Mechanistic home range models capture spatial patterns and dynamics of coyote territories in Yellowstone

Patterns of space-use by individuals are fundamental to the ecology of animal populations influencing their social organization, mating systems, demography and the spatial distribution of prey and competitors. To date, the principal method used to analyse the underlying determinants of animal home range patterns has been resource selection analysis (RSA), a spatially implicit approach that examines the relative frequencies of animal relocations in relation to landscape attributes. In this analysis, we adopt an alternative approach, using a series of mechanistic home range models to analyse observed patterns of territorial space-use by coyote packs in the heterogeneous landscape of Yellowstone National Park. Unlike RSAs, mechanistic home range models are derived from underlying correlated random walk models of individual movement behaviour, and yield spatially explicit predictions for patterns of space-use by individuals. As we show here, mechanistic home range models can be used to determine the underlying determinants of animal home range patterns, incorporating both movement responses to underlying landscape heterogeneities and the effects of behavioural interactions between individuals. Our analysis indicates that the spatial arrangement of coyote territories in Yellowstone is determined by the spatial distribution of prey resources and an avoidance response to the presence of neighbouring packs. We then show how the fitted mechanistic home range model can be used to correctly predict observed shifts in the patterns of coyote space-use in response to perturbation.     

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.     

Behavioral intermittence, Lévy patterns, and randomness in animal movement

The recent debate on both the existence and the cause of fractal (Lévy) patterns in animal movement resonates with much deeper and richer problems in movement ecology: (1) establishing mechanistic links between animal behavior and statistical patterns of movement, and (2) understanding what is the role of randomness (stochasticity) in animal motion. Here, the idea of behavioral intermittence is shown to be crucial to establish mechanistic connections between the behavior of organisms and the statistical properties they generate when moving. Attention is drawn to the fact that some random walk modeling procedures can impair the identification of intermittent biological mechanisms which could govern major statistical properties of movement. This fact, together with some misconceptions and prejudices regarding the role of randomness in animal motion may explain why stochastic processes have been disregarded as a potential source of adaptation in animal movement. In the near future, the advances in biotelemetry together with a more explicit consideration of behavioral intermittence, and the development of novel random walk approaches, could help us to set up the bases for a landscape-level behavioral ecology.     

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.     

Linking rates of diffusion and consumption in relation to resources

The functional response is a fundamental model of the relationship between consumer intake rate and resource abundance. The random walk is a fundamental model of animal movement and is well approximated by simple diffusion. Both models are central to our understanding of numerous ecological processes but are rarely linked in ecological theory. To derive a synthetic model, we draw on the common logical premise underlying these models and show how the diffusion and consumption rates of consumers depend on elementary attributes of naturally occurring consumer-resource interactions: the abundance, spatial aggregation, and traveling speed of resources as well as consumer handling time and directional persistence. We show that resource aggregation may lead to increased consumer diffusion and, in the case of mobile resources, reduced consumption rate. Resource-dependent movement patterns have traditionally been attributed to area-restricted search, reflecting adaptive decision making by the consumer. Our synthesis provides a simple alternative hypothesis that such patterns could also arise as a by-product of statistical movement mechanics.     

A diffusion-based theory of organism dispersal in heterogeneous populations

We develop a general theory of organism movement in heterogeneous populations that can explain the leptokurtic movement distributions commonly measured in nature. We describe population heterogeneity in a state-structured framework, employing advection-diffusion as the fundamental movement process of individuals occupying different movement states. Our general analysis shows that population heterogeneity in movement behavior can be defined as the existence of different movement states and among-individual variability in the time individuals spend in these states. A presentation of moment-based metrics of movement illustrates the role of these attributes in general dispersal processes. We also present a special case of the general theory: a model population composed of individuals occupying one of two movement states with linear transitions, or exchange, between the two states. This two-state "exchange model" can be viewed as a correlated random walk and provides a generalization of the telegraph equation. By exploiting the main result of our general analysis, we characterize the exchange model by deriving moment-based metrics of its movement process and identifying an analytical representation of the model's time-dependent solution. Our results provide general and specific theoretical explanations for empirical patterns in organism movement; the results also provide conceptual and analytical bases for extending diffusion-based dispersal theory in several directions, thereby facilitating mechanistic links between individual behavior and spatial population dynamics.     

Stochastic modelling of animal movement

Modern animal movement modelling derives from two traditions. Lagrangian models, based on random walk behaviour, are useful for multi-step trajectories of single animals. Continuous Eulerian models describe expected behaviour, averaged over stochastic realizations, and are usefully applied to ensembles of individuals. We illustrate three modern research arenas. (i) Models of home-range formation describe the process of an animal ‘settling down’, accomplished by including one or more focal points that attract the animal''s movements. (ii) Memory-based models are used to predict how accumulated experience translates into biased movement choices, employing reinforced random walk behaviour, with previous visitation increasing or decreasing the probability of repetition. (iii) Lévy movement involves a step-length distribution that is over-dispersed, relative to standard probability distributions, and adaptive in exploring new environments or searching for rare targets. Each of these modelling arenas implies more detail in the movement pattern than general models of movement can accommodate, but realistic empiric evaluation of their predictions requires dense locational data, both in time and space, only available with modern GPS telemetry.     

How a simple adaptive foraging strategy can lead to emergent home ranges and increased food intake

Animals often alternate between searching for food locally and moving over larger distances depending on the amount of food they find. This ability to switch between movement modes can have large implications on the fate of individuals and populations, and a mechanism that allows animals to find the optimal balance between alternative movement strategies is therefore selectively advantageous. Recent theory suggests that animals are capable of switching movement mode depending on heterogeneities in the landscape, and that different modes may predominate at different temporal scales. Here we develop a conceptual model that enables animals to use either an area‐concentrated food search behavior or undirected random movements. The model builds on the animals’ ability to remember the profitability and location of previously visited areas. In contrast to classical optimal foraging models, our model does not assume food to be distributed in large, well‐defined patches, and our focus is on animal movement rather than on how animals choose between foraging patches with known locations and value. After parameterizing the fine‐scale movements to resemble those of the harbor porpoise Phocoena phocoena we investigate whether the model is capable of producing emergent home ranges and use pattern‐oriented modeling to evaluate whether it can reproduce the large‐scale movement patterns observed for porpoises in nature. Finally we investigate whether the model enables animals to forage optimally. We found that the model was indeed able to produce either stable home ranges or movement patterns that resembled those of real porpoises. It enabled animals to maximize their food intake when fine‐tuning the memory parameters that controlled the relative contribution of area concentrated and random movements.     

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.     

Memory Effects on Movement Behavior in Animal Foraging

An individual’s choices are shaped by its experience, a fundamental property of behavior important to understanding complex processes. Learning and memory are observed across many taxa and can drive behaviors, including foraging behavior. To explore the conditions under which memory provides an advantage, we present a continuous-space, continuous-time model of animal movement that incorporates learning and memory. Using simulation models, we evaluate the benefit memory provides across several types of landscapes with variable-quality resources and compare the memory model within a nested hierarchy of simpler models (behavioral switching and random walk). We find that memory almost always leads to improved foraging success, but that this effect is most marked in landscapes containing sparse, contiguous patches of high-value resources that regenerate relatively fast and are located in an otherwise devoid landscape. In these cases, there is a large payoff for finding a resource patch, due to size, value, or locational difficulty. While memory-informed search is difficult to differentiate from other factors using solely movement data, our results suggest that disproportionate spatial use of higher value areas, higher consumption rates, and consumption variability all point to memory influencing the movement direction of animals in certain ecosystems.     

An allometric model of home range formation explains the structuring of animal communities exploiting heterogeneous resources

Understanding and predicting the composition and spatial structure of communities is a central challenge in ecology. An important structural property of animal communities is the distribution of individual home ranges. Home range formation is controlled by resource heterogeneity, the physiology and behaviour of individual animals, and their intra‐ and interspecific interactions. However, a quantitative mechanistic understanding of how home range formation influences community composition is still lacking. To explore the link between home range formation and community composition in heterogeneous landscapes we combine allometric relationships for physiological properties with an algorithm that selects optimal home ranges given locomotion costs, resource depletion and competition in a spatially‐explicit individual‐based modelling framework. From a spatial distribution of resources and an input distribution of animal body mass, our model predicts the size and location of individual home ranges as well as the individual size distribution (ISD) in an animal community. For a broad range of body mass input distributions, including empirical body mass distributions of North American and Australian mammals, our model predictions agree with independent data on the body mass scaling of home range size and individual abundance in terrestrial mammals. Model predictions are also robust against variation in habitat productivity and landscape heterogeneity. The combination of allometric relationships for locomotion costs and resource needs with resource competition in an optimal foraging framework enables us to scale from individual properties to the structure of animal communities in heterogeneous landscapes. The proposed spatially‐explicit modelling concept not only allows for detailed investigation of landscape effects on animal communities, but also provides novel insights into the mechanisms by which resource competition in space shapes animal communities.     

An empirically based estimate of home range

The minimum-convex-polygon method for estimating home-range area, in which the outermost points are connected in a particular way, is extremely sensitive to sample size. Existing methods for estimating home-range area that correct for sample size fail to encompass all the important kinds of biological variation in the home-range utilization. (The home-range utilization describes the relative degree to which different units of space are frequented by an animal.) Although previous methods have assumed specific unimodal distributions, such as the bivariate normal, home-range utilizations may resemble funnels or pies as well as hills. A regression method is introduced that uses data from well-sampled individuals whose true home ranges are assumed approximately known to predict home-range areas for less well-sampled individuals. Appendix 5 summarizes this method. Sizes of home ranges estimated by the regression method are half or less than sizes estimated by previous methods in which utilization distributions are assumed to be all of a particular statistical type.     

Diffusion and home range parameters from rodent population measurements in Panama

Simple random walk considerations are used to interpret rodent population data collected in Hantavirus-related investigations in Panama regarding the short-tailed cane mouse, Zygodontomys brevicauda. The diffusion constant of mice is evaluated to be of the order of (and larger than) 200 meters squared per day. The investigation also shows that the rodent mean square displacement saturates in time, indicating the existence of a spatial scale which could, in principle, be the home range of the rodents. This home range is concluded to be of the order of 70 meters. Theoretical analysis is provided for interpreting animal movement data in terms of an interplay of the home ranges, the diffusion constant, and the size of the grid used to monitor the movement. The study gives impetus to a substantial modification of existing theory of the spread of the Hantavirus epidemic which has been based on simple diffusive motion of the rodents, and additionally emphasizes the importance for developing more accurate techniques for the measurement of rodent movement.     

Animal space use: distinguishing a two‐level superposition of scale‐specific walks from scale‐free Lévy walk

How to differentiate between scale‐free space use like Lévy walk and a two‐level scale‐specific process like composite random walk (mixture of intra‐ and inter‐patch habitat movement) is surrounded by controversy. Composite random walk may under some parameter conditions appear Lévy walk‐like from the perspective of the path’s distribution of step lengths due to superabundance of very long steps relative to the expectation from a classic (single‐level) random walk. However, a more explicit focus on the qualitative differences between studying movement at a high resolution mechanistic (behavioral) level and the more coarse‐grained statistical mechanical level may contribute to resolving both this and other issues related to scaling complexity. Specifically, a re‐sampling of a composite random walk at larger time lags than the micro‐level unit time step for the simulation makes a Lévy‐look‐alike step length distribution re‐shaping towards a Brownian motion‐like pattern. Conversely, a true Levy walk maintains its scaling characteristics upon re‐sampling. This result illustrates how a confusing pattern at the mechanistic level may be resolved by changing observational scale from the micro level to the coarser statistical mechanical meso‐ or macro‐scale. The instability of the composite random walk pattern under rescaling is a consequence of influence of the central limit theorem. I propose that a coarse‐graining test – studying simulated animal paths at a coarsened temporal scale by re‐sampling a series – should be routinely performed prior to comparing theoretical results with those patterns generated from GPS data describing animal movement paths. Fixes from terrestrial mammals are often collected at hourly intervals or larger, and such a priori coarse‐grained series may thus comply better with the statistical mechanical meso‐ or macro‐level of analysis than the behavioral mechanics observed at finer resolutions typically in the range of seconds and minutes. If fixes of real animals are collected at this high frequency, coarse graining both the simulated and real series is advised in order to bring the analysis into a temporal scale domain where analytical methods from statistical mechanics can be applied.     

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.     

The home range fractal: From random walk to memory-dependent space use

We present theoretical developments of the multi-scaled random walk (MRW) model for cognitive map-influenced space use by animals. The extensions include a unified space–time scaling function, and further details with respect to statistical properties of the spatial distribution of a set of locations. Supported by numeric simulations we show how memory effects may open for a complex, multi-scaled and self-organized – i.e., intrinsically driven – habitat utilization pattern with fractal dimensional properties. These properties allow for testing for MRW compliance by using parameters from classic movement models like Brownian motion, correlated random walk and Levy walks as null models. In terms of applied ecology, empirical confirmation of memory-influenced space use by individuals will have consequences for interpretation and statistical analyses of habitat utilization. For example, under memory map influence, re-visits to intra-home range locations do not represent independent events. Further, the MRW formulation specifically implies home range movements over a continuum of process rates, spanning a range of spatio-temporal scales in parallel, in violation of the traditional low order Markovian (scale-specific) model architecture. The MRW approach requires an extension of classic Boltzmann–Gibbs statistical mechanics, which rests on the premise that spatio-temporal memory effects are averaged out beyond micro-scales. We suggest that the emergent coherence between spatial and temporal scaling from the MRW approach may open for a more realistic statistical mechanics theory for population processes under terms of memory-influenced space use by individuals.     

The home-range concept: are traditional estimators still relevant with modern telemetry technology?

Recent advances in animal tracking and telemetry technology have allowed the collection of location data at an ever-increasing rate and accuracy, and these advances have been accompanied by the development of new methods of data analysis for portraying space use, home ranges and utilization distributions. New statistical approaches include data-intensive techniques such as kriging and nonlinear generalized regression models for habitat use. In addition, mechanistic home-range models, derived from models of animal movement behaviour, promise to offer new insights into how home ranges emerge as the result of specific patterns of movements by individuals in response to their environment. Traditional methods such as kernel density estimators are likely to remain popular because of their ease of use. Large datasets make it possible to apply these methods over relatively short periods of time such as weeks or months, and these estimates may be analysed using mixed effects models, offering another approach to studying temporal variation in space-use patterns. Although new technologies open new avenues in ecological research, our knowledge of why animals use space in the ways we observe will only advance by researchers using these new technologies and asking new and innovative questions about the empirical patterns they observe.     

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.     

Step selection techniques uncover the environmental predictors of space use patterns in flocks of Amazonian birds

Understanding the behavioral decisions behind animal movement and space use patterns is a key challenge for behavioral ecology. Tools to quantify these patterns from movement and animal–habitat interactions are vital for transforming ecology into a predictive science. This is particularly important in environments undergoing rapid anthropogenic changes, such as the Amazon rainforest, where animals face novel landscapes. Insectivorous bird flocks are key elements of avian biodiversity in the Amazonian ecosystem. Therefore, disentangling and quantifying the drivers behind their movement and space use patterns is of great importance for Amazonian conservation. We use a step selection function (SSF) approach to uncover environmental drivers behind movement choices. This is used to construct a mechanistic model, from which we derive predicted utilization distributions (home ranges) of flocks. We show that movement decisions are significantly influenced by canopy height and topography, but depletion and renewal of resources do not appear to affect movement significantly. We quantify the magnitude of these effects and demonstrate that they are helpful for understanding various heterogeneous aspects of space use. We compare our results to recent analytic derivations of space use, demonstrating that the analytic approximation is only accurate when assuming that there is no persistence in the animals' movement. Our model can be translated into other environments or hypothetical scenarios, such as those given by proposed future anthropogenic actions, to make predictions of spatial patterns in bird flocks. Furthermore, our approach is quite general, so could potentially be used to understand the drivers of movement and spatial patterns for a wide variety of animal communities.