Calculating spatial statistics for velocity jump processes with experimentally observed reorientation parameters

Mathematical modelling of the directed movement of animals, microorganisms and cells is of great relevance in the fields of biology and medicine. Simple diffusive models of movement assume a random walk in the position, while more realistic models include the direction of movement by assuming a random walk in the velocity. These velocity jump processes, although more realistic, are much harder to analyse and an equation that describes the underlying spatial distribution only exists in one dimension. In this communication we set up a realistic reorientation model in two dimensions, where the mean turning angle is dependent on the previous direction of movement and bias is implicitly introduced in the probability distribution for the direction of movement. This model, and the associated reorientation parameters, is based on data from experiments on swimming microorganisms. Assuming a transport equation to describe the motion of a population of random walkers using a velocity jump process, together with this realistic reorientation model, we use a moment closure method to derive and solve a system of equations for the spatial statistics. These asymptotic equations are a very good match to simulated random walks for realistic parameter values.     

Characteristic spatial and temporal scales unify models of animal movement

Animal movements have been modeled with diffusion at large scales and with more detailed movement models at smaller scales. We argue that the biologically relevant behavior of a wide class of movement models can be efficiently summarized with two parameters: the characteristic temporal and spatial scales of movement. We define these scales so that they describe movement behavior both at short scales (through the velocity autocorrelation function) and at long scales (through the diffusion coefficient). We derive these scales for two types of commonly used movement models: the discrete-step correlated random walk, with either constant or random step intervals, and the continuous-time correlated velocity model. For a given set of characteristic scales, the models produce very similar trajectories and encounter rates between moving searchers and stationary targets. Thus, we argue that characteristic scales provide a unifying currency that can be used to parameterize a wide range of ecological phenomena related to movement.     

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.     

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.     

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.     

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.     

Some geometrical considerations on membrane insertion by exocytosis.

A two-dimensional random walk model is used to describe constitutive exocytic transport. Specifically, the possibility of attaining selective vesicle insertion by purely geometrical effects is studied. The distribution of vesicle insertion on the membrane and the average time of translocation are determined as a function of Golgi apparatus size and position, cell morphology and size, walk bias due to cytoplasmic streaming or differential movement on microtubules, and steric effects caused by other organelles. The simulations suggest that significant selectivity in targeting can result from adequate Golgi positioning, in accordance with the hypothesis of Golgi reorientation by the microtubular network (Singer and Kupfer, 1986, Annu. Rev. Cell Biol. 2, 337-365). The selectivity in targeting also depends on the cellular morphology. Segregation of vesicle insertion can be obtained in the transport to the cellular front end and lateral membrane surfaces of a quadrilateral cell, thus suggesting a geometrical component in the exocytic transport in polarized cells.     

Flow- and substratum-mediated movement by a stream insect

1. In streams subject to frequent hydrologic disturbance, the ability of benthic invertebrates to disperse within the channel is key to understanding the mechanisms of flow refugium use and population persistence. This study focuses on crawling invertebrates, the effects on movement of abiotic factors (specifically, flow near the stream bed and bed micro‐topography) and the consequences for dispersal. 2. In a large flume, we observed individual cased caddisflies, Potamophylax latipennis, moving in fully turbulent flows over a precise replica of a water‐worked surface. From maps of movement paths, we quantified crawling behaviour and entrainment, and the influence of bed micro‐topography. We manipulated discharge and tested its effect on movement, linear displacement and areal dispersal. The highest discharge treatment was a disturbance to the caddis; the lowest discharge was not. Crawling behaviours were used to parameterise random walk models and estimate population dispersal, and to test the effects of abiotic factors on movement. 3. Bed micro‐topography influenced crawling in several ways. Caddis spent most of their time at the junctions between proud particles and the adjacent plane bed. The frequency distribution of turn angles was bimodal, with modal values approximating the angle required to travel around median‐sized particles. Larvae generally crawled downstream, but crawling direction relative to the flow was skewed by bed micro‐topography and was not directly downstream, unlike drift. 4. Caddis crawled for most of the time and discharge affected almost every aspect of their movement. As discharge increased, caddis crawled less often, more slowly and over shorter distances; they also became entrained more frequently and over greater distances. With increased discharge, caddis spent proportionately less time at the junctions between proud particles and the adjacent plane bed, and more time on the tops and sides of proud clasts. This is curious as most entrainment occurred from the tops and sides of clasts and entrainment is generally considered to be disadvantageous during disturbances. 5. Linear displacement (drift and entrainment combined) was downstream, but the relation between total displacement and discharge was complex. Total displacement decreased at intermediate discharge as crawling decreased, but increased at high discharge as entrainment and drift played a greater role in movement. 6. Within‐stream dispersal via crawling contained elements of both a correlated random walk (we observed directional persistence in turn angles) and a biased random walk (we observed downstream bias in move direction angles) and was best described as a biased correlated random walk. Dispersal was inversely related to discharge, suggesting that the ability of P. latipennis to crawl into flow refugia on the streambed is reduced at high flow.     

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.     

An open spatial capture–recapture model for estimating density,movement, and population dynamics from line‐transect surveys

The purpose of many wildlife population studies is to estimate density, movement, or demographic parameters. Linking these parameters to covariates, such as habitat features, provides additional ecological insight and can be used to make predictions for management purposes. Line‐transect surveys, combined with distance sampling methods, are often used to estimate density at discrete points in time, whereas capture–recapture methods are used to estimate movement and other demographic parameters. Recently, open population spatial capture–recapture models have been developed, which simultaneously estimate density and demographic parameters, but have been made available only for data collected from a fixed array of detectors and have not incorporated the effects of habitat covariates. We developed a spatial capture–recapture model that can be applied to line‐transect survey data by modeling detection probability in a manner analogous to distance sampling. We extend this model to a) estimate demographic parameters using an open population framework and b) model variation in density and space use as a function of habitat covariates. The model is illustrated using simulated data and aerial line‐transect survey data for North Atlantic right whales in the southeastern United States, which also demonstrates the ability to integrate data from multiple survey platforms and accommodate differences between strata or demographic groups. When individuals detected from line‐transect surveys can be uniquely identified, our model can be used to simultaneously make inference on factors that influence spatial and temporal variation in density, movement, and population dynamics.     

Modeling observation error and its effects in a random walk/extinction model

This paper examines the consequences of observation errors for the "random walk with drift", a model that incorporates density independence and is frequently used in population viability analysis. Exact expressions are given for biases in estimates of the mean, variance and growth parameters under very general models for the observation errors. For other quantities, such as the finite rate of increase, and probabilities about population size in the future we provide and evaluate approximate expressions. These expressions explain the biases induced by observation error without relying exclusively on simulations, and also suggest ways to correct for observation error. A secondary contribution is a careful discussion of observation error models, presented in terms of either log-abundance or abundance. This discussion recognizes that the bias and variance in observation errors may change over time, the result of changing sampling effort or dependence on the underlying population being sampled.     

Integrating individual movement behaviour into dispersal functions

Dispersal functions are an important tool for integrating dispersal into complex models of population and metapopulation dynamics. Most approaches in the literature are very simple, with the dispersal functions containing only one or two parameters which summarise all the effects of movement behaviour as for example different movement patterns or different perceptual abilities. The summarising nature of these parameters makes assessing the effect of one particular behavioural aspect difficult. We present a way of integrating movement behavioural parameters into a particular dispersal function in a simple way. Using a spatial individual-based simulation model for simulating different movement behaviours, we derive fitting functions for the functional relationship between the parameters of the dispersal function and several details of movement behaviour. This is done for three different movement patterns (loops, Archimedean spirals, random walk). Additionally, we provide measures which characterise the shape of the dispersal function and are interpretable in terms of landscape connectivity. This allows an ecological interpretation of the relationships found.     

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.     

Dynamics of asynchronous random Boolean networks with asynchrony generated by stochastic processes

An asynchronous Boolean network with N nodes whose states at each time point are determined by certain parent nodes is considered. We make use of the models developed by Matache and Heidel [Matache, M.T., Heidel, J., 2005. Asynchronous random Boolean network model based on elementary cellular automata rule 126. Phys. Rev. E 71, 026232] for a constant number of parents, and Matache [Matache, M.T., 2006. Asynchronous random Boolean network model with variable number of parents based on elementary cellular automata rule 126. IJMPB 20 (8), 897-923] for a varying number of parents. In both these papers the authors consider an asynchronous updating of all nodes, with asynchrony generated by various random distributions. We supplement those results by using various stochastic processes as generators for the number of nodes to be updated at each time point. In this paper we use the following stochastic processes: Poisson process, random walk, birth and death process, Brownian motion, and fractional Brownian motion. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed-point analysis. The dynamics of the system show that the number of nodes to be updated at each time point is of great importance, especially for the random walk, the birth and death, and the Brownian motion processes. Small or moderate values for the number of updated nodes generate order, while large values may generate chaos depending on the underlying parameters. The Poisson process generates order. With fractional Brownian motion, as the values of the Hurst parameter increase, the system exhibits order for a wider range of combinations of the underlying parameters.     

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.     

Numerical Parameter Space Compression and Its Application to Biophysical Models

Physical models of biological systems can become difficult to interpret when they have a large number of parameters. But the models themselves actually depend on (i.e., are sensitive to) only a subset of those parameters. This phenomenon is due to parameter space compression (PSC), in which a subset of parameters emerges as “stiff” as a function of time or space. PSC has only been used to explain analytically solvable physics models. We have generalized this result by developing a numerical approach to PSC that can be applied to any computational model. We validated our method against analytically solvable models of a random walk with drift and protein production and degradation. We then applied our method to a simple computational model of microtubule dynamic instability. We propose that numerical PSC has the potential to identify the low-dimensional structure of many computational models in biophysics. The low-dimensional structure of a model is easier to interpret and identifies the mechanisms and experiments that best characterize the system.     

Estimation of the diffusion rate and crossing probability for biased edge movement between two different types of habitat

One of the fundamental goals of ecology is to examine how dispersal affects the distribution and dynamics of insects across natural landscapes. These landscapes are frequently divided into patches of habitat embedded in a matrix of several non-habitat regions, and dispersal behavior could vary within each landscape element as well as the edges between elements. Reaction–diffusion models are a common way of modeling dispersal and species interactions in such landscapes, but to apply these models we also need methods of estimating the diffusion rate and any edge behavior parameters. In this paper, we present a method of estimating the diffusion rate using the mean occupancy time for a circular region. We also use mean occupancy time to estimate a parameter (the crossing probability) that governs one type of edge behavior often used in these models, a biased random walk. These new methods have some advantages over other methods of estimating these parameters, including reduced computational cost and ease of use in the field. They also provide a method of estimating the diffusion rate for a particular location in space, compared to existing methods that represent averages over large areas. We further examine the statistical properties of the new method through simulation, and discuss how mean occupancy time could be estimated in field experiments.     

Multiscale analysis of collective motion and decision-making in swarms: An advection-diffusion equation with memory approach

We propose a (time) multiscale method for the coarse-grained analysis of collective motion and decision-making in self-propelled particle models of swarms comprising a mixture of ‘naïve’ and ‘informed’ individuals. The method is based on projecting the particle configuration onto a single ‘meta-particle’ that consists of the elongation of the flock together with the mean group velocity and position. We find that the collective states can be associated with the transient and asymptotic transport properties of the random walk followed by the meta-particle, which we assume follows a continuous time random walk (CTRW). These properties can be accurately predicted at the macroscopic level by an advection-diffusion equation with memory (ADEM) whose parameters are obtained from a mean group velocity time series obtained from a single simulation run of the individual-based model.