A general random walk model for the leptokurtic distribution of organism movement: Theory and application

The movement of organisms is usually leptokurtic in which some individuals move long distances while the majority remains at or near the area they are released. There has been extensive research into the origin of such leptokurtic movement, but one important aspect that has been overlooked is that the foraging behaviour of most organisms is not Brownian as assumed in most existing models. In this paper we show that such non-Brownian foraging indeed gives rise to leptokurtic distribution. We first present a general random walk model to describe the organism movement by breaking the foraging of each individual into events of active movement and inactive stationary period; its foraging behaviour is therefore fully characterized by a joint probability of how far the individual can move in each active movement and the duration it remains stationary between two consecutive movements. The spatio-temporal distribution of the organism can be described by a generalized partial differential equation, and the leptokurtic distribution is a special case when the stationary period is not exponentially distributed. Empirical observations of some organisms living in different habitats indicated that their rest time shows a power-law distribution, and we speculate that this is general for other organisms. This leads to a fractional diffusion equation with three parameters to characterize the distributions of stationary period and movement distance. A method to estimate the parameters from empirical data is given, and we apply the model to simulate the movement of two organisms living in different habitats: a stream fish (Cyprinidae: Nocomis leptocephalus) in water, and a root-feeding weevil, Sitona lepidus in the soil. Comparison of the simulations with the measured data shows close agreement. This has an important implication in ecology that the leptokurtic distribution observed at population level does not necessarily mean population heterogeneity as most existing models suggested, in which the population consists of different phenotypes; instead, a homogeneous population moving in homogeneous habitat can also lead to leptokurtic distribution.     

Analyzing animal movements using Brownian bridges

By studying animal movements, researchers can gain insight into many of the ecological characteristics and processes important for understanding population-level dynamics. We developed a Brownian bridge movement model (BBMM) for estimating the expected movement path of an animal, using discrete location data obtained at relatively short time intervals. The BBMM is based on the properties of a conditional random walk between successive pairs of locations, dependent on the time between locations, the distance between locations, and the Brownian motion variance that is related to the animal's mobility. We describe two critical developments that enable widespread use of the BBMM, including a derivation of the model when location data are measured with error and a maximum likelihood approach for estimating the Brownian motion variance. After the BBMM is fitted to location data, an estimate of the animal's probability of occurrence can be generated for an area during the time of observation. To illustrate potential applications, we provide three examples: estimating animal home ranges, estimating animal migration routes, and evaluating the influence of fine-scale resource selection on animal movement patterns.     

Using accelerometer,high sample rate GPS and magnetometer data to develop a cattle movement and behaviour model

The study described in this paper developed a model of animal movement, which explicitly recognised each individual as the central unit of measure. The model was developed by learning from a real dataset that measured and calculated, for individual cows in a herd, their linear and angular positions and directional and angular speeds. Two learning algorithms were implemented: a Hidden Markov model (HMM) and a long-term prediction algorithm. It is shown that a HMM can be used to describe the animal's movement and state transition behaviour within several “stay” areas where cows remained for long periods. Model parameters were estimated for hidden behaviour states such as relocating, foraging and bedding. For cows’ movement between the “stay” areas a long-term prediction algorithm was implemented. By combining these two algorithms it was possible to develop a successful model, which achieved similar results to the animal behaviour data collected. This modelling methodology could easily be applied to interactions of other animal species.     

Incorporating behaviour into simple models of dispersal using the biological control agent Dicyphus hesperus

We explored the utility of incorporating easily measured, biologically realistic movement rules into simple models of dispersal. We depart from traditional random walk models by designing an individual-based simulation model where we decompose animal movement into three separate processes: emigration, between-patch movement, and immigration behaviour. These processes were quantified using experiments on the omnivorous insect Dicyphus hesperus moving through a tomato greenhouse. We compare the predictions of the individual-based model, along with a series of biased random walk models, against an independent experimental release of D. hesperus. We find that in this system, the short-term dispersal of these insects is described well by our individual-based model, but can also be described by a 2D grid-based biased random walk model when mortality is accounted for.     

Bayesian analysis of Jolly-Seber type models

We propose the use of finite mixtures of continuous distributions in modelling the process by which new individuals, that arrive in groups, become part of a wildlife population. We demonstrate this approach using a data set of migrating semipalmated sandpipers (Calidris pussila) for which we extend existing stopover models to allow for individuals to have different behaviour in terms of their stopover duration at the site. We demonstrate the use of reversible jump MCMC methods to derive posterior distributions for the model parameters and the models, simultaneously. The algorithm moves between models with different numbers of arrival groups as well as between models with different numbers of behavioural groups. The approach is shown to provide new ecological insights about the stopover behaviour of semipalmated sandpipers but is generally applicable to any population in which animals arrive in groups and potentially exhibit heterogeneity in terms of one or more other processes.     

Diffusion about the mean drift location in a biased random walk

Random walks are used to model movement in a wide variety of contexts: from the movement of cells undergoing chemotaxis to the migration of animals. In a two-dimensional biased random walk, the diffusion about the mean drift position is entirely dependent on the moments of the angular distribution used to determine the movement direction at each step. Here we consider biased random walks using several different angular distributions and derive expressions for the diffusion coefficients in each direction based on either a fixed or variable movement speed, and we use these to generate a probability density function for the long-time spatial distribution. We demonstrate how diffusion is typically anisotropic around the mean drift position and illustrate these theoretical results using computer simulations. We relate these results to earlier studies of swimming microorganisms and explain how the results can be generalized to other types of animal movement.     

Kernel-based home range method for data with irregular sampling intervals

Studies of habitat selection and movements often use radio-tracking data for defining animal home ranges. Home ranges (HR) can be approximated by a utilization density distribution (UD) that instead of assuming uniform use of areas within HR boundary provides a probabilistic measure of animal space use. In reality, radio-tracking data contain periods of frequent autocorrelated observations interspersed with temporally more independent observations. Using such temporally irregular data directly may result in biased UD estimates, because areas that have been sampled intensively receive too much weight. The problem of autocorrelation has been tackled by resampling data with an appropriate time interval. However, resampling may cause a large reduction in the data set size along with a loss of information. Evidently, biased UD estimates or reduction in data may prejudice the results on animal habitat selection and movement. We introduce a new method for estimating UDs with temporally irregular data. The proposed method, called the time kernel, accounts for temporal aggregation of observations and gives less weight to temporally autocorrelated observations. A further extension of the method accounts also for spatially aggregated observations with relatively low weights given to observations that are both temporally and spatially aggregated. We test the behaviour of the time kernel method and its spatiotemporal version using simulated data. In addition, the method is applied to a data set of brown bear locations.     

Selecting the best home range model: an information-theoretic approach

Choosing an appropriate home range model is important for describing space use by animals and understanding the ecological processes affecting animal movement. Traditional approaches for choosing among home range models have not resulted in general, consistent, and unambiguous criteria that can be applied to individual data sets. We present a new application of information-theoretic model selection that overcomes many of the limitations of traditional approaches, as follows. (1) It alleviates the need to know the true home range to assess home range models, thus allowing performance to be evaluated with data on individual animals. (2) The best model can be chosen from a set of candidate models with the proper balance between fit and complexity. (3) If candidate home range models are based on underlying ecological processes, researchers can use the selected model not only to describe the home range, but also to infer the importance of various ecological processes affecting animal movements within the home range.     

Social networks and models for collective motion in animals

The theory of collective motion and the study of animal social networks have, each individually, received much attention. Currently, most models of collective motion do not consider social network structure. The implications for considering collective motion and social networks together are likely to be important. Social networks could determine how populations move in, split up into and form separate groups (social networks affecting collective motion). Conversely, collective movement could change the structure of social networks by creating social ties that did not exist previously and maintaining existing ties (collective motion affecting social networks). Thus, there is a need to combine the two areas of research and examine the relationship between network structure and collective motion. Here, we review different modelling approaches that combine social network structures and collective motion. Although many of these models have not been developed with ecology in mind, they present a current context in which a biologically relevant theory can be developed. We argue that future models in ecology should take inspiration from empirical observations and consider different mechanisms of how social preferences could be expressed in collectively moving animal groups.     

Pattern formation and individual-based models: The importance of understanding individual-based movement

This paper demonstrates that while pattern formation can stabilize individual-based models of predator–prey systems, the same individual-based models also allow for stabilization by alternate mechanisms, particularly localized consumption or diffusion limitation. The movement rules of the simulation are the critical feature which determines which of these mechanisms stabilizes any particular predator–prey individual-based model. In particular, systems from well-connected subpopulations, in each of which a predator can attack any prey, generally exhibit stabilization by pattern formation. In contrast, when restricted movement within a (sub-)population limits the ability of predators to consume prey, localized consumption or diffusion limitation can stabilize the system. Thus while the conclusions from differential equations on the role of pattern formation for stability may apply to discrete and noisy systems, it will take a detailed understanding of movement and scales of interaction to examine the role of pattern formation in real systems. Additionally, it will be important to link an understanding of both foraging and inter-patch movement, since by analogy to the models, both would be critical for understanding how real systems are stabilized by being discrete and spatial.     

Modeling and inference of animal movement using artificial neural networks

Movement of animals in relation to objects in their environment is important in many areas of ecology and wildlife conservation. Tools for analysis of movement data, however, still remain rather limited. In previous work, we developed nonlinear regression models for movement in relation to a single landscape feature. Here we greatly expand these previous models by using artificial neural networks. The new models add substantial flexibility and capabilities, including the ability to incorporate multiple factors and covariates. We devise a likelihood-based model fitting procedure that utilizes genetic algorithms and demonstrate the approach with movement data for red diamond rattlesnakes. The proposed methodology can be useful for global positioning system tracking data that are becoming more common in studies of animal movement behavior.     

Distinguishing between Lévy walks and strong alternative models

Lévy walks are a widely used but contentious model of animal movement patterns. They are contentious because they have been wrongly ascribed to some animal species through use of incorrect statistical methods and because they have not been adequately compared against strong alternative models, such as composite correlated random walks. This lack of comparison has been partly because the strong alternative models do not have simple likelihood functions. Here I show that power-spectra and the distribution of the first significant digits (the leading non-zero digits) of the step lengths can distinguish between Lévy walks and composite correlated random walks. Using these diagnostic tools, I bolster previous claims that honey bees use a movement strategy that can be approximated by Lévy walks when searching for their hive or for a food source.     

Delay dynamics of the slug Deroceras reticulatum,an agricultural pest

Slugs are devastating agricultural and horticultural pests. However, their population dynamics are not well understood and this hinders the construction of efficient control strategies. This is especially true with organic farming for which biological controls are preferred. Moreover, the dominant species, Deroceras reticulatum, does not follow a regular annual life cycle, as do the majority of the other slug species. Its dominance may be associated with this fact. In this paper, we investigate whether mechanisms associated with the slugs’ time-delayed population dynamics are responsible for the large variations in numbers, with particular emphasis on their sensitivity to environmental conditions. In order to do this, several versions of a non-autonomous delay differential equation model are developed in which we highlight some of the contentious issues in slug modelling. Analyses of the models are combined with numerical experiments using parameters based upon controlled laboratory experiments. In the absence of seasonal forcing, we find that the delay term may be neglected in the simplest models. However, the presence of a predator dramatically increases the impact of the delay term and may drive a delay induced instability. Notably, we find that in all cases the delay term is of considerable qualitative importance in models which incorporate seasonal fluctuations. We highlight the fact that the models are capable of producing a large range of solution behaviour and, furthermore, discuss the conditions for, and thus the likelihood of their relevance.     

Suspended particulate matter dynamics in a particle framework

Suspended particulate matter (SPM) dynamics in ocean models are usually treated with an advection–diffusion equation for one or more sediment size classes coupled to the hydrodynamical part of the model. Numerical solution of these additional partial differential equations unavoidably introduces numerical diffusion, i.e. in the case of sharp gradients the possible occurrence of artificial oscillations and non-positivity. A Lagrangian particle-tracking model has been developed to simulate short-term SPM dynamics. Modelling individual sediment particles allows a straightforward physical interpretation of the processes. The tracking of large numbers of individual and independent particles (up to 25 million in total in a single sediment class) can be achieved on high performance computer clusters, due to efficient parallelisation of particle tracking. The movement of the particles is described by a stochastic differential equation, which is consistent with the advection–diffusion equation. Here, the concentration profile is represented by a set of independent moving particles, which are advected according to the 3D velocity field, while the diffusive displacements of the particles are sampled from a random distribution, which is related to the eddy diffusivity field. To account for erosion a new parameterisation is proposed. Three numerical particle tracking schemes (EULER, MILSTEIN and HEUN) are presented and validated in idealised test cases. Finally, the particle tracking algorithms are applied to a realistic scenario, a severe winter storm in the East Frisian Wadden Sea (southern North Sea). The comparison with observations and an Eulerian SPM transport model seems to indicate a somewhat better fidelity of the Lagrangian approach.     

Spatiotemporal dynamics of habitat use by koalas: the checkerboard model

Animal movement patterns and use of space depend upon food and nonfood resources, as well as conspecific and heterospecific interactions, but models of habitat use often neglect to examine multiple factors and rarely include marsupials. We studied habitat use in an Australian population of koalas (Phascolarctos cinereus) over a 6-year period in order to determine how koalas navigate their environment and partition limited patchy food and nonfood resources. Tree selection among koalas appears to be mediated by folar chemistry, but nonfood tree selection exerts a major impact on home range use due to thermoregulatory constraints. Koalas moved on a daily basis, during both day and night, but daytime resting site was not necessarily in the same location as nighttime feeding site. Koalas had substantial home range overlap in the near absence of resource sharing with less than 1% of trees located in areas of overlap used by multiple koalas. We suggest that koala spatiotemporal distribution and habitat use are probably based upon a community structure of individuals, with a checkerboard model best describing overlap in home range area but not in resource use. Nonfood refugia and social networks should be incorporated into models of animal range and habitat use.     

Old models explain new observations of butterfly movement at patch edges

Understanding movement in heterogeneous environments is central to predicting how landscape changes affect animal populations. Several recent studies point out an intriguing and distinctive looping behavior by butterflies at habitat patch edges and hypothesize that this behavior requires a new framework for analyzing animal movement. We show that this looping behavior could be caused by a longstanding movement model, biased correlated random walk, with bias toward habitat patches. The ability of this longstanding model to explain recent observations reinforces the point that butterflies respond to habitat heterogeneity and do not move randomly through heterogeneous environments. We discuss the implications of different movement models for predicting butterfly responses to landscape change, and our rationale for retaining longstanding movement models, rather than developing new modeling frameworks for looping behavior at patch edges.     

Bivariate geostatistical modelling: a review and an application to spatial variation in radon concentrations

We present a comprehensive review of multivariate geostatistical models, focusing on the bivariate case. We compare in detail three approaches, the linear model of coregionalisation, the common component model and the kernel convolution approach, and discuss similarities between them. We demonstrate the merits of the common component class of models as a flexible means for modelling bivariate geostatistical data of the type that frequently arises in environmental applications. In particular, we show how kernel convolution can be used to approximate the common component model, and demonstrate the method using a data-set of calcium and magnesium concentrations in soil samples. We then apply the model to a study of domestic radon concentrations in the city of Winnipeg, Canada, in which exposure was measured at two sites (bedroom and basement) in each residential location. Our analysis demonstrates that in this study the correlation between the two sites within each house dominates the short-range spatial correlation typical of the distribution of radon.