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Understanding animal movement is a key challenge in ecology and conservation biology. Relocation data often represent a complex mixture of different movement behaviors, and reliably decomposing this mix into its component parts is an unresolved problem in movement ecology. Traditional approaches, such as composite random walk models, require that the timescales characterizing the movement are all similar to the usually arbitrary data-sampling rate. Movement behaviors such as long-distance searching and fine-scale foraging, however, are often intermixed but operate on vastly different spatial and temporal scales. An approach that integrates the full sweep of movement behaviors across scales is currently lacking. Here we show how the semivariance function (SVF) of a stochastic movement process can both identify multiple movement modes and solve the sampling rate problem. We express a broad range of continuous-space, continuous-time stochastic movement models in terms of their SVFs, connect them to relocation data via variogram regression, and compare them using standard model selection techniques. We illustrate our approach using Mongolian gazelle relocation data and show that gazelle movement is characterized by ballistic foraging movements on a 6-h timescale, fast diffusive searching with a 10-week timescale, and asymptotic diffusion over longer timescales.

Climatic variability, resource availability, and anthropogenic impacts heavily influence an animal's home range. This makes home range size an effective metric for understanding how variation in environmental factors alter the behavior and spatial distribution of animals. In this study, we estimated home range size of African elephants (Loxodonta africana) across four sites in Namibia, along a gradient of precipitation and human impact, and investigated how these gradients influence the home range size on regional and site scales. Additionally, we estimated the time individuals spent within protected area boundaries. The mean 50% autocorrelated kernel density estimate for home range was 2200 km2 [95% CI:1500–3100 km2]. Regionally, precipitation and vegetation were the strongest predictors of home range size, accounting for a combined 53% of observed variation. However, different environmental covariates explained home range variation at each site. Precipitation predicted most variation (up to 74%) in home range sizes (n = 66) in the drier western sites, while human impacts explained 71% of the variation in home range sizes (n = 10) in Namibia's portion of the Kavango‐Zambezi Transfrontier Conservation Area. Elephants in all study areas maintained high fidelity to protected areas, spending an average of 85% of time tracked on protected lands. These results suggest that while most elephant space use in Namibia is driven by natural dynamics, some elephants are experiencing changes in space use due to human modification. We analyze the largest dataset of collared elephants ever recorded in Namibia to determine the relative importance of humans and environmental variables in shaping elephant movements. We found that human activities are influencing elephant space use at some sites, though regional trends are still driven by precipitation and vegetation productivity.

Background Periodicity in activity level (rest/activity cycles) is ubiquitous in nature, but whether and how these periodicities translate into periodic patterns of space use by animals is much less documented. Here we introduce an analytical protocol based on the Lomb-Scargle periodogram (LSP) to facilitate exploration of animal tracking datasets for periodic patterns. The LSP accommodates missing observations and variation in the sampling intervals of the location time series. Results We describe a new, fast algorithm to compute the LSP. The gain in speed compared to other R implementations of the LSP makes it tractable to analyze long datasets (>10 6 records). We also give a detailed primer on periodicity analysis, focusing on the specificities of movement data. In particular, we warn against the risk of flawed inference when the sampling schedule creates artefactual periodicities and we introduce a new statistical test of periodicity that accommodates temporally autocorrelated background noise. Applying our LSP-based analytical protocol to tracking data from three species revealed that an ungulate exhibited periodicity in its movement speed but not in its locations, that a central place-foraging seabird tracked moon phase, and that the movements of a range-resident canid included a daily patrolling component that was initially masked by the stochasticity of the movements. Conclusion The new, fast algorithm tailored for movement data analysis and now available in the R-package ctmm makes the LSP a convenient exploratory tool to detect periodic patterns in animal movement data.

An animal's trajectory is a fundamental object of interest in movement ecology, as it directly informs a range of topics from resource selection to energy expenditure and behavioral states. Optimally inferring the mostly unobserved movement path and its dynamics from a limited sample of telemetry observations is a key unsolved problem, however. The field of geostatistics has focused significant attention on a mathematically analogous problem that has a statistically optimal solution coined after its inventor, Krige. Kriging revolutionized geostatistics and is now the gold standard for interpolating between a limited number of autocorrelated spatial point observations. Here we translate Kriging for use with animal movement data. Our Kriging formalism encompasses previous methods to estimate animal's trajectories—the Brownian bridge and continuous‐time correlated random walk library—as special cases, informs users as to when these previous methods are appropriate, and provides a more general method when they are not. We demonstrate the capabilities of Kriging on a case study with Mongolian gazelles where, compared to the Brownian bridge, Kriging with a more optimal model was 10% more precise in interpolating locations and 500% more precise in estimating occurrence areas.

Accurately estimating home range and understanding movement behavior can provide important information on ecological processes. Advances in data collection and analysis have improved our ability to estimate home range and movement parameters, both of which have the potential to impact species conservation. Fitting continuous-time movement model to data and incorporating the autocorrelated kernel density estimator (AKDE), we investigated range residency of forty-four jaguars fit with GPS collars across five biomes in Brazil and Argentina. We assessed home range and movement parameters of range resident animals and compared AKDE estimates with kernel density estimates (KDE). We accounted for differential space use and movement among individuals, sex, region, and habitat quality. Thirty-three (80%) of collared jaguars were range resident. Home range estimates using AKDE were 1.02 to 4.80 times larger than KDE estimates that did not consider autocorrelation. Males exhibited larger home ranges, more directional movement paths, and a trend towards larger distances traveled per day. Jaguars with the largest home ranges occupied the Atlantic Forest, a biome with high levels of deforestation and high human population density. Our results fill a gap in the knowledge of the species' ecology with an aim towards better conservation of this endangered/critically endangered carnivore-the top predator in the Neotropics.

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An open quantum system is a quantum system that interacts with some environment whose degrees of freedom have been coarse grained away. This model describes non-equilibrium processes more general than scattering-matrix formulations. Furthermore, the microscopically-derived environment provides a model of noise, dissipation and decoherence far more general than Markovian (white noise) models. The latter are fully characterized by Lindblad equations and can be motivated phenomenologically. Non-Markovian processes consistently account for backreaction with the environment and can incorporate effects such as finite temperature and spatial correlations. We consider linear systems with bilinear coupling to the environment, or quantum Brownian motion, and nonlinear systems with weak coupling to the environment. For linear systems we provide exact solutions with analytical results for a variety of spectral densities. Furthermore, we point out an important mathematical subtlety which led to incorrect master-equation coefficients in earlier derivations, given nonlocal dissipation. For nonlinear systems we provide perturbative solutions by translating the formalism of canonical perturbation theory into the context of master equations. It is shown that unavoidable degeneracy causes an unfortunate reduction in accuracy between perturbative master equations and their solutions. We also extend the famous theorem of Lindblad, Gorini, Kossakowski and Sudarshan on completely positivity to non-Markovian master equations. Our application is primarily to model atoms interacting via a common electromagnetic field. The electromagnetic field contains correlations in both space and time, which are related to its relativistic (photon-mediated) nature. As such, atoms residing in the same field experience different environmental effects depending upon their relative position and orientation. Our more accurate solutions were necessary to assess sudden death of entanglement at zero temperature. In contrast to previous claims, we found that all initial states of two-level atoms undergo finite-time disentanglement. We were also able to access regimes which cannot be described by Lindblad equations and other simpler methods, such as near resonance. Finally we revisit the infamous Abraham-Lorentz force, wherein a single particle in motion experiences backreaction from the electromagnetic field. This leads to a number of well-known problems including pre-acceleration and runaway solutions. We found a more a more-suitable open-system treatment of the nonrelativistic particle to be perfectly causal and dissipative without any extraneous requirements for finite size of the particle, weak coupling to the field, etc.

The amount of space organisms use is thought to be tightly linked to the availability of resources within their habitats, such that organisms living in productive habitats generally require less space than those in resource-poor habitats. This hypothesis has widespread em-pirical support, but existing studies have focused primarily on responses to the mean amount of resources, while responses to the variance around the mean are still largely unknown. This is not a trivial oversight. Organisms adjust to variable environmental conditions, so failing to consider the effects of resource (un)predictability can result in a limited understanding of an organism’s range size, which challenges ecological theory and applied conservation alike. In this study, we leverage the available literature to provide a unifying framework and hypotheses for the effect of mean and variance in resources on range sizes. Next, we use simulated movement data to demonstrate how the combined effects of mean and variance in resource abundance interact to shape predictable patterns in range size. Finally, we use real-world tracking data on a lowland tapir (Tapirus terrestris) from the Brazilian Cerrado to show how this framework can be applied to better understand the movement ecology of free-ranging animals.

We introduce a class of maximum-entropy states that naturally includes within it all of the major continuous-time stochastic processes that have been applied to animal movement, including Brownian motion, Ornstein-Uhlenbeck motion, integrated Ornstein-Uhlenbeck motion, a recently discovered hybrid of the previous models, and a new model that describes central-place foraging. We are also able to predict a further hierarchy of new models that will emerge as data quality improves to better resolve the underlying continuity of animal movement. Finally, we also show that Langevin equations must obey a fluctuation-dissipation theorem to generate processes that fall from this class of maximum-entropy distributions.