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Sempra Energy funded this study. There are no patents, products in development or marketed products to declare. This does not alter the authors' adherence to all of the PLOS ONE policies on sharing data and materials.

Conceived and designed the experiments: JAT JS JZ. Performed the experiments: JAT JS JZ FW. Analyzed the data: JAT JS. Contributed reagents/materials/analysis tools: JAT JS. Wrote the paper: JAT JS JZ FW RRS RNF. Field data: JS FW.

Advances in digital biotelemetry technologies are enabling the collection of bigger and more accurate data on the movements of free-ranging wildlife in space and time. Although many biotelemetry devices record 3D location data with

Biologists have sought to understand the patterns of space use of individual animals for decades.

Burt

Animal space use can be characterized by the (

Left: California condor with a GPS biologger attached to its patagium. Center: A giant panda telemetered with a GPS collar. Right: A dugong fitted with a tail mounted GPS biologger.

We present a novel 3D movement-based kernel density estimator (MKDE) of animal home ranges and demonstrate the application and value of these estimators using biotelemetry data acquired from endangered animals that occupy aerial, terrestrial, and aquatic spatial domains. We also present a novel MKDE-based approach for calculating the spatio-temporal interaction between two individuals. We show that analyses and visualization using 3D MKDEs can be more informative and yield greater ecological insights than traditional 2D estimators in representing the space use of animals that have a substantive vertical component (

A utilization distribution (UD) describes the probability of an animal location at an arbitrary time during which the animal was observed

LKDEs are criticized for excluding areas that have been used by animals with large data sets (type I errors,

We use animal location data in which each observation includes an _{m}, y_{m}, z_{m}_{m}_{m}_{m}_{m}_{m}_{m-1} < t_{m} < t_{m+1}^{th}_{m}_{m+1}_{m}_{m}^{th}

Projecting a terrestrial animal's UD onto a 2D plane systematically underestimates the area used if the terrain is not a level, flat surface. As the curvature of the terrain increases, the underestimation becomes more severe. Thus, we demonstrate a 2.5D approach for computing home range area that essentially uses a 2D MKDE draped over a 2D elevation raster

We estimate the 3D MKDE using a trivariate normal kernel integrated over time for each observed move step. The kernel describing the probability density at time

where the vector of means is _{m}_{m+1}_{m}_{m+1}^{2})/time. Thus, when preparing the data, we express time in smaller time units such as minutes, which reduces the magnitude of the parameter and helps avoid numerical issues. The move variance parameters ^{2}^{2}

Because we assume that the off-diagonal elements of the covariance matrix are zero and that the variances in the

The density for the Brownian bridge for the ^{th}

Often, an animal's movement is limited in the

An MKDE can be integrated to compute the probability for any area in 2D space or volume in 3D space, but to visualize the 3D MKDEs across space we compute the probability for every voxel (3D cell) on a regular grid in

We index rows as _{x}_{y}_{z}

The probability of an individual being in voxel

In practice, when we set a lower or upper bound on the 3D MKDE in the

Describing the interaction between two individuals is often of interest to ecologists because it relates to social interaction, transmission of infectious disease, and other individual-level ecological processes. Various approaches have been developed for describing interactions between individuals based on movement data and UDs in 2D space _{A}_{B}

All calculations are performed using code written by the authors in R and C++, and use the raster, Rcpp, and ggplot2 packages in R

We determined the voxel or cell probabilities corresponding to contours delineating the minimum volume or area containing a user-specified proportion (e.g., 0.99) of the total UD as follows. First, vectorize the 3D or 2D array containing the voxel or cell probabilities (so that it becomes a 1D array). Second, sort the resulting 1D array in ascending order using a fast sorting algorithm such as quicksort. Third, create a 1D array of the same length containing a cumulative sum of the sorted values in step 2. Fourth, find the index of the entry in the array created in step 3 that most closely matches one minus the user-specified proportion. Fifth, return the voxel or cell probability at that index in the sorted array created in step 2. Every voxel or cell with a probability greater-than-or-equal to this value is included within the contour and excluded otherwise.

We applied 2.5D MKDE to data for a free-ranging giant panda (^{2}. Terrain surface area was computed using a 32 × 25 km, 30-meter resolution digital elevation model (DEM) with elevation ranging from 875 to 3,035 meters in the

We compared home range areas estimated by the 2D MKDE versus our 2.5D MKDE that incorporates this rugged topography. We performed this comparison using all panda locations, locations in the summer at higher elevation, and locations in the winter at lower elevation. We allowed a maximum time between locations (

We applied the 3D MKDE to study free-ranging California condors (^{2} and a Rayleigh distribution to obtain a vertical measurement error variance of ^{2}. The units are programmed with an hourly fix rate between 6:00am and 8:00pm and transmit location data via the Argos network. Research on California condors was approved by the United States Fish and Wildlife Service, the San Diego Zoo IACUC animal welfare committee (Project ID#11-014) and the Instituto Nacional de Ecología, México.

We present several examples using data from California condors reintroduced to Baja, Mexico. In all examples, a 2.5-minute integration time step was used. The avian MKDE was bounded below by elevation based on a DEM raster

First, we illustrate the process of constructing a 3D MKDE. In this example, we used 2,760 GPS location fixes from a 5-year-old adult female California condor collected over 214 days from December 2009 to July 2010 to compute a 3D MKDE at a 216.55 meter resolution. In a second example, we compared 3D MKDEs for an adult female and adult male condor that formed a breeding pair following reintroduction. These condors were tracked during 10 January – 9 March 2011, with locations being collected at the same times at one-hour intervals. In all, 1,132 temporally-matched locations were available for analysis. We used these data to compute 3D MKDEs at 108.28 meter resolution for each individual separately and also the probability of both members of the pair occurring in the same voxel at the same time using the spatio-temporal interaction 3D MKDE. We also use the data from the female of this pair to compare results from the 2D MKDE and 3D MKDE approaches. We compare contours of the 3D MKDE projected onto the 2D plane to contours of a corresponding 2D MKDE. Next, motivated by a 2D MKDE approach by Lewis et al.

We applied our 3D MKDE to study a free-ranging dugong (^{2} core seagrass habitat ^{2}. An Mk9 timed-depth recorder (TDR, Wildlife Computers) was fitted to the satellite tag harness at the base of the dugong's tail to record its dive profile (^{2}) and an accuracy of ±1% and a continuous 1-second sampling interval. The dugong MKDE was bounded below by a 4.3×6.8 km, 10 m resolution bathymetric surface of the core seagrass habitat in depth below the mean sea level, which ranged from 6.1 m to -1.2 m

To examine dugong space use patterns in relation to tidal height, the data set was divided into locations that occurred in five tidal height ranges of 0.5–1.0 m, 1.0–1.5 m, 1.5–2.0 m, 2.0–2.5 m, and 2.5–3.0 m. For each tidal height range, the dugong 3D MKDE was bounded below by the raster describing bathymetry and above by a constant water level above low tide based on the upper limit of the tidal height range. Thus, in this example, we accounted for the temporally dynamic space that dugongs inhabit. We allowed a maximum time between locations (

When using all locations, 650 of the 1,916 total observed locations satisfied the conditions for use in the 2D MKDE and 678 were removed because we concluded that the following location did not represent a large enough displacement to be considered a move step. Based on the 650 move steps, we estimated ^{2}/min. For locations in the winter range, 167 of the 479 total observed locations satisfied the conditions for use in the 2D MKDE and 177 were removed. Based on the 167 move steps, we estimated ^{2}/min. For locations in the summer range, 479 of the 1425 total observed locations satisfied the conditions for use in the 2D MKDE and 500 were removed. Based on the 479 move steps, we estimated ^{2}/min. Twelve locations were not used because they represented migratory movements between the winter and summer ranges.

^{2} (mean = 1024.9 m^{2}, sd = 158.8 m^{2}) and showed a 12.09% to 16.82% relative increase over the estimates based on 2D MKDE (

In A, giant panda GPS locations in its summer (red points) and winter (blue points) ranges are shown in relation to a digital elevation model (DEM). Using the DEM, the surface area of each raster cell is calculated (B). The surface area increases as the color gradient changes from green to red. In C, the observed summer range locations and interpolated move paths (red points and lines) are shown against 2D MKDE contours draped over the DEM. In D, the observed winter range locations and interpolated move paths (blue points and lines) are shown against 2D MKDE contours draped over the DEM. 2D MKDE 99%, 95%, 75%, 50% contours are shown with colors ranging from light to dark green.

Area km^{2} |
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Season | MKDE | 99% | 95% | 75% | 50% | n |

2D | 8.159 | 5.030 | 1.940 | 0.821 | 650 | |

All | 2.5D | 9.285 | 5.746 | 2.214 | 0.940 | |

Change | 13.80 | 14.23 | 14.17 | 14.57 | ||

2D | 1.173 | 0.810 | 0.347 | 0.159 | 167 | |

Winter | 2.5D | 1.370 | 0.944 | 0.402 | 0.183 | |

Change | 16.82 | 16.55 | 15.67 | 14.96 | ||

2D | 5.592 | 3.405 | 1.352 | 0.572 | 479 | |

Summer | 2.5D | 6.289 | 3.826 | 1.515 | 0.642 | |

Change | 12.48 | 12.38 | 12.09 | 12.25 |

In ^{2}/min and ^{2}/min. The kernel was then integrated over time and over each voxel to estimate the utilization probability (

The 3D MKDE is constructed from observed 3D locations and a digital elevation model that sets the lower bound on the MKDE. The expected location (gray points) at each unobserved time is determined by linear interpolation (white lines) between the observations (A). The 3D MKDE is then constructed by integrating a trivariate normal distribution, possibly constrained above or below in the z-dimension, over time along the interpolated movement path (B). The variance of the kernel increases as it moves further from the times of the observed locations. The contours of the final 3D MKDE is shown in C. In B and C, the 99%, 95%, and 50% 3D MKDE volumes are shown in transparent white, orange, and red, respectively.

Next, we considered the breeding pair of condors. Of the 1,132 total observed locations for the female condor, 501 satisfied the conditions for use in the 3D MKDE. Based on the 501 move steps, we estimated ^{2}/min and ^{2}/min. Of the 1132 total observed locations for the male condor, 550 satisfied the conditions for use in the 3D MKDE. Based on the 550 move steps, we estimated ^{2}/min and ^{2}/min. Comparing the side-view of the 3D MKDEs for each condor shows that the female tended to be at lower altitudes when the pair moved into the lower-elevation areas to the east and west of the mountain range where they were most active (

First we illustrate the 99% contours for the female (orange) and male (yellow), shown as a profile view (A) and an overhead view (B). The MKDEs overlap considerably, but the male appears to spend more time at lower elevations when the pair moves into lower elevation areas. The contours for the voxels that contribute 75% and 50% of the total spatio-temporal interaction of the pair are shown in medium and deep purple (B, C). These areas correspond to the reintroduction site where condors were provisioned with carcasses following reintroduction and the nesting site for the pair. When the 99% (white) and 95% (light purple) contours shown, several other areas are included which may also be of ecological interest.

Next, we consider the interaction between historical movements of a subadult female condor and placement of a proposed wind farm. Of the 333 total observed locations, 134 satisfied the conditions for use in the 3D MKDE and 74 were removed because we concluded that the following location did not represent a large enough displacement to consider it a move step. Based on the 134 move steps, we estimated ^{2}/min and ^{2}/min. The 3D MKDE in relation to the proposed wind farm project area is shown in

In 2007, a subadult female condor made an exploratory movement through a proposed wind energy development (A). The proposed wind turbine locations are shown in yellow, and the 99% contour for the condor is shown in red. When approximating the condor's move path by linearly interpolating between observed locations (red lines, A–E), the path passes through the proposed locations of the wind turbines (B). The 3D models of 120 wind turbines are shown (B) in their proposed locations and size (Vestas V112-3.3 turbines with a 84 meter hub height and a 56 meter rotor radius). Using 2D and 3D MKDEs, we estimated the probability that the condor would have passed through cells (54 meters square, C) and voxels (54 meters cubed, D–E) intersecting each turbine. The 99%, 95%,75%,and 50% contours are shown for the 2D MKDE, and the height of blue 3D bars at each turbine location indicate the probability that the condor passed through cells intersected by the turbines (C). The 95%, 75%,and 50% contour volumes are shown for the 3D MKDE (D, the 99% contour was omitted because it covered most of the topography). For comparison to (C), the 99%, 95%,75%,and 50% contours for the three levels of voxels closest to the ground (the approximate height of the turbines) are shown, and red 3D bars at each turbine location indicate the probability that the condor passed through voxels intersected by the turbines (E). The height of the bars in (C) and (E) are on the same relative scale. In general, the probabilities based on the 3D MKDE are lower and more closely related to the observed altitudes of the condor, the possible altitudes it may be at when it is not observed, and the terrain.

We estimated the 2D and 3D MKDEs using the single move step through site (^{2}/min and ^{2}/min. From these MKDEs, we extracted the probabilities associated with each wind turbine to assess the risk the wind farm would have posed to the condor. Based on the 2D MKDE, the turbine encounter probabilities ranged from 5.180e-06 to 8.861e-06, with a mean of 7.988e-06 and standard deviation of 7.141e-07 (

In the final condor example, we relate MKDE probabilities to predicted wind speed using 292 of 433 total locations that satisfied the conditions for use in the 3D MKDE, where 118 move steps were removed because we concluded that they were non-movements. Based on the 292 move steps, we estimated ^{2}/min and ^{2}/min. The resulting 3D MKDE is illustrated in

Like 2D utilization distributions (UDs), 3D UDs can be related to 2D and 3D habitat covariates. In A, we show the 99% and 95% contour volumes for an adult female condor during the month of November, 2009. In B, we show a volume rendering for predicted mean wind speed (meters/second) for November 2009 in voxels 250 m by 250 m by 10 m (x, y, z, respectively) for 0 to 150 m above the ground. Wind speed increases as the color transitions from pale yellow to red. In C, we relate the probability of the condor being in a voxel to the predicted voxel wind speed for each voxel within 150 meters of the Earth's surface. Rug plots (in red) show the marginal distribution of each variable.

Dugong 3D MKDE density is visualized in relation to bathymetry (A). The 99% contour volumes for 3D MKDEs based on locations when tidal heights ranged from 0.5–1.0 (red), 1.0–1.5 (orange), 1.5–2.0 (yellow), 2.0–2.5 (light green), and 2.5–3.0 (green) meters are shown. Based on the 3D MKDEs for each tidal height category, we computed the probability that the dugong would have been at different water depths, grouped in 0.5 meter bins (B). The value on the y-axis is the upper depth value for each 0.5 meter bin (i.e. 0 indicates 0.0–0.5 m depth).

Variance Estimates | Volume (m^{3}) |
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Tidal Height (m) | 99% | 95% | 90% | 75% | 50% | n | ||

0.5–1.0 | 1.197E+02 | 6.329E-05 | 1.185E+06 | 7.296E+05 | 5.370E+05 | 2.836E+05 | 1.130E+05 | 100 |

1.0–1.5 | 3.212E+02 | 5.894E-05 | 3.200E+06 | 1.993E+06 | 1.475E+06 | 7.838E+05 | 2.923E+05 | 145 |

1.5–2.0 | 2.942E+02 | 2.197E-04 | 3.422E+06 | 2.173E+06 | 1.598E+06 | 8.141E+05 | 2.945E+05 | 145 |

2.0–2.5 | 2.043E+02 | 2.527E-03 | 1.772E+06 | 1.074E+06 | 7.849E+05 | 4.144E+05 | 1.623E+05 | 113 |

2.5–3.0 | 8.694E+01 | 2.182E-04 | 4.305E+05 | 2.757E+05 | 2.051E+05 | 1.084E+05 | 4.100E+04 | 45 |

Our 3D home range estimators and visualizations offer considerable theoretical benefits over traditional 2D techniques. First, we were able to visually explore the 3D MKDE volumes of each example species to more intuitively understand how they spatially related to the environmental covariates and bounding layers within their ranges, such as bathymetry or topography. Second, by integrating the vertical component of animal movements into home range estimates, 3D estimators are more accurate and biologically realistic than their 2D counterparts. For example, the giant panda 2D MKDE had a much lower estimate of home range surface area than the 2.5D MKDE that took terrain into account. Home range size is positively associated with extinction risk

The 2.5D MKDEs allow more realistic inferences to be drawn regarding giant panda habitat use. Pandas are known to make use of seasonal food resources that vary with elevation

Dugongs exhibit strong spatial association with seagrass patches of relatively elevated nutritional quality and quantity

Volumetric MKDE home ranges enable condor spatial behaviors to be matched with the environmental covariates, such as wind speed or other climatic conditions that modify flight behaviors, in 3D (

The benefits of 3D home range estimation can extend to other circumstances where there is a vertical component to animal movement. For example, understanding the role of vertical stratification in resource partitioning in arboreal species is an especially promising future application of this technique, and may help explain the high biodiversity found in tropical forests

3D MKDEs can enhance the ecological basis of conservation management strategies for mitigating anthropogenic impacts on threatened populations of vagile wildlife. For example, analyzing the 3D movements of avifauna in relation to the spatio-temporal distribution of aircraft flight paths, power lines, or buildings will provide more accurate estimates of collision risk than 2D models. Improved understanding of the 3D spatial behaviors of the many aquatic animals currently being tracked with biologgers, such as marine turtles, would help managers to minimize their incidental capture by fisheries. 3D MKDEs could also be incorporated into predictive models of wildlife exposure to soil, air and water borne contaminants, and used in simulations of the effects of changing water temperatures, currents or acidification on threatened populations. Although no home range estimator is uniformly superior, our 3D MKDE is a significant step towards Burt's original 1943 concept of a home range and timely leap out of 2D “Flatland”. Wildlife biologists and conservation managers may now analyze their biotelemetry data sets in all three spatial dimensions to visualize and estimate animal space use that can be more realistic, accurate, and informative than those calculated using 2D methods.

We acknowledge the support of Foping National Nature Reserve and field staff. Giant panda research was funded by the National Natural Science Foundation of China (31230011), Wildlife Experimental Platform of Chinese Academy of Sciences and San Diego Zoo Global. We acknowledge the following California condor funding agencies and collaborators: United States Fish and Wildlife Service, Instituto Nacional de Ecologia, Comision Nacional Para El Conocimiento y Uso de la Biodiversidad, Secretaria de Medio Ambiente y Recursos Naturales, Wildcoast/Costasalvaje, Sempra Energy, Michael Wallace and the SDZG condor field team. Funding and in-kind support for dugong research was provided by CRC Reef, Australian Research Council LIEF Scheme, and James Cook University. We thank Helene Marsh, Ivan Lawler and Rhondda Jones and our many field assistants. We thank Jesse Lewis for his helpful discussions related to Brownian bridge models. We thank James Stalker, Regional Earth System Predictability Research (RESPR), Inc., New Mexico for providing wind simulation predictions. We also thank Lisa Nordstrom of San Diego Zoo Global. We thank Julie Yee (USGS), Erin Boydston (USGS), Emily Eisner, Juan Abella, and two anonymous reviewers for their helpful comments. Any use of trade, product, or firm names is for descriptive purposes only and does not imply an endorsement by the U.S. Government.