lévy flight pattern were assessed for each track after incrementing levels of location error were applied at each spatial scale, with additional assessments of the effect of error on scale-dependent movement patterns measured using fractal mean dimension and first-passage time (FPT) analyses. The accuracy of parameter estimation (Lévy mu, fractal mean D, and variance in FPT) declined precipitously at threshold errors relative to each spatial scale. At 100-km maximum daily step lengths, error standard deviations of > or = 10 km seriously eroded the biological patterns evident in the simulated tracks, with analogous thresholds at the 10-km and 1-km scales (error SD > or = 1.3 km and 0.07 km, respectively). Temporal subsampling of the simulated tracks maintained some elements of the biological signals depending on error level and spatial scale. Failure to account for large errors relative to the scale of movement can produce substantial biases in the interpretation of movement patterns. This study provides researchers with a framework for understanding the limitations of their data and identifies how temporal subsampling can help to reduce the influence of spatial error on their conclusions.     

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.     

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.     

Quasi-Poisson vs. negative binomial regression: how should we model overdispersed count data?

Quasi-Poisson and negative binomial regression models have equal numbers of parameters, and either could be used for overdispersed count data. While they often give similar results, there can be striking differences in estimating the effects of covariates. We explain when and why such differences occur. The variance of a quasi-Poisson model is a linear function of the mean while the variance of a negative binomial model is a quadratic function of the mean. These variance relationships affect the weights in the iteratively weighted least-squares algorithm of fitting models to data. Because the variance is a function of the mean, large and small counts get weighted differently in quasi-Poisson and negative binomial regression. We provide an example using harbor seal counts from aerial surveys. These counts are affected by date, time of day, and time relative to low tide. We present results on a data set that showed a dramatic difference on estimating abundance of harbor seals when using quasi-Poisson vs. negative binomial regression. This difference is described and explained in light of the different weighting used in each regression method. A general understanding of weighting can help ecologists choose between these two methods.     

Integration of distance, direction and habitat into a predictive migratory movement model for blue-winged teal (Anas discors)

Historically, the migration of birds has been poorly understood in comparison to other life stages during the annual cycle. The goal of our research is to present a novel approach to predict the migratory movement of birds. Using a blue-winged teal case study, our process incorporates not only constraints on habitat (temperature, precipitation, elevation, and depth to water table), but also approximates the likely bearing and distance traveled from a starting location. The method allows for movement predictions to be made from unsampled areas across large spatial scales. We used USGS’ Bird Banding Laboratory database as the source of banding and recovery locations. We used recovery locations from banding sites with multiple within-30-day recoveries were used to build core maximum entropy models. Because the core models encompass information regarding likely habitat, distance, and bearing, we used core models to project (or forecast) probability of movement from starting locations that lacked sufficient data for independent predictions. The final model for an unsampled area was based on an inverse-distance weighted averaged prediction from the three nearest core models. To illustrate this approach, three unsampled locations were selected to probabilistically predict where migratory blue-wing teals would stopover. These locations, despite having little or none data, are assumed to have populations. For the blue-winged teal case study, 104 suitable locations were identified to generate core models. These locations ranged from 20 to 228 within-30-day recoveries, and all core models had AUC scores greater than 0.80. We can infer based on model performance assessment, that our novel approach to predicting migratory movement is well-grounded and provides a reasonable approximation of migratory movement.     

Geolocation of free-ranging fish on the European continental shelf as determined from environmental variables II. Reconstruction of plaice ground tracks

Between December 1993 and February 1997, 302 electronic data storage tags (DSTs), programmed to record depth at 10-min intervals and temperature daily, were attached to mature female plaice, Pleuronectes platessa, and released in the southern North Sea. Fifty tags were returned, 38 of which functioned fully and recorded 2,955 days of data. Twenty-seven tags recorded data over the full period at liberty, and 34 geographical ground tracks were reconstructed. Reconstruction was performed using a two-dimensional tidal stream simulation model that translated vertical movement of fish, recorded by DSTs, into horizontal movement assuming an initial down-tide swimming speed of 0.6 body lengths s^–1. Geographical accuracy of reconstructed tracks was assessed based on closeness of fit between (1) reconstruction endpoint and reported recapture position; (2) reconstructed locations and corresponding locations based on tidal data recorded by DSTs using the tidal location method (TLM; location of areas with similar tidal range and time of high water); and (3) DST temperature records and corresponding averaged sea surface temperature data records for corresponding locations. The results demonstrate that the assumptions of the tidal stream simulation model were sufficient to reconstruct geographically accurate representations of the migrations of individual plaice, which have in turn provided new information on the extent, duration, and directionality of movement. Our study demonstrates how DSTs can provide fishery-independent data with direct management applications in behaviourally driven, individual-based predictive models of fish migration.Communicated by J.P. Thorpe, Port Erin     

Inferring linear feature use in the presence of GPS measurement error

Global Positioning System (GPS) collars are increasingly used to study animal movement and habitat use. Measurement error is defined as the difference between the observed and true value being measured. In GPS data measurement error is referred to as location error and leads to misclassification of observed locations into habitat types. This is particularily true when studying habitats of small spatial extent with large amounts of edge, such as linear features (e.g. roads and seismic lines). However, no consistent framework exists to address the effect of measurement error on habitat classification of observed locations and resulting biological inference. We developed a mechanistic, empirically-based method for buffering linear features that minimizes the underestimation of animal use introduced by GPS measurement error. To do this we quantified the distribution of measurement error and derived an explicit formula for buffer radius which incorporated the error distribution, the width of the linear feature, and a predefined amount of acceptable type I error in location classification. In our empirical study we found the GPS measurement error of the Lotek GPS_3300 collar followed a bivariate Laplace distribution with parameter ρ = 0.1123. When we applied our method to a simulated landscape, type I error was reduced by 57%. This study highlights the need to address the effect of GPS measurement error in animal location classification, particularily for habitats of small spatial extent.     

Estimating behavioral parameters in animal movement models using a state-augmented particle filter

Data on fine-scale animal movement are being collected worldwide, with the number of species being tagged and the resolution of data rapidly increasing. In this study, a general methodology is proposed to understand the patterns in these high-resolution movement time series that relate to marine animal behavior. The approach is illustrated with dive data from a northern fur seal (Callorhinus ursinus) tagged on the Pribilof Islands, Alaska, USA. We apply a state-space model composed of a movement model and corresponding high-resolution vertical movement data. The central goal is to estimate parameters of this movement model, particularly their variation on appropriate time scales, thereby providing a direct link to behavior. A particle filter with state augmentation is used to jointly estimate the movement parameters and the state. A multiple iterated filter using overlapping data segments is implemented to match the parameter time scale with the behavioral inference. The time variation in the auto-covariance function facilitates identification of a movement model, allows separation of observation and process noise, and provides for validation of results. The analysis yields fitted parameters that show distinct time-evolving changes in fur seal behavior over time, matching well what is observed in the original data set.     

Estimated home ranges can misrepresent habitat relationships on patchy landscapes

Home ranges of animals are generally structured by the selective use of resource-bearing patches that comprise habitat. Based on this concept, home ranges of animals estimated from location data are commonly used to infer habitat relationships. Because home ranges estimated from animal locations are largely continuous in space, the resource-bearing patches selected by an animal from a fragmented distribution of patches would be difficult to discern; unselected patches included in the home range estimate would bias an understanding of important habitat relationships. To evaluate potential for this bias, we generated simulated home ranges based on optimal selection of resource-bearing patches across a series of simulated resource distributions that varied in the spatial continuity of resources. For simulated home ranges where selected patches were spatially disjunct, we included interstitial, unselected cells most likely to be traveled by an animal moving among selected patches. We compared characteristics of the simulated home ranges with and without interstitial patches to evaluate how insights derived from field estimates can differ from actual characteristics of home ranges, depending on patchiness of landscapes. Our results showed that contiguous home range estimates could lead to misleading insights on the quality, size, resource content, and efficiency of home ranges, proportional to the spatial discontinuity of resource-bearing patches. We conclude the potential bias of including unselected, largely irrelevant patches in the field estimates of home ranges of animals can be high, particularly for home range estimators that assume uniform use of space within home range boundaries. Thus, inferences about the habitat relationships that ultimately define an animal's home range can be misleading where animals occupy landscapes with patchily distributed resources.     

A new index for measuring perpendicularity of animal movements in relation to patch boundaries

We introduce a new index for measuring perpendicularity of animal movements with respect to a boundary (e.g., a habitat patch edge), and provide a computer algorithm for its calculation. Our index, η, improves on an approach that measures perpendicularity with respect to a fixed boundary direction. This is because η accounts for moment-to-moment trajectories relative to nearest-neighbor boundary attributes at the scale of an animal's movement. Our algorithm prp calculates η efficiently and accurately with both synthetic data and large telemetry datasets. In addition, we have included routines in prp which account for scenarios inherently problematic to perpendicularity estimators.     

Detecting an orientation component in animal paths when the preferred direction is individual-dependent

An orientation component leads to directionally biased paths, with major consequences in animal population redistribution. Classical orientation analyses, which focus on the overall direction of motion, are useless for detecting such a component when the preferred direction is not common to the whole population, but differs from one path to another. In-depth path analyses are required in this case. They consist of determining whether paths are more suitably represented as biased or unbiased random walks. The answer is not easy because most animals' paths show some forward persistence propensity that acts as a purely local directional bias and, hence, blurs the possible occurrence of an additional, consistent bias in a preferred direction. I highlight the key differences between biased and unbiased random walks and the different ways orientation mechanisms can generate a consistent directional bias. I then examine the strength and weakness of the available methods likely to detect it. Finally, I introduce a new procedure based on the backward evolution of the beeline distance, from the end of the path, which might correspond to a goal toward which the animal orients itself, to each of the animal's preceding locations. This new procedure proves to be very efficient, as it requires only a small sample of short paths for detecting a possible orientation component.     

Reconstructing the movements of free-ranging demersal fish in the North Sea: a data-matching and simulation method

Demersal fish cannot be readily tracked using data loggers that provide satellite-based or light-based geolocation. Moreover, fish that are highly mobile within the water column cannot readily be located with other methods, such as the tidal location method (TLM). As an alternative, we describe a process that provides estimates of geographic location by simulating movement paths through geographic locations that match temperature and depth data recorded by data loggers. Depths and temperatures recorded by data loggers were compared with a North Sea temperature and depth database to identify all locations with matching data. A movement rate filter was then applied to eliminate spurious locations and simulations of possible movement paths through the remaining positions were used to generate estimates of the likelihood of a particular location having been occupied. The performance of the technique was assessed by reconstructing movement paths of artificial migrations and by using depth and temperature data collected at known locations in the North Sea. Estimates of the positional accuracy and error were comparable to the North Sea TLM. Reconstructions of the migrations of cod tagged and released in the North Sea were successfully achieved with the method. This method has application in defining the movements and migrations of commercial species in any sea area where databases of commonly measured environmental variables are available.     

Mitigation of Habitat "Take": Application to Habitat Conservation Planning

One of the most important provisions of the U.S. Endangered Species Act precludes the "taking" of listed species on both public and private land. In past Endangered Species Act litigation, take has been broadly interpreted to include the destruction or modification of habitats as well as the direct killing of animals. This requirement created an extensive burden on private landowners to provide habitats for listed species. This burden was substantially lessened when the ESA was modified in 1982 to allow incidental takings conditioned on preparation of a satisfactory "habitat conservation plan." Because the majority of listed species are imperiled due to habitat modification, most habitat conservation plans must demonstrate defensible methods to mitigate against incidental habitat loss. A review of HCPs for the Northern Spotted Owl ( Strix occidentalis), and other species, indicates that mitigation solutions are often arbitrary, lacking an empirical foundation in the species' life history requirements. Based on data from the Spotted Owl, we illustrate a biologically based method for estimating the areal requirements necessary to mitigate against the take of essential habitats. Toward this goal we adopt the concept of "core area," that portion of an animal's home range that receives disproportionate use. We estimated core areas by means of the adaptive kernel density function and tested against a null distribution of animal use that assumes a bivariate, uniform distribution of locations within the home range. The method we illustrate, which is defensible, repeatable, and empirical, is a clear improvement over the ad hoc methods used in many habitat conservation plans. Further, the methods we propose should be applicable to a large number of terrestrial species for which home range is a meaningful concept.     

Minimum area requirements for an at‐risk butterfly based on movement and demography

Determining the minimum area required to sustain populations has a long history in theoretical and conservation biology. Correlative approaches are often used to estimate minimum area requirements (MARs) based on relationships between area and the population size required for persistence or between species’ traits and distribution patterns across landscapes. Mechanistic approaches to estimating MAR facilitate prediction across space and time but are few. We used a mechanistic MAR model to determine the critical minimum patch size (CMP) for the Baltimore checkerspot butterfly (Euphydryas phaeton), a locally abundant species in decline along its southern range, and sister to several federally listed species. Our CMP is based on principles of diffusion, where individuals in smaller patches encounter edges and leave with higher probability than those in larger patches, potentially before reproducing. We estimated a CMP for the Baltimore checkerspot of 0.7–1.5 ha, in accordance with trait‐based MAR estimates. The diffusion rate on which we based this CMP was broadly similar when estimated at the landscape scale (comparing flight path vs. capture‐mark‐recapture data), and the estimated population growth rate was consistent with observed site trends. Our mechanistic approach to estimating MAR is appropriate for species whose movement follows a correlated random walk and may be useful where landscape‐scale distributions are difficult to assess, but demographic and movement data are obtainable from a single site or the literature. Just as simple estimates of lambda are often used to assess population viability, the principles of diffusion and CMP could provide a starting place for estimating MAR for conservation.     

Inferring ecological and behavioral drivers of African elephant movement using a linear filtering approach

Understanding the environmental factors influencing animal movements is fundamental to theoretical and applied research in the field of movement ecology. Studies relating fine-scale movement paths to spatiotemporally structured landscape data, such as vegetation productivity or human activity, are particularly lacking despite the obvious importance of such information to understanding drivers of animal movement. In part, this may be because few approaches provide the sophistication to characterize the complexity of movement behavior and relate it to diverse, varying environmental stimuli. We overcame this hurdle by applying, for the first time to an ecological question, a finite impulse-response signal-filtering approach to identify human and natural environmental drivers of movements of 13 free-ranging African elephants (Loxodonta africana) from distinct social groups collected over seven years. A minimum mean-square error (MMSE) estimation criterion allowed comparison of the predictive power of landscape and ecological model inputs. We showed that a filter combining vegetation dynamics, human and physical landscape features, and previous movement outperformed simpler filter structures, indicating the importance of both dynamic and static landscape features, as well as habit, on movement decisions taken by elephants. Elephant responses to vegetation productivity indices were not uniform in time or space, indicating that elephant foraging strategies are more complex than simply gravitation toward areas of high productivity. Predictions were most frequently inaccurate outside protected area boundaries near human settlements, suggesting that human activity disrupts typical elephant movement behavior. Successful management strategies at the human-elephant interface, therefore, are likely to be context specific and dynamic. Signal processing provides a promising approach for elucidating environmental factors that drive animal movements over large time and spatial scales.     

Effects of Weighting Schemes on the Identification of Wildlife Corridors Generated with Least‐Cost Methods

The importance of movement corridors for maintaining connectivity within metapopulations of wild animals is a cornerstone of conservation. One common approach for determining corridor locations is least‐cost corridor (LCC) modeling, which uses algorithms within a geographic information system to search for routes with the lowest cumulative resistance between target locations on a landscape. However, the presentation of multiple LCCs that connect multiple locations generally assumes all corridors contribute equally to connectivity, regardless of the likelihood that animals will use them. Thus, LCCs may overemphasize seldom‐used longer routes and underemphasize more frequently used shorter routes. We hypothesize that, depending on conservation objectives and available biological information, weighting individual corridors on the basis of species‐specific movement, dispersal, or gene flow data may better identify effective corridors. We tested whether locations of key connectivity areas, defined as the highest 75th and 90th percentile cumulative weighted value of approximately 155,000 corridors, shift under different weighting scenarios. In addition, we quantified the amount and location of private land that intersect key connectivity areas under each weighting scheme. Some areas that appeared well connected when analyzed with unweighted corridors exhibited much less connectivity compared with weighting schemes that discount corridors with large effective distances. Furthermore, the amount and location of key connectivity areas that intersected private land varied among weighting schemes. We believe biological assumptions and conservation objectives should be explicitly incorporated to weight corridors when assessing landscape connectivity. These results are highly relevant to conservation planning because on the basis of recent interest by government agencies and nongovernmental organizations in maintaining and enhancing wildlife corridors, connectivity will likely be an important criterion for prioritization of land purchases and swaps. Efectos de los Esquemas de Ponderación sobre la Identificación de Corredores para Vida Silvestre Generados con Métodos Menos Costosos     

Markov Modulated Poisson Processes for Clustered Line Transect Data

We model the points of the detection along the transect line by a Markov modulated Poisson process (MMPP). The MMPP can accommodate the spatial cluster structure typical of many line transect surveys. The basic idea is that animal density switches between a low and a high level according to a latent Markov process. The MMPP is attractive from a mathematical point of view, as it provides an explicit expression for the likelihood function and other important quantities. We focus on estimating the level of overdispersion in the number of detected animals, as this is important for quantifying the precision of the line transect estimator of animal abundance. The approach is illustrated using both simulated data and data from a minke whale sighting survey conducted in the North Atlantic. Received: August 2004 / Revised: August 2005