Justin M. Calabrese

A comparison‐shopper's guide to connectivity metrics

Connectivity is an important but inconsistently defined concept in spatial ecology and conservation biology. Theoreticians from various subdisciplines of ecology argue over its definition and measurement, but no consensus has yet emerged. Despite this disagreement, measuring connectivity is an integral part of many resource management plans. A more practical approach to understanding the many connectivity metrics is needed. Instead of focusing on theoretical issues surrounding the concept of connectivity, we describe a data-dependent framework for classifying these metrics. This framework illustrates the data requirements, spatial scales, and information yields of a range of different connectivity measures. By highlighting the costs and benefits associated with using alternative metrics, this framework allows practitioners to make more informed decisions concerning connectivity measurement.

Statistical inference for stochastic simulation models – theory and application

Statistical models are the traditional choice to test scientific theories when observations, processes or boundary conditions are subject to stochasticity. Many important systems in ecology and biology, however, are difficult to capture with statistical models. Stochastic simulation models offer an alternative, but they were hitherto associated with a major disadvantage: their likelihood functions can usually not be calculated explicitly, and thus it is difficult to couple them to well-established statistical theory such as maximum likelihood and Bayesian statistics. A number of new methods, among them Approximate Bayesian Computing and Pattern-Oriented Modelling, bypass this limitation. These methods share three main principles: aggregation of simulated and observed data via summary statistics, likelihood approximation based on the summary statistics, and efficient sampling. We discuss principles as well as advantages and caveats of these methods, and demonstrate their potential for integrating stochastic simulation models into a unified framework for statistical modelling.

Statistical inference for stochastic simulations models : theory and application

Statistical models are the traditional choice to test scientific theories when observations, processes or boundary conditions are subject to stochasticity. Many important systems in ecology and biology, however, are difficult to capture with statistical models. Stochastic simulation models offer an alternative, but they were hitherto associated with a major disadvantage: their likelihood functions can usually not be calculated explicitly, and thus it is difficult to couple them to well-established statistical theory such as maximum likelihood and Bayesian statistics. A number of new methods, among them Approximate Bayesian Computing and Pattern-Oriented Modelling, bypass this limitation. These methods share three main principles: aggregation of simulated and observed data via summary statistics, likelihood approximation based on the summary statistics, and efficient sampling. We discuss principles as well as advantages and caveats of these methods, and demonstrate their potential for integrating stochastic simulation models into a unified framework for statistical modelling.

Lost in Time, Lonely, and Single: Reproductive Asynchrony and the Allee Effect

Identifying linkages between life‐history traits and small population processes is essential to effective multispecies conservation. Reproductive asynchrony, which occurs when individuals are reproductively active for only a portion of the population‐level breeding period, may provide one such link. Traditionally, reproductive asynchrony has been considered from evolutionary perspectives as an advantageous bet‐hedging strategy in temporally unpredictable environments. Here, we explore the dynamic consequences of reproductive asynchrony as a density‐dependent life‐history trait. To examine how asynchrony affects population growth rate and extinction risk, we used a general model of reproductive timing to quantify the temporal overlap of opposite‐sex individuals and to simulate population dynamics over a range of initial densities and empirical estimates of reproductive asynchrony. We also considered how protandry, a sexually selected life‐history strategy that often accompanies asynchrony, modulates the population‐level effects of reproductive asynchrony. We found that asynchrony decreases the number of males a female overlaps with, decreases the average probability of mating per male/female pair that does overlap, and leaves some females completely isolated in time. This loss of reproductive potential, which is exacerbated by protandry, reduces population growth rate at low density and can lead to extinction via an Allee effect. Thus reproductive asynchrony and protandry, both of which can be evolutionarily advantageous at higher population densities, may prove detrimental when population density declines.

From Fine-Scale Foraging to Home Ranges: A Semivariance Approach to Identifying Movement Modes across Spatiotemporal Scales

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.

Reproductive asynchrony in natural butterfly populations and its consequences for female matelessness.

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.

The independent and interactive effects of tree-tree establishment competition and fire on savanna structure and dynamics.

Savanna ecosystems are widespread and economically important and harbor considerable biodiversity. Despite extensive study, the mechanisms regulating savanna tree populations are not well understood. Recent empirical work suggests that both tree‐tree competition and fire are key factors in semiarid to mesic savannas, but the potential for competition to structure savannas, particularly in interaction with fire, has received little theoretical attention. We develop a minimalistic and analytically tractable stochastic cellular automaton to study the individual and combined effects of these two factors on savannas. We find that while competition often substantially depresses tree density, fire generally has little effect but can drive tree extinction in extreme scenarios. When combined, competition and fire interact nonlinearly with strong negative consequences for tree density. This novel result may help explain observed variability among apparently similar savannas in their response to fire. Paradoxically, this interaction could also render the presence of competition more difficult to detect in empirical studies because fire can override the characteristic regular spacing driven by competition and lead instead to clustering.

ctmm: an r package for analyzing animal relocation data as a continuous‐time stochastic process

Summary Movement ecology has developed rapidly over the past decade, driven by advances in tracking technology that have largely removed data limitations. Development of rigorous analytical tools has lagged behind empirical progress, and as a result, relocation data sets have been underutilized. Discrete-time correlated random walk models (CRW) have long served as the foundation for analyzing relocation data. Unfortunately, CRWs confound the sampling and movement processes. CRW parameter estimates thus depend sensitively on the sampling schedule, which makes it difficult to draw sampling-independent inferences about the underlying movement process. Furthermore, CRWs cannot accommodate the multiscale autocorrelations that typify modern, finely sampled relocation data sets. Recent developments in modelling movement as a continuous-time stochastic process (CTSP) solve these problems, but the mathematical difficulty of using CTSPs has limited their adoption in ecology. To remove this roadblock, we introduce the ctmm package for the R statistical computing environment. ctmm implements all of the CTSPs currently in use in the ecological literature and couples them with powerful statistical methods for autocorrelated data adapted from geostatistics and signal processing, including variograms, periodograms and non-Markovian maximum likelihood estimation. ctmm is built around a standard workflow that begins with visual diagnostics, proceeds to candidate model identification, and then to maximum likelihood fitting and AIC-based model selection. Once an accurate CTSP for the data has been fitted and selected, analyses that require such a model, such as quantifying home range areas via autocorrelated kernel density estimation or estimating occurrence distributions via time-series Kriging, can then be performed. We use a case study with African buffalo to demonstrate the capabilities of ctmm and highlight the steps of a typical CTSP movement analysis workflow.

How far to go? Determinants of migration distance in land mammals.

Animal migration is a global phenomenon, but few studies have examined the substantial within- and between-species variation in migration distances. We built a global database of 94 land migrations of large mammalian herbivore populations ranging from 10 to 1638 km. We examined how resource availability, spatial scale of resource variability and body size affect migration distance among populations. Resource availability measured as normalised difference vegetation index had a strong negative effect, predicting a tenfold difference in migration distances between low- and high-resource areas and explaining 23% of the variation in migration distances. We found a weak, positive effect of the spatial scale of resource variability but no effect of body size. Resource-poor environments are known to increase the size of mammalian home ranges and territories. Here, we demonstrate that for migratory populations as well, animals living in resource-poor environments travel farther to fulfil their resource needs.

Non-Markovian maximum likelihood estimation of autocorrelated movement processes

Summary By viewing animal movement paths as realizations of a continuous stochastic process, we introduce a rigorous likelihood method for estimating the statistical parameters of movement processes. This method makes no assumption of a hidden Markov property, places no special emphasis on the sampling rate, is insensitive to irregular sampling and data gaps, can produce reasonable estimates with limited sample sizes and can be used to assign AIC values to a vast array of qualitatively different models of animal movement at the individual and population levels. To develop our approach, we consider the likelihood of the first two cumulants of stochastic processes, the mean and autocorrelation functions. Together, these measures provide a considerable degree of information regarding searching, foraging, migration and other aspects of animal movement. As a specific example, we develop the likelihood analyses necessary to contrast performance of animal movement models based on Brownian motion, the Ornstein–Uhlenbeck process and a generalization of the Ornstein–Uhlenbeck process that includes ballistic bouts. We then show how our framework also provides a new and more accurate approach to home-range estimation when compared to estimators that neglect autocorrelation in the movement path. We apply our methods to a data set on Mongolian gazelles (Procapra gutturosa) to identify the movement behaviours and their associated time and length scales that characterize the movement of each individual. Additionally, we show that gazelle annual ranges are vastly larger than those of other non-migratory ungulates.