Turn designation, sampling rate and the misidentification of power laws in movement path data using maximum likelihood estimates

Many authors have claimed to observe animal movement paths that appear to be Lévy walks, i.e. a random walk where the distribution of move lengths follows an inverse power law. A Lévy walk is known to be the optimal search strategy of a particular class of random walks in certain environments; hence, it is important to distinguish correctly between Lévy walks and other types of random walks in observed animal movement paths. Evidence of a power law distribution in the step length distribution of observed animal movement paths is often used to classify a particular movement path as a Lévy walk. However, there is some doubt about the accuracy of early studies that apparently found Lévy walk behaviour. A recently accepted method to determine whether a movement path truly exhibits Lévy walk behaviour is based on an analysis of move lengths with a maximum likelihood estimate using Akaike weights. Here, we show that simulated (non-Lévy) random walks representing different types of animal movement behaviour (a composite correlated random walk; pooled data from a set of random walks with different levels of correlation and three-dimensional correlated random walks projected into one dimension) can all show apparent power law behaviour typical of Lévy walks when using the maximum likelihood estimation method. The probability of the movement path being identified as having a power law step distribution is related to both the sampling rate used by the observer and the way that ‘turns’ or ‘reorientations’ in the movement path are designated. However, identification is also dependent on the nature and properties of the simulated path, and there is currently no standard method of observation and analysis that is robust for all cases. Our results indicate that even apparently robust maximum likelihood methods can lead to a mismatch between pattern and process, as paths arising from non-Lévy walks exhibit Lévy-like patterns.     

Sampling rate effects on measurements of correlated and biased random walks

When observing the two-dimensional movement of animals or microorganisms, it is usually necessary to impose a fixed sampling rate, so that observations are made at certain fixed intervals of time and the trajectory is split into a set of discrete steps. A sampling rate that is too small will result in information about the original path and correlation being lost. If random walk models are to be used to predict movement patterns or to estimate parameters to be used in continuum models, then it is essential to be able to quantify and understand the effect of the sampling rate imposed by the observer on real trajectories. We use a velocity jump process with a realistic reorientation model to simulate correlated and biased random walks and investigate the effect of sampling rate on the observed angular deviation, apparent speed and mean turning angle. We discuss a method of estimating the values of the reorientation parameters used in the original random walk from the rediscretized data that assumes a linear relation between sampling time step and the parameter values.     

Animal movement rates as behavioural bouts

Johnson et al . ( Journal of Animal Ecology , 2002, 71 , 225–235) have proposed a new technique for identifying scales of movement in animals. Animals are located at certain time intervals, and movement rates between successive animal relocations are calculated. The null model of a nonscalar response predicts a decreasing linear relationship between log(frequency) vs. movement rate, while a scalar response predicts a monotonically decreasing curve with an inflection point at the separation between the processes. I tested this technique using three types of simulated movement paths: correlated random walks, directed walks, and movements in patchy habitat. None of the simulations showed the results expected by the technique. This occurs because the technique assumes that movement rates are exponentially distributed, which is highly unlikely. Thus before this technique can be applied to animal movement data we need to understand how spatial and temporal scale, as well as sampling interval, affect the frequency histogram of animal movement rates.     

Improving Accuracy and Precision in Estimating Fractal Dimension of Animal movement paths

It is difficult to watch wild animals while they move, so often biologists analyse characteristics of animal movement paths. One common path characteristic used is tortuousity, measured using the fractal dimension (D). The typical method for estimating fractal D, the divider method, is biased and imprecise. The bias occurs because the path length is truncated. I present a method for minimising the truncation error. The imprecision occurs because sometimes the divider steps land inside the bends of curves, and sometimes they miss the curves. I present three methods for minimising this variation and test the methods with simulated correlated random walks. The traditional divider method significantly overestimates fractal D when paths are short and the range of spatial scales is narrow. The best method to overcome these problems consists of walking the dividers forwards and backwards along the path, and then estimating the path length remaining at the end of the last divider step.     

Scales of orientation, directed walks and movement path structure in sharks

1. Animal search patterns reflect sensory perception ranges combined with memory and knowledge of the surrounding environment. 2. Random walks are used when the locations of resources are unknown, whereas directed walks should be optimal when the location of favourable habitats is known. However, directed walks have been quantified for very few species. 3. We re-analysed tracking data from three shark species to determine whether they were using directed walks, and if so, over which spatial scales. Fractal analysis was used to quantify how movement structure varied with spatial scale and determine whether the sharks were using patches. 4. Tiger sharks performed directed walks at large spatial scales (at least 6-8 km). Thresher sharks also showed directed movement (at scales of 400-1900 m), and adult threshers were able to orient at greater scales than juveniles, which may suggest that learning improves the ability to perform directed walks. Blacktip reef sharks had small home ranges, high site fidelity and showed no evidence of oriented movements at large scales. 5. There were inter- and intraspecific differences in path structure and patch size, although most individuals showed scale-dependent movements. Furthermore, some individuals of each species performed movements similar to a correlated random walk. 6. Sharks can perform directed walks over large spatial scales, with scales of movements reflecting site fidelity and home range size. Understanding when and where directed walks occur is crucial for developing more accurate population-level dispersal models.     

Are there general mechanisms of animal home range behaviour? A review and prospects for future research

Home range behaviour is a common pattern of space use, having fundamental consequences for ecological processes. However, a general mechanistic explanation is still lacking. Research is split into three separate areas of inquiry - movement models based on random walks, individual-based models based on optimal foraging theory, and a statistical modelling approach - which have developed without much productive contact. Here we review recent advances in modelling home range behaviour, focusing particularly on the problem of identifying mechanisms that lead to the emergence of stable home ranges from unbounded movement paths. We discuss the issue of spatiotemporal scale, which is rarely considered in modelling studies, as well as highlighting the need to consider more closely the dynamical nature of home ranges. Recent methodological and theoretical advances may soon lead to a unified approach, however, conceptually unifying our understanding of linkages among home range behaviour and ecological or evolutionary processes.     

A discrete space model for continuous space dispersal processes

The simulation of dispersal processes in landscapes over large spatial extents is challenging because of the large difference in geographical scale between overwhelmingly dominant localised dispersal events, and rare long-distance dispersal events which typically drive overall rates of spread. While localised dispersal may point to high resolution individual level models, long-distance dispersal events are likely to involve much coarser grid-based models. In this paper we propose a discrete space (i.e., grid-based) model for dispersal processes in continuous space. We start by illustrating the behaviour of continuous space walks when their movement is discretised to a grid. The importance of short time period cell-to-cell moves which return a walk to its previous grid cell location is identified. A conceptual model which uses a Markov chain buffer phase between cells to replicate the observed behaviour of discretised continuous space walks is proposed. Analysis of the Markov chain shows that it can be parameterised using just two parameters in addition to the dispersal kernel. An algorithm for implementation of the proposed model is presented. Empirical results demonstrate that the proposed mechanism produces good matches to continuous space dispersal processes with both exponential and heavy-tailed dispersal kernels.     

The evolution of an 'intelligent' dispersal strategy: biased, correlated random walks in patchy landscapes

Theoretical work exploring dispersal evolution focuses on the emigration rate of individuals and typically assumes that movement occurs either at random to any other patch or to one of the nearest‐neighbour patches. There is a lack of work exploring the process by which individuals move between patches, and how this process evolves. This is of concern because any organism that can exert control over dispersal direction can potentially evolve efficiencies in locating patches, and the process by which individuals find new patches will potentially have major effects on metapopulation dynamics and gene flow. Here, we take an initial step towards filling this knowledge gap. To do this we constructed a continuous space population model, in which individuals each carry heritable trait values that specify the characteristics of the biased correlated random walk they use to disperse from their natal patch. We explore how the evolution of the random walk depends upon the cost of dispersal, the density of patches in the landscape, and the emigration rate. The clearest result is that highly correlated walks always evolved (individuals tended to disperse in relatively straight lines from their natal patch), reflecting the efficiency of straight‐line movement. In our models, more costly dispersal resulted in walks with higher correlation between successive steps. However, the exact walk that evolved also depended upon the density of suitable habitat patches, with low density habitat evolving more biased walks (individuals which orient towards suitable habitat at quite large distances from that habitat). Thus, low density habitat will tend to develop individuals which disperse efficiently between adjacent habitat patches but which only rarely disperse to more distant patches; a result that has clear implications for metapopulation theory. Hence, an understanding of the movement behaviour of dispersing individuals is critical for robust long‐term predictions of population dynamics in fragmented landscapes.     

First capture success in two dimensions: The search for prey by a random walk predator in a comprehensive space of random walks

Random walk models are an important tool used for understanding how complex organisms redistribute themselves through space and time in search of targets such as food, shelter, or mates. These walks are easily studied with agent-based models, which can be used to ask which search strategy is best according to some efficiency metric. Current studies however, generally do not consider the full range of potential random walks, success metrics, and constraints on the walker, and implementation details vary widely. It is therefore difficult to compare results across studies. In this paper, we investigate predator search behaviour in a comprehensive space of key movement variables that allows the predator to select from a continuum of random walks ranging from Brownian walks (BWs) to correlated random walks (CorRWs) which include directional persistence, to composite random walks (ComRWs) which feature intensive and extensive search modes (ISMs and ESMs), and finally to more complex correlated composite random walks (CCRWs). We specifically focus on the search behaviour of a predator between the initiation of a search for a prey item and the first successful acquisition of a prey target: we call this interval the “search-to-capture” event. We measure the predator's success against three metrics of energetic cost: (1) the time elapsed, (2) the distance travelled, and (3) an equally weighted combination of time and distance. In addition, we explore the effect of three different constraints on the predator: (1) hunting success in the extensive search mode, (2) detection radius when in the extensive search mode, and (3) prey density. Our work confirms the broadly held notion that CCRW movement patterns should always outperform BWs, but find instructive cases where other walks are superior. We also show that, within the CCRW category, there is a wide range of possible walks and rank these according to measures of energetic cost. Our work also offers insights into the evolutionary pressures surrounding the “search-to-capture” event, and suggests that CCRW predators with low hunting success in one movement mode experience higher evolutionary pressures and are thus more likely to adopt a nearly optimal random walk. Our work highlights the need for comprehensive studies that examine several aspects of random walks simultaneously.     

From random walks to informed movement

The analysis of animal movement is a large and continuously growing field of research. Detailed knowledge about movement strategies is of crucial importance for understanding eco‐evolutionary dynamics at all scales – from individuals to (meta‐)populations. This and the availability of detailed movement and dispersal data motivated Nathan and colleagues to published their much appreciated call to base movement ecology on a more thorough mechanistic basis. So far, most movement models are based on random walks. However, even if a random walk might describe real movement patterns acceptably well, there is no reason to assume that animals move randomly. Therefore, mechanistic models of foraging strategies should be based on information use and memory in order to increase our understanding of the processes that lead to animal movement decisions. We present a mechanistic movement model of an animal with a limited perceptual range and basic information storage capacities. This ‘spatially informed forager’ constructs an internal map of its environment by using perception, memory and learned or evolutionarily acquired assumptions about landscape attributes. We analyse resulting movement patterns and search efficiencies and compare them to area restricted search strategies (ARS) and biased correlated random walks (BCRW) of omniscient individuals. We show that, in spite of their limited perceptual range, spatially informed individuals boost their foraging success and may perform much better than the best ARS. The construction of an internal map and the use of spatial information results in the emergence of a highly correlated walk between patches and a rather systematic search within resource clusters. Furthermore, the resulting movement patterns may include foray search behaviour. Our work highlights the strength of mechanistic modelling approaches and sets the stage for the development of more sophisticated models of memory use for movement decisions and dispersal.     

How to reliably estimate the tortuosity of an animal's path: straightness, sinuosity, or fractal dimension?

The tortuosity of an animal's path is a key parameter in orientation and searching behaviours. The tortuosity of an oriented path is inversely related to the efficiency of the orientation mechanism involved, the best mechanism being assumed to allow the animal to reach its goal along a straight line movement. The tortuosity of a random search path controls the local searching intensity, allowing the animal to adjust its search effort to the local profitability of the environment. This paper shows that (1) the efficiency of an oriented path can be reliably estimated by a straightness index computed as the ratio between the distance from the starting point to the goal and the path length travelled to reach the goal, but such a simple index, ranging between 0 and 1, cannot be applied to random search paths; (2) the tortuosity of a random search path, ranging between straight line movement and Brownian motion, can be reliably estimated by a sinuosity index which combines the mean cosine of changes of direction with the mean step length; and (3) in the current state of the art, the fractal analysis of animals' paths, which may appear as an alternative and promising way to measure the tortuosity of a random search path as a fractal dimension ranging between 1 (straight line movement) and 2 (Brownian motion), is only liable to generate artifactual results. This paper also provides some help for distinguishing between oriented and random search paths, and depicts a general, comprehensive framework for analysing individual animals' paths in a two-dimensional space.     

Searching on patch networks using correlated random walks: space usage and optimal foraging predictions using Markov chain models

We describe a novel representation of a discrete correlated random walk as the transition matrix of a Markov chain with the displacements as the states. Such a representation makes it possible to utilize results from the theory of absorbing Markov chains, to make biologically interesting predictions without having to resort to Monte Carlo simulations. Our motivation for constructing such a representation is to explore the relationship between the movement strategy of an animal searching for resources upon a network of patches, and its consequent utilization of space and foraging success. As an illustrative case study, we have determined the optimal movement strategy and the consequent usage of space for a central place forager utilizing a continuous movement space which is discretized as a hexagonal lattice. The optimal movement strategy determines the size of the optimal home range. In this example, the animal uses mnemokinesis, which is a sinuosity regulating mechanism, to return it to the central place. The movement strategy thus refers to the choice of the intrinsic path sinuosity and the strength of the mnemokinetic mechanism. Although the movement space has been discretized as a regular lattice in this example, the method can be readily applied to naturally compartmentalized movement spaces, such as forest canopy networks. This paper is thus an attempt at incorporating results from the theory of random walk-based animal movements into Foraging Theory.     

Planning connectivity at multiple scales for large mammals in a human-dominated biodiversity hotspot

Connectivity for large mammals across human-altered landscapes results from movement by individuals that can be described via nested spatial scales as linkages (or zones or areas) with compatible land use types, constrictions that repeatedly funnel movement (as corridors) or impede it (as barriers), and the specific paths (or routes) across completely anthropogenic features (such as highways). Mitigation to facilitate animal movement through such landscapes requires similar attention to spatial scale, particularly when they involve complex topography, diverse types of human land use, and transportation infrastructure. We modeled connectivity for Asian elephant (Elephas maximus) and gaur (Bos gaurus) in the Shencottah Gap, a multiple-use region separating two tiger reserves in the Western Ghats, India. Using 840 km of surveys for animal signs within a region of 621 km², we modeled landscape linkages via resource selection functions integrated across two spatial resolutions, and then potential dispersal corridors within these linkages using circuit theoretical models. Within these corridors, we further identified potential small-scale movement paths across a busy transportation route via least-cost paths and evaluated their viability. Both elephants and gaur avoided human-dominated habitat, resulting in broken connectivity across the Shencottah Gap. Predicted corridor locations were sensitive to analysis resolution, and corridors derived from scale-integrated habitat models correlated best with habitat quality. Less than 1% of elephant and gaur detections occurred in habitat that was poorer in quality than the lowest-quality component of the movement path across the transportation route, suggesting that connectivity will require habitat improvement. Only 28% of dispersal corridor area and 5% of movement path length overlapped with the upper 50% quantile of the landscape linkage; thus, jointly modeling these three components enabled a more nuanced evaluation of connectivity than any of them in isolation.     

Biologically meaningful update rules increase the critical connectivity of generalized Kauffman networks

We generalize random Boolean networks by softening the hard binary discretization into multiple discrete states. These multistate networks are generic models of gene regulatory networks, where each gene is known to assume a finite number of functionally different expression levels. We analytically determine the critical connectivity that separates the biologically unfavorable frozen and chaotic regimes. This connectivity is inversely proportional to a parameter which measures the heterogeneity of the update rules. Interestingly, the latter does not necessarily increase with the mean number of discrete states per node. Still, allowing for multiple states decreases the critical connectivity as compared to random Boolean networks, and thus leads to biologically unrealistic situations.Therefore, we study two approaches to increase the critical connectivity. First, we demonstrate that each network can be kept in its frozen regime by sufficiently biasing the update rules. Second, we restrict the randomly chosen update rules to a subclass of biologically more meaningful functions. These functions are characterized based on a thermodynamic model of gene regulation. We analytically show that their usage indeed increases the critical connectivity. From a general point of view, our thermodynamic considerations link discrete and continuous models of gene regulatory networks.     

Equivalence between Step Selection Functions and Biased Correlated Random Walks for Statistical Inference on Animal Movement

Animal movement has a fundamental impact on population and community structure and dynamics. Biased correlated random walks (BCRW) and step selection functions (SSF) are commonly used to study movements. Because no studies have contrasted the parameters and the statistical properties of their estimators for models constructed under these two Lagrangian approaches, it remains unclear whether or not they allow for similar inference. First, we used the Weak Law of Large Numbers to demonstrate that the log-likelihood function for estimating the parameters of BCRW models can be approximated by the log-likelihood of SSFs. Second, we illustrated the link between the two approaches by fitting BCRW with maximum likelihood and with SSF to simulated movement data in virtual environments and to the trajectory of bison (Bison bison L.) trails in natural landscapes. Using simulated and empirical data, we found that the parameters of a BCRW estimated directly from maximum likelihood and by fitting an SSF were remarkably similar. Movement analysis is increasingly used as a tool for understanding the influence of landscape properties on animal distribution. In the rapidly developing field of movement ecology, management and conservation biologists must decide which method they should implement to accurately assess the determinants of animal movement. We showed that BCRW and SSF can provide similar insights into the environmental features influencing animal movements. Both techniques have advantages. BCRW has already been extended to allow for multi-state modeling. Unlike BCRW, however, SSF can be estimated using most statistical packages, it can simultaneously evaluate habitat selection and movement biases, and can easily integrate a large number of movement taxes at multiple scales. SSF thus offers a simple, yet effective, statistical technique to identify movement taxis.     

Using animal movement paths to measure response to spatial scale

Animals live in an environment that is patchy and hierarchical. I present a method of detecting the scales at which animals perceive their world. The hierarchical nature of habitat causes movement path structure to vary with spatial scale, and the patchy nature of habitat causes movement path structure to vary throughout space. These responses can be measured by a combination of path tortuousity (measured with fractal dimension) versus spatial scale, the variation in tortuousity of small path segments along the movement path, and the correlation between tortuousities of adjacent path segments. These statistics were tested using simulated animal movements. When movement paths contained no spatial heterogeneity, then fractal D and variance continuously increased with scale, and correlation was zero at all scales. When movement paths contained spatial heterogeneity, then fractal D sometimes showed a discontinuity at transitions between domains of scale, variation showed peaks at transitions, and correlations showed a statistically significant positive value at scales smaller than patch size, decreasing to below zero at scales greater than patch size. I illustrated these techniques with movement paths from deer mice and red-backed voles. These new analyses should help understand how animals perceive and react to their landscape structure at various spatial scales, and to answer questions about how habitat structure affects animal movement patterns.     

Quantitative analysis of bottlenose dolphin movement patterns and their relationship with foraging

1. Broad-scale telemetry studies have greatly improved our understanding of the ranging patterns and habitat-use of many large vertebrates. However, there often remains considerable uncertainty over the function of different areas or the factors influencing habitat selection. Further insights into these processes can be obtained through analyses of finer scale movement patterns. For example, search behaviour may be modified in response to prey distribution and abundance. 2. In this study, quantitative analysis techniques are applied to the movements of bottlenose dolphins, recorded from land using a theodolite, to increase our understanding of their foraging strategies. Movements were modelled as a correlated random walk (CRW) and a biased random walk (BRW) to identify movement types and using a first-passage time (FPT) approach, which quantifies the time allocated to different areas and identifies the location and spatial scale of intensive search effort. 3. Only a quarter of the tracks were classed as CRW movement. Turning angle and directionality appeared to be key factors in determining the type of movement adopted. A high degree of overlap in search effort between separate movement paths indicated that there were small key sites (0.3 km radius) within the study area (4 km(2)). Foraging behaviour occurred mainly within these intensive search areas, indicating that they were feeding sites. 4. This approach provides a quantitative method of identifying important foraging areas and their spatial scale. Such techniques could be applied to movement paths for a variety of species derived from telemetry studies and increase our understanding of their foraging strategies.     

Landscape and fine-scale movements of a leaf beetle: the importance of boundary behaviour

Movement underpins animal spatial ecology and is often modelled as habitat-dependent correlated random walks. Here, we develop such a model for the flightless tansy leaf beetle Chrysolina graminis moving within and between patches of its host plant tansy Tanacetum vulgare. To parameterize the model, beetle movement paths on timescales of minutes were observed in uniform plots of tansy and inter-patch matrix (meadow) vegetation. Movement lasted longer, covered greater distances and had narrower turning angles in the matrix. Simulations of the model emulated an independent two-season multi-patch mark–resight study at daily timescales and included variable boundary-mediated behaviour affecting the probability of leaving habitat patches. As boundaries in the model became stronger there were disproportionately large decreases in net displacements, inter-patch movements and the proportion of beetles in the matrix. The model produced realistic patterns of population-level displacement over periods up to 13 days with fully permeable boundaries for one dataset and strong boundaries for the other. This may be explained by the heights of the tansy patches in each study, as beetles will be unable to cross the boundary near the top of a patch that emerges from the matrix. The simulations demonstrate the important effects of boundary behaviour on displacement patterns and indicate temporal and spatial variability in permeability. Realistic models of movement must therefore include behaviour at habitat boundaries. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.