Defining null expectations for animal site fidelity.

Site fidelity-the tendency to return to previously visited locations-is widespread across taxa. Returns may be driven by several mechanisms, including memory, habitat selection, or chance; however, pattern-based definitions group different generating mechanisms under the same label of 'site fidelity', often assuming memory as the main driver. We propose an operational definition of site fidelity as patterns of return that deviate from a null expectation derived from a memory-free movement model. First, using agent-based simulations, we show that without memory, intrinsic movement characteristics and extrinsic landscape characteristics are key determinants of return patterns and that even random movements may generate substantial probabilities of return. Second, we illustrate how to implement our framework empirically to establish ecologically meaningful, system-specific null expectations for site fidelity. Our approach provides a conceptual and operational framework to test hypotheses on site fidelity across systems and scales.

Return Probability of Quantum and Correlated Random Walks.

The analysis of the return probability is one of the most essential and fundamental topics in the study of classical random walks. In this paper, we study the return probability of quantum and correlated random walks in the one-dimensional integer lattice by the path counting method. We show that the return probability of both quantum and correlated random walks can be expressed in terms of the Legendre polynomial. Moreover, the generating function of the return probability can be written in terms of elliptic integrals of the first and second kinds for the quantum walk.

Three-dimensional random walk models of individual animal movement and their application to trap counts modelling.

BACKGROUND: Random walks (RWs) have proved to be a powerful modelling tool in ecology, particularly in the study of animal movement. An application of RW concerns trapping which is the predominant sampling method to date in insect ecology and agricultural pest management. A lot of research effort has been directed towards modelling ground-dwelling insects by simulating their movement in 2D, and computing pitfall trap counts, but comparatively very little for flying insects with 3D elevated traps. METHODS: We introduce the mathematics behind 3D RWs and present key metrics such as the mean squared displacement (MSD) and path sinuosity, which are already well known in 2D. We develop the mathematical theory behind the 3D correlated random walk (CRW) which involves short-term directional persistence and the 3D Biased random walk (BRW) which introduces a long-term directional bias in the movement so that there is an overall preferred movement direction. In this study, we focus on the geometrical aspects of the 3D trap and thus consider three types of shape; a spheroidal trap, a cylindrical trap and a rectangular cuboidal trap. By simulating movement in 3D space, we investigated the effect of 3D trap shapes and sizes and of movement diffusion on trapping efficiency. RESULTS: We found that there is a non-linear dependence of trap counts on the trap surface area or volume, but the effect of volume appeared to be a simple consequence of changes in area. Nevertheless, there is a slight but clear hierarchy of trap shapes in terms of capture efficiency, with the spheroidal trap retaining more counts than a cylinder, followed by the cuboidal type for a given area. We also showed that there is no effect of short-term persistence when diffusion is kept constant, but trap counts significantly decrease with increasing diffusion. CONCLUSION: Our results provide a better understanding of the interplay between the movement pattern, trap geometry and impacts on trapping efficiency, which leads to improved trap count interpretations, and more broadly, has implications for spatial ecology and population dynamics.

Novel step selection analyses on energy landscapes reveal how linear features alter migrations of soaring birds.

Human modification of landscapes includes extensive addition of linear features, such as roads and transmission lines. These can alter animal movement and space use and affect the intensity of interactions among species, including predation and competition. Effects of linear features on animal movement have seen relatively little research in avian systems, despite ample evidence of their effects in mammalian systems and that some types of linear features, including both roads and transmission lines, are substantial sources of mortality. Here, we used satellite telemetry combined with step selection functions designed to explicitly incorporate the energy landscape (el-SSFs) to investigate the effects of linear features and habitat on movements and space use of a large soaring bird, the golden eagle Aquila chrysaetos, during migration. Our sample consisted of 32 adult eagles tracked for 45 spring and 39 fall migrations from 2014 to 2017. Fitted el-SSFs indicated eagles had a strong general preference for south-facing slopes, where thermal uplift develops predictably, and that these areas are likely important aspects of migratory pathways. el-SSFs also provided evidence that roads and railroads affected movement during both spring and fall migrations, but eagles selected areas near roads to a greater degree in spring compared to fall and at higher latitudes compared to lower latitudes. During spring, time spent near linear features often occurred during slower-paced or stopover movements, perhaps in part to access carrion produced by vehicle collisions. Regardless of the behavioural mechanism of selection, use of these features could expose eagles and other soaring species to elevated risk via collision with vehicles and/or transmission lines. Linear features have previously been documented to affect the ecology of terrestrial species (e.g. large mammals) by modifying individuals' movement patterns; our work shows that these effects on movement extend to avian taxa.

Movement responses to environment: fast inference of variation among southern elephant seals with a mixed effects model.

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.

Correlated random walks inside a cell: actin branching and microtubule dynamics.

Correlated random walks (CRW) have been explored in many settings, most notably in the motion of individuals in a swarm or flock. But some subcellular systems such as growth or disassembly of bio-polymers can also be described with similar models and understood using related mathematical methods. Here we consider two examples of growing cytoskeletal elements, actin and microtubules. We use CRW or generalized CRW-like PDEs to model their spatial distributions. In each case, the linear models can be reduced to a Telegrapher's equation. A combination of explicit solutions (in one case) and numerical solutions (in the other) demonstrates that the approach to steady state can be accompanied by (decaying) waves.

Navigational efficiency in a biased and correlated random walk model of individual animal movement.

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.

A general model of forager search: Adaptive encounter-conditional heuristics outperform Lévy flights in the search for patchily distributed prey.

A theoretical and applied literature has suggested that foragers search using Lévy flights, since Lévy flights can maximize the efficiency of search in the absence of information on the location of randomly distributed prey. Foragers, however, often have available to them at least some information about the distribution of prey, gained either through evolved mechanisms, experience and memory, or social transmission of information. As such, we might expect selection for heuristics that make use of such information to further improve the efficiency of random search. Here we present a general model of random search behavior that includes as special cases: area-restricted search, correlated random walks, Brownian search, and Lévy flights. This generative model allows foragers to adjust search parameters based on encounter-conditional and other heuristics. Using a simulation model, we demonstrate the efficiency gains of these search heuristics, and illustrate the resulting differences in the distributions of step-size and heading angle change they imply, relative to Lévy flights. We conclude by presenting a statistical model that can be fit to empirical data and a set of testable, quantitative predictions that contrast our model of adaptive search with the Lévy flight foraging hypothesis.

Auto-correlated directional swimming can enhance settlement success and connectivity in fish larvae.

Larvae of coastal-marine fishes have been shown repeatedly to swim directionally in the pelagic environment. Yet, biophysical models of larval dispersal typically impose a Simple Random Walk (SRW) algorithm to simulate non-directional movement in the open ocean. Here we investigate the use of a Correlated Random Walk (CRW) algorithm; imposing auto-correlated directional swimming onto simulated larvae within a high-resolution 3D biophysical model of the Gulf of Aqaba, the Red Sea. Our findings demonstrate that implementation of auto-correlated directional swimming can result in an increase of up to ×2.7 in the estimated success rate of larval-settlement, as well as an increase in the extent of connectivity. With accumulating empirical support for the capacity for directional-swimming during the pelagic phase, we propose that CRW should be applied in biophysical models of dispersal by coastal marine fish-larvae.

Does movement behaviour predict population densities? A test with 25 butterfly species.

Diffusion, which approximates a correlated random walk, has been used by ecologists to describe movement, and forms the basis for many theoretical models. However, it is often criticized as too simple a model to describe animal movement in real populations. We test a key prediction of diffusion models, namely, that animals should be more abundant in land cover classes through which they move more slowly. This relationship between density and diffusion has rarely been tested across multiple species within a given landscape. We estimated diffusion rates and corresponding densities of 25 Israeli butterfly species from flight path data and visual surveys. The data were collected across 19 sites in heterogeneous landscapes with four land cover classes: semi-natural habitat, olive groves, wheat fields and field margins. As expected from theory, species tended to have higher densities in land cover classes through which they moved more slowly and lower densities in land cover classes through which they moved more quickly. Two components of movement (move length and turning angle) were not associated with density, nor was expected net squared displacement. Move time, however, was associated with density, and animals spent more time per move step in areas with higher density. The broad association we document between movement behaviour and density suggests that diffusion is a good first approximation of movement in butterflies. Moreover, our analyses demonstrate that dispersal is not a species-invariant trait, but rather one that depends on landscape context. Thus, land cover classes with high diffusion rates are likely to have low densities and be effective conduits for movement.

Allometric and temporal scaling of movement characteristics in Galapagos tortoises.

Understanding how individual movement scales with body size is of fundamental importance in predicting ecological relationships for diverse species. One-dimensional movement metrics scale consistently with body size yet vary over different temporal scales. Knowing how temporal scale influences the relationship between animal body size and movement would better inform hypotheses about the efficiency of foraging behaviour, the ontogeny of energy budgets, and numerous life-history trade-offs. We investigated how the temporal scaling of allometric patterns in movement varies over the course of a year, specifically during periods of motivated (directional and fast movement) and unmotivated (stationary and tortuous movement) behaviour. We focused on a recently diverged group of species that displays wide variation in movement behaviour - giant Galapagos tortoises (Chelonoidis spp.) - to test how movement metrics estimated on a monthly basis scaled with body size. We used state-space modelling to estimate seven different movement metrics of Galapagos tortoises. We used log-log regression of the power law to evaluate allometric scaling for these movement metrics and contrasted relationships by species and sex. Allometric scaling of movement was more apparent during motivated periods of movement. During this period, allometry was revealed at multiple temporal intervals (hourly, daily and monthly), with values observed at daily and monthly intervals corresponding most closely to the expected one-fourth scaling coefficient, albeit with wide credible intervals. We further detected differences in the magnitude of scaling among taxa uncoupled from observed differences in the temporal structuring of their movement rates. Our results indicate that the definition of temporal scales is fundamental to the detection of allometry of movement and should be given more attention in movement studies. Our approach not only provides new conceptual insights into temporal attributes in one-dimensional scaling of movement, but also generates valuable insights into the movement ecology of iconic yet poorly understood Galapagos giant tortoises.

Combining animal movements and behavioural data to detect behavioural states.

Animal movement paths show variation in space caused by qualitative shifts in behaviours. I present a method that (1) uses both movement path data and ancillary sensor data to detect natural breakpoints in animal behaviour and (2) groups these segments into different behavioural states. The method can also combine analyses of different path segments or paths from different individuals. It does not assume any underlying movement mechanism. I give an example with simulated data. I also show the effects of random variation, # of states and # of segments on this method. I present a case study of a fisher movement path spanning 8 days, which shows four distinct behavioural states divided into 28 path segments when only turning angles and speed were considered. When accelerometer data were added, the analysis shows seven distinct behavioural states divided into 41 path segments.

Animal path integration: a model of positional uncertainty along tortuous paths.

Exact closed form mathematical solutions are reported which quantify the dynamic uncertainty resulting from path integration (PI) along tortuous paths. Based on a correlated random walk model, the derived results quantify positional estimation error moments with and without a compass, in discrete and continuous time. Consistent with earlier studies on attempted straight-line navigation, using a compass significantly reduces the uncertainty during PI, making purely idiothetic PI biologically implausible except over short distances. Examples are used to illustrate the contributions of angular noise, linear noise and path tortuosity, under different conditions. Linear noise is shown to be relatively more important with a compass while angular noise is more important without. It is shown that increasing path tortuosity decreases positional uncertainty, true for long and short journeys, irrespective of whether a compass is used, or the level of noise. In contrast, reducing angular noise also reduces uncertainty, but only below some critical level of noise. Using canonical equations of PI, it is shown that polar PI using a compass accumulates uncertainty in a manner similar to Cartesian PI without a compass. Issues of data sampling bias and intermittent use of a compass are also considered for PI along tortuous paths.

The Lévy flight foraging hypothesis in a pelagic seabird.

Lévy flight foraging represents an innovative paradigm for the analysis of animal random search by including models of heavy-tailed distribution of move length, which complements the correlated random walk paradigm that is founded on Brownian walks. Theory shows that the efficiency of the different foraging tactics is a function of prey abundance and dynamics with Lévy flight being especially efficient in poor prey fields. Lévy flights have been controversial in some quarters, because they previously have been wrongly ascribed to many species through the employment of inappropriate statistical techniques and by misunderstanding movement pattern data. More recent studies using state-of-the-art statistical tools have, however, provided seemingly compelling evidence for Lévy flights. In this study, we employ these maximum-likelihood methods and their Bayesian equivalents by analysing both turning angles and move length distributions. We tested, for compliance with Lévy flight foraging, a set of 77 independent foraging trajectories of Cory's shearwaters Calonectris diomedea diomedea. Birds were tagged with high-resolution GPS loggers in two Mediterranean colonies (Linosa and Tremiti) during both incubation and chick rearing. We found that the behaviour of six birds was fitted by a correlated random walk; the movement of 32 birds was better represented by adaptive correlated random walks by switching from intensive to extensive searches; and the trajectories of 36 birds were fitted by a Lévy flight pattern of movement. The probability of performing Lévy flights was higher for trips during chick provisioning when shearwaters were forced to forage in suboptimal areas. This study supports Lévy flight foraging as an appropriate framework to analyse search tactics in this pelagic bird species and highlights that the adoption of a given search strategy is a function of biological and ecological constraints.

Experimental arena settings might lead to misinterpretation of movement properties.

Movement is an important animal behavior contributing to reproduction and survival. Animal movement is often examined in arenas or enclosures under laboratory conditions. We used the red flour beetle (Tribolium castaneum) to examine here the effect of the arena size, shape, number of barriers, access to the arena's center, and illumination on six movement properties. We demonstrate great differences among arenas. For example, the beetles moved over longer distances in clear arenas than in obstructed ones. Movement along the arena's perimeter was greater in smaller arenas than in larger ones. Movement was more directional in round arenas than in rectangular ones. In general, the beetles stopped moving closer to the perimeter and closer to corners (in the square and rectangular arenas) than expected by chance. In some cases, the arena properties interacted with the beetle sex to affect several movement properties. All these suggest that arena properties might also interact with experimental manipulations to affect the outcome of studies and lead to results specific to the arena used. In other words, instead of examining animal movement, we in fact examine the animal interaction with the arena structure. Caution is therefore advised in interpreting the results of studies on movement in arenas under laboratory conditions and we recommend paying attention also to barriers or obstacles in field experiments. For instance, movement along the arena's perimeter is often interpreted as centrophobism or thigmotaxis but the results here show that such movement is arena dependent.

Absolute direction in organelle movement.

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.

Caterpillar movement mediates spatially local interactions and determines the relationship between population density and contact.

BACKGROUND: While interactions in nature are inherently local, ecological models often assume homogeneity across space, allowing for generalization across systems and greater mathematical tractability. Density-dependent disease models are a prominent example of models that assume homogeneous interactions, leading to the prediction that disease transmission will scale linearly with population density. In this study, we examined how the scale of larval butterfly movement interacts with the resource landscape to influence the relationship between larval contact and population density in the Baltimore checkerspot (Euphydryas phaeton). Our study was inspired by the recent discovery of a viral pathogen that is transmitted horizontally among Baltimore checkerspot larvae. METHODS: We used multi-year larvae location data across six Baltimore checkerspot populations in the eastern U.S. to test whether larval nests are spatially clustered. We then integrated these spatial data with larval movement data in different resource contexts to investigate whether heterogeneity in spatially local interactions alters the assumed linear relationship between larval nest density and contact. We used Correlated Random Walk (CRW) models and field observations of larval movement behavior to construct Probability Distribution Functions (PDFs) of larval dispersal, and calculated the overlap in these PDFs to estimate conspecific contact within each population. RESULTS: We found that all populations exhibited significant spatial clustering in their habitat use. Subsequent larval movement rates were influenced by encounters with host plants and larval age, and under many movement scenarios, the scale of predicted larval movement was not sufficient to allow for the "homogeneous mixing" assumed in density dependent disease models. Therefore, relationships between population density and larval contact were typically non-linear. We also found that observed use of available habitat patches led to significantly greater contact than would occur if habitat use were spatially random. CONCLUSIONS: These findings strongly suggest that incorporating larval movement and spatial variation in larval interactions is critical to modeling disease outcomes in E. phaeton. Epidemiological models that assume a linear relationship between population density and larval contact have the potential to underestimate transmission rates, especially in small populations that are already vulnerable to extinction.

Turn costs change the value of animal search paths.

The tortuosity of the track taken by an animal searching for food profoundly affects search efficiency, which should be optimised to maximise net energy gain. Models examining this generally describe movement as a series of straight steps interspaced by turns, and implicitly assume no turn costs. We used both empirical- and modelling-based approaches to show that the energetic costs for turns in both terrestrial and aerial locomotion are substantial, which calls into question the value of conventional movement models such as correlated random walk or Lévy walk for assessing optimum path types. We show how, because straight-line travel is energetically most efficient, search strategies should favour constrained turn angles, with uninformed foragers continuing in straight lines unless the potential benefits of turning offset the cost.

Selection pressures give composite correlated random walks Lévy walk characteristics.

Composite correlated random walks have been posited as a strong alternative to Lévy walks as models of multi-scale forager movement patterns. Here it is shown that if plastic then intrinsic composite correlated random walks will, under selection pressures, evolve to resemble optimal Lévy walks when foraging is non-destructive. The fittest composite correlated random walkers are found to be those that come closest to being optimal Lévy walkers. This may explain why such a diverse range of foragers have movement patterns that can be approximated by optimal Lévy walks and shows that the 'Lévy-flight foraging' hypothesis has a broad hinterland. The new findings are consistent with recent observations of mussels Mytilus edulis and the Australian desert ant Melophorus bagoti which suggest that animals approximate a Lévy walk by adopting an intrinsic composite movement strategy with different modes.

Not looking where you are leaping: a novel method of oriented travel in the caterpillar Calindoea trifascialis (Moore) (Lepidoptera: Thyrididae).

The prepupation caterpillar of the Southeast Asian moth Calindoea trifascialis constructs a leaf shelter that jumps across the ground using a jumping method novel among the insects. We found that movement path direction was correlated to the direction opposite to the most intense light. Correlated random walk (CRW) analyses found net squared displacements higher than predicted by a CRW, and fractal dimension analysis indicated straighter paths at large spatial scales. Rearing experiments showed high mortality from predation on the ground, but higher mortality resulted from sun exposure. We interpret jumping path orientation as an efficient search strategy to find shade in a variable landscape, given limited perception, in the presence of overheating and desiccation risks.