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.     

Territorial dynamics and stable home range formation for central place foragers

Uncovering the mechanisms behind territory formation is a fundamental problem in behavioural ecology. The broad nature of the underlying conspecific avoidance processes are well documented across a wide range of taxa. Scent marking in particular is common to a large range of terrestrial mammals and is known to be fundamental for communication. However, despite its importance, exact quantification of the time-scales over which scent cues and messages persist remains elusive. Recent work by the present authors has begun to shed light on this problem by modelling animals as random walkers with scent-mediated interaction processes. Territories emerge as dynamic objects that continually change shape and slowly move without settling to a fixed location. As a consequence, the utilisation distribution of such an animal results in a slowly increasing home range, as shown for urban foxes (Vulpes vulpes). For certain other species, however, home ranges reach a stable state. The present work shows that stable home ranges arise when, in addition to scent-mediated conspecific avoidance, each animal moves as a central place forager. That is, the animal's movement has a random aspect but is also biased towards a fixed location, such as a den or nest site. Dynamic territories emerge but the probability distribution of the territory border locations reaches a steady state, causing stable home ranges to emerge from the territorial dynamics. Approximate analytic expressions for the animal's probability density function are derived. A programme is given for using these expressions to quantify both the strength of the animal's movement bias towards the central place and the time-scale over which scent messages persist. Comparisons are made with previous theoretical work modelling central place foragers with conspecific avoidance. Some insights into the mechanisms behind allometric scaling laws of animal space use are also given.     

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.     

Climate alters the movement ecology of a non‐migratory bird

Global climate change is causing increased climate extremes threatening biodiversity and altering ecosystems. Climate is comprised of many variables including air temperature, barometric pressure, solar radiation, wind, relative humidity, and precipitation that interact with each other. As movement connects various aspects of an animal''s life, understanding how climate influences movement at a fine‐temporal scale will be critical to the long‐term conservation of species impacted by climate change. The sedentary nature of non‐migratory species could increase some species risk of extirpation caused by climate change. We used Northern Bobwhite (Colinus virginianus; hereafter bobwhite) as a model to better understand the relationship between climate and the movement ecology of a non‐migratory species at a fine‐temporal scale. We collected movement data on bobwhite from across western Oklahoma during 2019–2020 and paired these data with meteorological data. We analyzed movement in three different ways (probability of movement, hourly distance moved, and sinuosity) using two calculated movement metrics: hourly movement (displacement between two consecutive fixes an hour apart) and sinuosity (a form of tortuosity that determines the amount of curvature of a random search path). We used generalized linear‐mixed models to analyze probability of movement and hourly distance moved, and used linear‐mixed models to analyze sinuosity. The interaction between air temperature and solar radiation affected probability of movement and hourly distance moved. Bobwhite movement increased as air temperature increased beyond 10°C during low solar radiation. During medium and high solar radiation, bobwhite moved farther as air temperature increased until 25–30°C when hourly distance moved plateaued. Bobwhite sinuosity increased as solar radiation increased. Our results show that specific climate variables alter the fine‐scale movement of a non‐migratory species. Understanding the link between climate and movement is important to determining how climate change may impact a species’ space use and fitness now and in the future.     

Memory keeps you at home: a mechanistic model for home range emergence

Despite its central place in animal ecology no general mechanistic movement model with an emergent home-range pattern has yet been proposed. Random walk models, which are commonly used to model animal movement, show diffusion instead of a bounded home range and therefore require special modifications. Current approaches for mechanistic modeling of home ranges apply only to a limited set of taxa, namely territorial animals and/or central place foragers. In this paper we present a more general mechanistic movement model based on a biased correlated random walk, which shows the potential for home-range behavior. The model is based on an animal tracking a dynamic resource landscape, using a biologically plausible two-part memory system, i.e. a reference- and a working-memory. Our results show that by adding these memory processes the random walker produces home-range behavior as it gains experience, which also leads to more efficient resource use. Interestingly, home-range patterns, which we assessed based on home-range overlap and increase in area covered with time, require the combined action of both memory components to emerge. Our model has the potential to predict home-range size and can be used for comparative analysis of the mechanisms shaping home-range patterns.     

Some effects of energy costs on foraging strategies

We consider the effect of including energy costs on the optimal strategy for animals exploiting a depleting food resource. In the context of central place foraging this leads to the problem of what load size should be brought back to the central place. Two strategies are discussed: (i) maximize gross rate of energy delivery and (ii) maximize net rate of energy delivery. The optimal load size (or optimal patch time) for net maximizers is not always larger than for gross maximizers, as has been claimed. Instead, the difference in optimal load size has the same sign as the difference between metabolic rates of travelling and foraging. We point out that the influence of costs has not always been correctly incorporated in experimental tests of the theory.     

Geometry for mutualistic and selfish herds: the limited domain of danger

We present a two-dimensional individual-based model of aggregation behaviour in animals by introducing the concept of a "limited domain of danger", which represents either a limited detection range or a limited attack range of predators. The limited domain of danger provides a suitable framework for the analysis of individual movement rules under real-life conditions because it takes into account the predator's prey detection and capture abilities. For the first time, a single geometrical construct can be used to analyse the predation risk of both peripheral and central individuals in a group. Furthermore, our model provides a conceptual framework that can be equally applied to aggregation behaviour and refuge use and thus presents a conceptual advance on current theory that treats these antipredator behaviours separately. An analysis of individual movement rules using limited domains of danger showed that the time minimization strategy outcompetes the nearest neighbour strategy proposed by Hamilton's (J. Theor. Biol. 31 (1971) 295) selfish herd model, whereas a random strategy confers no benefit and can even be disadvantageous. The superior performance of the time minimization strategy highlights the importance of taking biological constraints, such as an animal's orientation relative to its neighbours, into account when searching for efficient movement rules underlying the aggregation process.     

Intervention in Context-Sensitive Probabilistic Boolean Networks Revisited

An approximate representation for the state space of a context-sensitive probabilistic Boolean network has previously been proposed and utilized to devise therapeutic intervention strategies. Whereas the full state of a context-sensitive probabilistic Boolean network is specified by an ordered pair composed of a network context and a gene-activity profile, this approximate representation collapses the state space onto the gene-activity profiles alone. This reduction yields an approximate transition probability matrix, absent of context, for the Markov chain associated with the context-sensitive probabilistic Boolean network. As with many approximation methods, a price must be paid for using a reduced model representation, namely, some loss of optimality relative to using the full state space. This paper examines the effects on intervention performance caused by the reduction with respect to various values of the model parameters. This task is performed using a new derivation for the transition probability matrix of the context-sensitive probabilistic Boolean network. This expression of transition probability distributions is in concert with the original definition of context-sensitive probabilistic Boolean network. The performance of optimal and approximate therapeutic strategies is compared for both synthetic networks and a real case study. It is observed that the approximate representation describes the dynamics of the context-sensitive probabilistic Boolean network through the instantaneously random probabilistic Boolean network with similar parameters.     

Fitting probability distributions to animal movement trajectories: using artificial neural networks to link distance, resources, and memory

Animal movement paths are often thought of as a confluence of behavioral processes and landscape patterns. Yet it has proven difficult to develop frameworks for analyzing animal movement that can test these interactions. Here we describe a novel method for fitting movement models to data that can incorporate diverse aspects of landscapes and behavior. Using data from five elk (Cervus canadensis) reintroduced to central Ontario, we employed artificial neural networks to estimate movement probability kernels as functions of three landscape-behavioral processes. These consisted of measures of the animals' response to the physical spatial structure of the landscape, the spatial variability in resources, and memory of previously visited locations. The results support the view that animal movement results from interactions among elements of landscape structure and behavior, motivating context-dependent movement probabilities, rather than from successive realizations of static distributions, as some traditional models of movement and resource selection assume. Flexible, nonlinear models may thus prove useful in understanding the mechanisms controlling animal movement patterns.     

A hidden Markov movement model for rapidly identifying behavioral states from animal tracks

Electronic telemetry is frequently used to document animal movement through time. Methods that can identify underlying behaviors driving specific movement patterns can help us understand how and why animals use available space, thereby aiding conservation and management efforts. For aquatic animal tracking data with significant measurement error, a Bayesian state‐space model called the first‐Difference Correlated Random Walk with Switching (DCRWS) has often been used for this purpose. However, for aquatic animals, highly accurate tracking data are now becoming more common. We developed a new hidden Markov model (HMM) for identifying behavioral states from animal tracks with negligible error, called the hidden Markov movement model (HMMM). We implemented as the basis for the HMMM the process equation of the DCRWS, but we used the method of maximum likelihood and the R package TMB for rapid model fitting. The HMMM was compared to a modified version of the DCRWS for highly accurate tracks, the DCRWS, and to a common HMM for animal tracks fitted with the R package moveHMM. We show that the HMMM is both accurate and suitable for multiple species by fitting it to real tracks from a grey seal, lake trout, and blue shark, as well as to simulated data. The HMMM is a fast and reliable tool for making meaningful inference from animal movement data that is ideally suited for ecologists who want to use the popular DCRWS implementation and have highly accurate tracking data. It additionally provides a groundwork for development of more complex modeling of animal movement with TMB. To facilitate its uptake, we make it available through the R package swim.     

Linking animal movement to site fidelity

Site fidelity, the recurrent visit of an animal to a previously occupied area is a wide-spread behavior in the animal kingdom. The relevance of site fidelity to territoriality, successful breeding, social associations, optimal foraging and other ecological processes, demands accurate quantification. Here we generalize previous theory that connects site fidelity patterns to random walk parameters within the framework of the space-time fractional diffusion equation. In particular, we describe the site fidelity function in terms of animal movement characteristics via the Lévy exponent, which controls the step-length distribution of the random steps at each turning point, and the waiting time exponent that controls for how long an animal awaits before actually moving. The analytical results obtained will provide a rigorous benchmark for empirically driven studies of animal site fidelity.     

Emergence of adaptive global movement from a subjective inference about local resource distribution

In animal foraging, the optimal search strategy in an unknown environment varies depending on the context, such as the resource density and season. When food is distributed sparsely and uniformly, superdiffusive walks outperform normal-diffusive walks. However, superdiffusive walks are no longer advantageous when random walkers forage in resource-rich environments. It is not currently clear whether a relationship exists between an agent's use of local information to make subjective inferences about global food distribution and the optimal random walk strategy. Therefore, I investigated how flexible exploration is achieved if an agent alters its directional rule based on the local resource distribution. In the proposed model, the agent, a Brownian-like walker, estimates whether an abundant or sparse area is nearby using local resource patterns and then makes a decision by altering its movement rules. I show that the agent can behave like a non-Brownian walker if it interacts with a prey distribution. The agent can adaptively switch between diffusive properties depending on the resource density. This leads to a more effective resource-searching performance than a simple random-walk model. These results demonstrate that optimal searching is a context-dependent process.     

Estimating animal behavior and residency from movement data

We present a process‐based approach to estimate residency and behavior from uncertain and temporally correlated movement data collected with electronic tags. The estimation problem is formulated as a hidden Markov model (HMM) on a spatial grid in continuous time, which allows straightforward implementation of barriers to movement. Using the grid to explicitly resolve space, location estimation can be supplemented by or based entirely on environmental data (e.g. temperature, daylight). The HMM method can therefore analyze any type of electronic tag data. The HMM computes the joint posterior probability distribution of location and behavior at each point in time. With this, the behavioral state of the animal can be associated to regions in space, thus revealing migration corridors and residence areas. We demonstrate the inferential potential of the method by analyzing satellite‐linked archival tag data from a southern bluefin tuna Thunnus maccoyii where longitudinal coordinates inferred from daylight are supplemented by latitudinal information in recorded sea surface temperatures.     

A Minimum‐Cost Strategy for Cluster Recruitment

It is becoming increasingly common for the design of a clinical study to involve cluster samples. Very few researches investigated the appropriate number of clusters. None of them treat cluster size and the number of clusters as random variables. In reality, the recruitment of clusters can not be reached at one time and the cluster sizes are usually random. The longer the recruitment takes the more expensive the total study costs will be. This paper provides a strategy for sequential recruitment of clusters, which can minimize the total study cost. By treating the number of additional observational subjects required at each time point as a Markov Chain, we derive an iterative procedure for optimal strategy and study the property of this strategy, especially the duration of the cluster recruitment. This strategy is also extended to search for an optimal number of centers in a multi‐center clinical trial.     

Prey processing in central place foragers

The importance of prey processing as an integral part of foraging behaviour has long been acknowledged, but little theoretical consideration has been given to the optimization of the processing behaviour itself. Processing renders food down to ingestible, palatable portions, and also removes non-essential mass thus reducing transport costs. Here, several models of processing are developed for a central place forager. When the forager has to make a simple choice between processing the prey and not, a critical distance from the central place can be calculated, beyond which it is optimal to process prey. If the forager also decides on how much of the prey to remove, the optimal amount to be removed can also be calculated. Imposing a ceiling on overall metabolic expenditure is shown to reduce the distances at which processing becomes the optimal strategy. The models are tested using parameters derived for a provisioning merlin, Falco columbarius, and alternative explanations as to why observed behaviours should differ from the optimal behaviour predicted are discussed.     

Should I Stay or Should I Go? A Habitat-Dependent Dispersal Kernel Improves Prediction of Movement

The analysis of animal movement within different landscapes may increase our understanding of how landscape features affect the perceptual range of animals. Perceptual range is linked to movement probability of an animal via a dispersal kernel, the latter being generally considered as spatially invariant but could be spatially affected. We hypothesize that spatial plasticity of an animal''s dispersal kernel could greatly modify its distribution in time and space. After radio tracking the movements of walking insects (Cosmopolites sordidus) in banana plantations, we considered the movements of individuals as states of a Markov chain whose transition probabilities depended on the habitat characteristics of current and target locations. Combining a likelihood procedure and pattern-oriented modelling, we tested the hypothesis that dispersal kernel depended on habitat features. Our results were consistent with the concept that animal dispersal kernel depends on habitat features. Recognizing the plasticity of animal movement probabilities will provide insight into landscape-level ecological processes.     

The Mass-Longevity Triangle: Pareto Optimality and the Geometry of Life-History Trait Space

When organisms need to perform multiple tasks they face a fundamental tradeoff: no phenotype can be optimal at all tasks. This situation was recently analyzed using Pareto optimality, showing that tradeoffs between tasks lead to phenotypes distributed on low dimensional polygons in trait space. The vertices of these polygons are archetypes—phenotypes optimal at a single task. This theory was applied to examples from animal morphology and gene expression. Here we ask whether Pareto optimality theory can apply to life history traits, which include longevity, fecundity and mass. To comprehensively explore the geometry of life history trait space, we analyze a dataset of life history traits of 2105 endothermic species. We find that, to a first approximation, life history traits fall on a triangle in log-mass log-longevity space. The vertices of the triangle suggest three archetypal strategies, exemplified by bats, shrews and whales, with specialists near the vertices and generalists in the middle of the triangle. To a second approximation, the data lies in a tetrahedron, whose extra vertex above the mass-longevity triangle suggests a fourth strategy related to carnivory. Each animal species can thus be placed in a coordinate system according to its distance from the archetypes, which may be useful for genome-scale comparative studies of mammalian aging and other biological aspects. We further demonstrate that Pareto optimality can explain a range of previous studies which found animal and plant phenotypes which lie in triangles in trait space. This study demonstrates the applicability of multi-objective optimization principles to understand life history traits and to infer archetypal strategies that suggest why some mammalian species live much longer than others of similar mass.     

Inference in MCMC step selection models

Habitat selection models are used in ecology to link the spatial distribution of animals to environmental covariates and identify preferred habitats. The most widely used models of this type, resource selection functions, aim to capture the steady-state distribution of space use of the animal, but they assume independence between the observed locations of an animal. This is unrealistic when location data display temporal autocorrelation. The alternative approach of step selection functions embed habitat selection in a model of animal movement, to account for the autocorrelation. However, inferences from step selection functions depend on the underlying movement model, and they do not readily predict steady-state space use. We suggest an analogy between parameter updates and target distributions in Markov chain Monte Carlo (MCMC) algorithms, and step selection and steady-state distributions in movement ecology, leading to a step selection model with an explicit steady-state distribution. In this framework, we explain how maximum likelihood estimation can be used for simultaneous inference about movement and habitat selection. We describe the local Gibbs sampler, a novel rejection-free MCMC scheme, use it as the basis of a flexible class of animal movement models, and derive its likelihood function for several important special cases. In a simulation study, we verify that maximum likelihood estimation can recover all model parameters. We illustrate the application of the method with data from a zebra.     

A model of animal movements in a bounded space

Most studies describing animal movements have been developed in the framework of population dispersion or population dynamics, and have mainly focused on movements in open spaces. During their trips, however, animals are likely to encounter physical heterogeneities that guide their movements and, as a result, influence their spatial distribution. In this paper, we develop a statistical model of individual movement in a bounded space. We introduced cockroaches in a circular arena and quantified accurately the behaviors underlying their movement in a finite space. Close to the edges, we considered that the animals exhibit a linear movement mode with a constant probability per unit time to leave the edge and enter the central zone of the arena. Far from the walls cockroaches were assumed to move according to a diffusive random walk which enabled us to overcome the inherent problem of the quantification of the turning angle distribution. A numerical model implementing the behavioral rules derived from our experiments, confirms that the pattern of the spatial distribution of animals observed can be reliably accounted for by wall-following behaviors combined with a diffusive random walk. The approach developed in this study can be applied to model the movements of animals in various environment under consideration of spatial structure.