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Monitoring technologies now provide real-time animal location information, which opens up the possibility of developing forecasting systems to fuse these data with movement models to predict future trajectories. State-space modeling approaches are well established for retrospective location estimation and behavioral inference through state and parameter estimation. Here we use a statespace model within a comprehensive data assimilative framework for probabilistic animal movement forecasting. Real-time location information is combined with stochastic movement model predictions to provide forecasts of future animal locations and trajectories, as well as estimates of key behavioral parameters. Implementation uses ensemble-based sequential Monte Carlo methods (a particle filter). We first apply the framework to an idealized case using a nondimensional animal movement model based on a continuous-time random walk process. A set of numerical forecasting experiments demonstrates the workflow and key features, such as the online estimation of behavioral parameters using state augmentation, the use of potential functions for habitat preference, and the role of observation error and sampling frequency on forecast skill. For a realistic demonstration, we adapt the framework to short-term forecasting of the endangered southern resident killer whale (SRKW) in the Salish Sea using visual sighting information wherein the potential function reflects historical habitat utilization of SRKW. We successfully estimate whale locations up to 2.5 h in advance with a moderate prediction error (<5 km), providing reasonable lead-in time to mitigate vessel–whale interactions. It is argued that this forecasting framework can be used to synthesize diverse data types and improve animal movement models and behavioral understanding and has the potential to lead to important advances in movement ecology.

The paper investigates angular regressions that express the angles of an animal's motion in terms of time varying directions and distances to environmental features that could influence its displacement. The mean direction proposed is a compromise between several possible targets. Conditions for the identifiability of the regression parameters are provided. Maximum likelihood estimators for the parameters are derived under two von Mises error structures. Robust sandwich estimators of the parameter variance—covariance matrix are obtained. The statistical methodology proposed is first used to reanalyse a classical data set on periwinkle movement. A second application investigates how bison trails are shaped by meadows and canopy gaps in Saskatchewan's Prince Albert National Park.

The presence of temporal correlations in random movement trajectories is a widespread phenomenon across biological, chemical and physical systems. The ubiquity of persistent and anti-persistent motion in many natural and synthetic systems has led to a large literature on the modelling of temporally correlated movement paths. Despite the substantial body of work, little progress has been made to determine the dynamical properties of various transport related quantities, including the first-passage or first-hitting probability to one or multiple absorbing targets when space is bounded. To bridge this knowledge gap we generalise the renewal theory of first-passage and splitting probabilities to correlated discrete variables. We do so in arbitrary dimensions on a lattice for the so-called correlated or persistent random walk, the one step non-Markovian extension of the simple lattice random walk in bounded and unbounded space. We focus on bounded domains and consider both persistent and anti-persistent motion in hypercubic lattices as well as the hexagonal lattice. The discrete formalism allows us to extend the notion of the first-passage to that of the directional first-passage, whereby the walker must reach the target from a prescribed direction for a hitting event to occur. As an application to spatio-temporal observations of correlated moving cells that may be either repelled or attracted to hard surfaces, we compare the first-passage statistics to a target within a reflecting domain depending on whether an interaction with the reflective interface invokes a reversal of the movement direction or not. With strong persistence we observe multi-modality in the first-passage distribution in the former case, which instead is greatly suppressed in the latter.

Like many species, movement patterns of southern elephant seals (Mirounga leonina) are being influenced by long-term environmental change. These seals migrate up to 4,000 km from their breeding colonies, foraging for months in a variety of Southern Ocean habitats. Understanding how movement patterns vary with environmental features and how these relationships differ among individuals employing different foraging strategies can provide insight into foraging performance at a population level. We apply new fast-estimation tools to fit mixed effects within a random walk movement model, rapidly inferring among-individual variability in southern elephant seal environment–movement relationships. We found that seals making foraging trips to the sea ice on or near the Antarctic continental shelf consistently reduced speed and directionality (move persistence) with increasing sea-ice coverage but had variable responses to chlorophyll a concentration, whereas seals foraging in the open ocean reduced move persistence in regions where circumpolar deep water shoaled. Given future climate scenarios, open-ocean foragers may encounter more productive habitat but sea-ice foragers may see reduced habitat availability. Our approach is scalable to large telemetry data sets and allows flexible combinations of mixed effects to be evaluated via model selection, thereby illuminating the ecological context of animal movements that underlie habitat usage.

Recent developments in animal tracking technology have permitted the collection of detailed data on the movement paths of individuals from many species. However, analysis methods for these data have not developed at a similar pace, largely due to a lack of suitable candidate models, coupled with the technical difficulties of fitting such models to data. To facilitate a general modeling framework, we propose that complex movement paths can be conceived as a series of movement strategies among which animals transition as they are affected by changes in their internal and external environment. We synthesize previously existing and novel methodologies to develop a general suite of mechanistic models based on biased and correlated random walks that allow different behavioral states for directed (e.g., migration), exploratory (e.g., dispersal), area-restricted (e.g., foraging), and other types of movement. Using this \"toolbox\" of nested model components, multistate movement models may be custom-built for a wide variety of species and applications. As a unified state-space modeling framework, it allows the simultaneous investigation of numerous hypotheses about animal movement from imperfectly observed data, including time allocations to different movement behavior states, transitions between states, the use of memory or navigation, and strengths of attraction (or repulsion) to specific locations. The inclusion of covariate information permits further investigation of specific hypotheses related to factors driving different types of movement behavior. Using reversible-jump Markov chain Monte Carlo methods to facilitate Bayesian model selection and multi-model inference, we apply the proposed methodology to real data by adapting it to the natural history of the grey seal ( Halichoerus grypus ) in the North Sea. Although previous grey seal studies tended to focus on correlated movements, we found overwhelming evidence that bias toward haul-out or foraging locations better explained seal movement than did simple or correlated random walks. Posterior model probabilities also provided evidence that seals transition among directed, area-restricted, and exploratory movements associated with haul-out, foraging, and other behaviors. With this intuitive framework for modeling and interpreting animal movement, we believe that the development and application of custom-made movement models will become more accessible to ecologists and non-statisticians.

Our previous work presented explicit formulas for the generalized zeta function and the generalized Ihara zeta function corresponding to the Grover walk and the positive-support version of the Grover walk on the regular graph via the Konno–Sato theorem, respectively. This paper extends these walks to a class of walks including random walks, correlated random walks, quantum walks, and open quantum random walks on the torus by the Fourier analysis.

Understanding how an individual animal is able to navigate through its environment is a key question in movement ecology that can give insight into observed movement patterns and the mechanisms behind them. Efficiency of navigation is important for behavioral processes at a range of different spatio-temporal scales, including foraging and migration. Random walk models provide a standard framework for modeling individual animal movement and navigation. Here we consider a vector-weighted biased and correlated random walk (BCRW) model for directed movement (taxis), where external navigation cues are balanced with forward persistence. We derive a mathematical approximation of the expected navigational efficiency for any BCRW of this form and confirm the model predictions using simulations. We demonstrate how the navigational efficiency is related to the weighting given to forward persistence and external navigation cues, and highlight the counter-intuitive result that for low (but realistic) levels of error on forward persistence, a higher navigational efficiency is achieved by giving more weighting to this indirect navigation cue rather than direct navigational cues. We discuss and interpret the relevance of these results for understanding animal movement and navigation strategies.

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.

In movement analysis, correlated random walk (CRW) models often use so‐called turning angles, which are measured relative to the previous movement direction. To segregate between different movement modes, hidden Markov models (HMMs) describe movements as piecewise stationary CRWs in which the distributions of turning angles and step sizes depend on the underlying state. This typically allows for the segregation of movement modes that show different movement speeds. We show that in some cases, it may be interesting to investigate absolute angles, that is, biased random walks (BRWs) instead of turning angles. In particular, while discrimination between states in the turning angle setting can only rely on movement speed, models with absolute angles can be used to discriminate between sections of different movement directions. A preprocessing algorithm is provided that enables the analysis of absolute angles in the existing R package moveHMM. In a data set of movements of cell organelles, models using not the turning angle but the absolute angle could capture interesting additional properties. Goodness‐of‐fit was increased for HMMs with absolute angles, and HMMs with absolute angles tended to choose a higher number of states, suggesting the existence and relevance of prominent directional changes in the present data set. These results suggest that models with absolute angles can provide important information in the analysis of movement patterns if the existence and frequency of directional changes is of biological importance. In movement analysis, models often use so‐called turning angles, which are measured relative to the previous movement direction. We use hidden Markov models (HMMs) to show that the analysis of absolute angles instead of turning angles can provide important additional information if the existence and frequency of directional changes is of biological relevance. In a data set of movements of cell organelles, goodness‐of‐fit was most often increased for HMMs with absolute angles, and these could therefore be used to investigate important additional properties, such as the switching of movement between periods of different movement directions.

The Atlantic salmon, Salmo salar Linnaeus 1758, is a charismatic, anadromous species that has faced dramatic declines throughout its range. There is currently a lack of information on the effect of free-standing bodies of water on a key life event, sea migration, for the species. This study extends our understanding in this area by combining acoustic telemetry with a correlated random walk model to try to examine potential morphological and behavioural factors that differentiate successful from unsuccessful migrants through Scotland’s largest lake. Consistent with other studies, we found that smolts experienced a high rate of mortality in the lake (~ 43%), with approximately 14% potentially predated upon by birds and 4% by Northern pike. Migration speed in the lake was slow (the mean minimum movement speed between centres of activity was 0.13 m/s), and pathways frequently deviated away from the outlet river. There was no evidence of a morphological or behavioural trait or migratory pathway that distinguished successful from unsuccessful smolts. This suggests that migration movement direction in the main body of Loch Lomond appeared to be random. This was further supported by the output of a correlated random walk model which closely resembled the pathway and migration speed and distance patterns displayed by successful migrants. However, once successful smolts came within ~2 km of the lake exit, a high proportion remained in this region prior to entering the River Leven. We suggest that this “goldilocks zone” is where directional cues become apparent to migrating fish. Future studies should combine random walk models with environmental variables to determine if external factors are driving the apparently random movement patterns exhibited by smolts in lakes.