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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.

Quantifying animals' home ranges is a key problem in ecology and has important conservation and wildlife management applications. Kernel density estimation (KDE) is a workhorse technique for range delineation problems that is both statistically efficient and nonparametric. KDE assumes that the data are independent and identically distributed (IID). However, animal tracking data, which are routinely used as inputs to KDEs, are inherently autocorrelated and violate this key assumption. As we demonstrate, using realistically autocorrelated data in conventional KDEs results in grossly underestimated home ranges. We further show that the performance of conventional KDEs actually degrades as data quality improves, because autocorrelation strength increases as movement paths become more finely resolved. To remedy these flaws with the traditional KDE method, we derive an autocorrelated KDE (AKDE) from first principles to use autocorrelated data, making it perfectly suited for movement data sets. We illustrate the vastly improved performance of AKDE using analytical arguments, relocation data from Mongolian gazelles, and simulations based upon the gazelle's observed movement process. By yielding better minimum area estimates for threatened wildlife populations, we believe that future widespread use of AKDE will have significant impact on ecology and conservation biology.

We investigated the relationships between search efficiency, movement strategy, and nonlocal communication in the biological context of animal foraging. We considered situations where the members of a population of foragers perform either Gaussian jumps or Lévy flights, and show that the search time is minimized when communication among individuals occurs at intermediate ranges, independently of the type of movement. Additionally, while Brownian strategies are more strongly influenced by the communication mechanism, Lévy flights still result in shorter overall search durations.

We propose a model equation for the dynamics of tree density in mesic savannas which considers long-range competition among trees and the effect of fire indirectly acting as a local facilitation mechanism. Despite the fact that we take short-range facilitation to the local-range limit, the standard full spectrum of spatial structures already obtained in self-organization models of vegetation is recovered. Nonlocal competition, in the limit of infinitesimally short facilitation, promotes the clustering of trees. The long time coexistence between trees and grass, and how fires affect the survival of trees as well as the maintenance of the patterns is studied. The influence of demographic noise is analyzed. The stochastic system, under the parameter constraints typical of mesic savannas, shows non-homogeneous patterns characteristic of realistic situations. The coexistence of trees and grass still remains at reasonable noise intensities., (Copyright © 2013 Elsevier Ltd. All rights reserved.)

Mate finding, which is essential to both population growth and gene exchange, involves both spatial and temporal components. From a population dynamics perspective, spatial mate-finding problems are well studied, and decreased mate-finding efficiency at low population densities is a well-recognized mechanism for the Allee effect. Temporal aspects of mate finding have been rarely considered, but reproductive asynchrony may engender an Allee effect in which some females go mateless by virtue of temporal isolation. Here we develop and explore a model that unifies previously disparate theoretical considerations of spatial and temporal aspects of mate finding. Specifically, we develop a two-sex reaction-diffusion system to examine the interplay between reproductive asynchrony and the dispersal of individuals out of a patch. We also consider additional behavioral complications, including several alternative functional forms for mating efficiency and advective movements in which males actively seek out females. By calculating the fraction of females expected to go mateless as a joint function of reproductive asynchrony and patch size, we find that the population-level reproductive rates necessary to offset female matelessness may be quite high. These results suggest that Allee effects engendered by reproductive asynchrony will be greatly exacerbated in spatially isolated populations.

1. Reproductive asynchrony, where individuals in a population are short-lived relative to the population-level reproductive period, has been identified recently as a theoretical mechanism of the Allee effect that could operate in diverse plant and insect species. The degree to which this effect impinges on the growth potential of natural populations is not yet well understood. 2. Building on previous models of reproductive timing, we develop a general framework that allows a detailed, quantitative examination of the reproductive potential lost to asynchrony in small natural populations. 3. Our framework includes a range of biologically plausible submodels that allow details of mating biology of different species to be incorporated into the basic reproductive timing model. 4. We tailor the parameter estimation methods of the full model (basic model plus mating biology submodels) to take full advantage of data from detailed field studies of two species of Parnassius butterflies whose mating status may be assessed easily in the field. 5. We demonstrate that for both species, a substantial portion of the female population (6.5-18.6%) is expected to die unmated. These analyses provide the first direct, quantitative evidence of female reproductive failure due to asynchrony in small natural populations, and suggest that reproductive asynchrony exerts a strong and largely unappreciated influence on the population dynamics of these butterflies and other species with similarly asynchronous reproductive phenology.