Kinds of Data.

The most basic set of data that can be collected on the movement path of an individual is a sequence of positions (x_i, y_i) at points t_i, i = 0, 1, 2, …, n: that is, the set 𝒟 = {u₀, …, u_n} with u_i = (x_i, y_i), where x_i = x(t_i) and y_i = y(t_i). For convenience, we set t₀ = 0 but do not necessarily require all of the time intervals τ_i = [t_i−1, t_i] to be of equal length. In fact, in some analyses, the focus is on the distribution of waiting times τ_i associated with the events of first being in position (x_i−1, y_i−1) and then next in position (x_i, y_i) (14). These data can be transformed into polar coordinates (6) to obtain a set of vectors z_i = (t_i, d_i, θ_i) with and

A considerable body of diffusion and stochastic process theory exists to analyze such data in the context of featureless landscapes, including uncorrelated and correlated random walks (14⇓⇓–17) and super- and subdiffusive Lévy walks (18⇓⇓–21) and Lévy modulated correlated walks (22).

Brief Review of Random Walks and Diffusion.

Consider the distribution of waiting times τ_i, velocities v_i, and absolute displacements d_i associated with a set of displacement data 𝒟. If the distributions of both waiting times and velocities have finite mean and variance, then the stochastic process associated with the data is said to be diffusive. From theory (16⇓⇓⇓–20), this implies that over an ensemble of K sets 𝒟_k, k = 1, …, K, where each set is one realization of the same stochastic process, the mean-square displacement (msd) ψ_i of the series d_ik averaged over all K sets is asymptotically linear, that is ψ_i = (1/K)Σ_k=1^Kd_ik² ∼ t. By contrast, if upon plotting this relationship we find that ψ_i ∼ t^p for p > 1 or < 1, then the walk is respectively referred to as superdiffusion or subdiffusion. As we demonstrate below, using simulated data drawn from a modified Pareto power law distribution (23) (Fig. S3) and plotted on a log–log scale, such plots can be misleading if time is not sufficiently large. The reason is that, initially, the relationship is affected by the fact that the distance moved in the first sampling interval is controlled by the actual step-size distribution, but from the second step onwards the distance moved from the origin is now also affected by turning angles.

Movement Pathways and Diffusion.

Over the past decade, analyses of the movement paths of several organisms, including albatrosses (24) and spider monkeys (14), have concluded that the associated movement processes are superdiffusive, although a re-analysis of these data refute this finding (ref. 1, but see ref. 25). A possible source of error in estimating p in the msd relationship ψ_i ∼ t^p (Fig. 1) is t must be sufficiently large for the asymptotic value to emerge. Another source of error is that superdiffusion predicts fractal looking movement paths (14, 20, 24). Using one of several different methods to estimate the fractal dimension (26) of such paths, some organisms have been pronounced as superdiffusive (14). In these organisms, however, repeated fractal patterns occur at no more than two or three particular scales and are more a feature of the way resources are distributed across the landscape than of a genuine superdiffusive process. Thus, it is important to understand how animal movement is influenced by landscape features and to assess the extent to which step-size distributions are modified by landscape heterogeneity.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 1.

The logarithm of mean-square displacements (msd) ψ versus the logarithm of time t average over 10,000 simulations are plotted for a random walk with step lengths drawn from modified Pareto distributions (Upper Left, q = 2; Upper Right, q = 1.5; Lower Left, q = 1) and directions for each step completely random. From the lines inserted by “eye” (red, small t; blue, large t), Upper Left represents diffusion (P = 1 for large t), and Upper Right and Lower Left represent super diffusion (respectively, P = 1.15 > 1 and P = 2 for large t). In Lower Right, for the case q = 1, the length of these excursions are truncated at a step size of 100 (biologically, an upper bound is set by the maximum velocity of the organism multiplied by the length of the time interval), which is far out in the tails of the distribution (two orders of magnitude beyond the mode; see Fig. S3). In this case, the noisy superdiffusive behavior is completely tamed even though, initially, it looks superdiffusive (ψ ∼ t^1.4 for t ≤ 2).

We note here that because all bounded step-size distributions produce Gaussian random walks when turning angles are uncorrelated, critical information in the step-size distribution, such as multimodality arising from mixed distributions of FMEs, is lost when using the statistics of emergent global characteristics such as msd as a function of time. This stresses the importance of knowing the actual step-size distributions when deconstructing movement paths and trying to understand the causal processes creating local path structures.

FMEs and CAMs.

The framework presented here is formulated in the context of a group of N known individuals indexed by k. This specificity allows us to account for the following: (i) species, gender, and age-specific differences; (ii) unique memory and knowledge of landscape; and (iii) cues and vectors individuals use to select a new position on the landscape. Further, we assume that each individual has n_m^k FMEs (Table 1), each with its own characteristic speed s_j^k, j = 1, …, n_m^k, from which its movement track is generated. The set of FMEs constitute the basic motion capacity Ω^k (7). Furthermore, in any segment of the movement path (Fig. S1), these FMEs are mixed in various proportions to constitute a set of A_r^k CAMs (Table 1), r = 1, …, n_a^k best characterized by distributions of speeds (equivalently distances) with means and standard deviations s̄_r and σ_τr for each r (Fig. S2). We note, however, that when the sample intervals τ_i^k = τ^k are fixed over all intervals i, for reasons discussed below only the standard deviation (and not the mean) depends on sampling frequency 1/τ^k.

As an example, Ω^k in horses has been defined in terms of five FMEs or gaits (27) of increasing speeds: stationarity (s₁ = 0), walking (a four beat gait with s₂ ≈ 4 mph), trotting (a two beat gait with s₃ ≈ 8 mph on average), cantoring (a three beat gait for which s₄ > s₃), and galloping (a four beat gait for which s₅ ≈ 25–30 mph, varying across horses). Beyond these natural gaits, some horses have been bred to implement so-called “ambling gaits” that provide a smoother ride, but not all horses can execute these, thereby providing an example of selection at work on the motion capacity component Ω of our guiding conceptual model (7).

For each of the n_a^k, CAMs associated with individual k, the distribution of speeds (equivalently distances or “step sizes”) associated with a particular CAM is affected by the value of τ: if τ is small an individual will generally be in one movement mode or another in the mix that constitutes the activity, whereas if τ is large an individual will more likely be executing some mix of modes over one interval (Fig. S2). Thus, even though the proportion of different movement mode events used to construct the movement tracks may be quite stable with regard to a specific activity, the directions of heading from one event to the next make the relationship between the length of a track and displacement quite complicated. Only for very simple cases, such as tracks generated from one type of FME, is the relationship between length-of-track and displacement easily cracked by analytical methods. For the rest, the easiest way to generate the distribution of speeds (distances) A_τr(s) as a function of sample interval size τ is to use Monte Carlo simulation.

The proportions of FMEs in a particular CAM is likely to vary with changes in landscape. For example, the mix of stationary, walking, and running modes used by a foraging African antelope will differ in an open savannah compared with thick bushveld. This problem can be dealt with by assuming that the parameters of a CAM distribution depend on external landscape factors in addition to the frequency of sampling along a movement path.

Internal States and Goals.

One of the goals in an analysis of empirical data is to see how cleanly a set of CAM distribution of step-size FMEs can be extracted from the movement track to explain the actual activity producing different segments of the movement track. Some segments may reflect a pure activity (e.g., foraging) while others are a mix of activities (e.g., foraging and resting) or even a compromise between competing activities such as an individual foraging as it heads to a known water source or home. To be able to account for such mixed activities, as well as assess factors that may lead to an individuals switching from one CAM to another, we must infer that the individual has a multidimensional internal state that drives the behavior (7). The current state of this driver can in turn be associated with a goal emerging from an individual's internal state, which in general will vary with time. The relationship between an individual's internal state (i.e., the vector w—see Fig. S4) and its current goal state can be treated in various ways. One way is to construct a mapping of a continuous n-dimensional vector space to a discrete space of g goals. Another is to consider the internal state as a weighting vector that produces a goal-modified “ideal” distribution G_τi^k(s) of speeds (distances) s for individual k at time t_i = iτ with the weighted sum of its n_a^k CAMs: i.e., In the case of organisms that have no internal mechanism for generating goals (e.g., plants), the internal state may, for example, represent elements controlling dehiscence of seed dispersal structures.

Spatial Explication.

The spatial information embedded in the distribution G_τi^k(s) needs to be made explicit before other relevant spatially explicit geographical and biological information can be incorporated into the movement process. The distribution G_τi^k(s) is defined on s ∈ [0, s_max]. On a flat structureless landscape—that is, in the absence of all cues and other directional biasing factors beyond the context of a correlated walk—an individual is equally likely to move in any direction. Thus, the ideal (i.e., featureless landscape) distribution G_τi^k(s) can be given an explicit spatial dimension by rotating it around the point [0, 0] on a plane parameterized by coordinates (α, β) to obtain a radially symmetric distribution 𝒢_τi^k(α, β) that has a top-half-of-a-donut-like structure (Fig. S5). The distribution can now be relocated so that its center of symmetry is the current point of location u_i^k = (x_i^k, y_i^k) of organism k.

Landscape Raster.

We incorporate landscape features into each individual's goal as follows. First, we cover the landscape with a raster of rectangular cells C^αβ. Second, we locate the position u_i^k = (x_i^k, y_i^k) of each individual within the cell containing this point: in general, this allows several individuals of one or more types to be located in each cell. Also, we associate an n_c-dimensional external state to account for all factors relevant to the movement of each of the N individuals on the landscape, including other individuals. Third, the one-dimensional CAM distributions A_τr^k(s) are used, as described above, to generate two-dimensional radially symmetric, but discretized, distributions 𝒜_τr^k(α, β) ≥ 0 with Σ_(α,β)𝒜_τr^k(α, β) = 1 for all k and r = 1, …, n_a^k. Fourth, we incorporate the landscape effects through a set of n_a^k landscape modifier matrices (LMM) L_ir^k (i.e., specific to individual k, their activity r, and time iτ) with elements ℓ_ir,αβ^k constructed from those elements of the external state vector that are applicable for the activity in question. These elements ℓ_ir,αβ^k are used to modify the CAM distributions to reflect the preference that each individual has for each of the landscape cells while involved in one of its CAMs (e.g., one cell may be the most desirable from a foraging point of view, whereas another is desirable as a target when heading for water). These LMMs are used to modify the movement distribution 𝒢_τi^k(α, β) to account for the state of the landscape, as well as individual specific information that inter alia relates to the location of places [remembered, sensed, and inferred through landscape cues, including the earth's magnetic field (28)] associated with heading activities, the distribution of resources across the landscape, and the location of other individuals on the landscape; and hence the elements of these matrices must be updated each time step.

The simplest way to incorporate this landscape information into the movement process is to use the activity-specific LMM elements ℓ_ir,αβ^k (Table 1) as a way to weight the terms in the idealized goal determined movement distribution G_τi^k(s) to produce a composite realized movement distribution, with spatial matrix elements for all α, β, i, and k = 1, where normalizes the discrete elements of the probability distribution over the cells C^αβ for each individual k and all time i.

Decision Mechanism.

The final component of the movement process is how the organism selects the particular cell (α, β) that will determine its next position u_i+1^k = (x_i+1^k, y_i+1^k) at time t = (i + 1)τ. The simplest rule is for an organism to move to cell (α, β) with largest value M_τi^k(α, β), effectively without making any interim decision on its way to the new cell. For the case of an organism driven purely by stochastic landscape processes (such as wind), movement to the next cell can be regarded as random with the probability of selecting a particular cell (α, β) equal to the values M_τi^k(α, β). A third possibility is to select cells with a probability that is proportional to some power of M_τi^k(α, β), a solution that is intermediate between the first two if this power is >1. If this power is <1, the solution is intermediate between the second mechanism and a purely random solution that gives no weight to the relative values M_τi^k(α, β).