The Biophysical Framework’s Potential for Behavioural Ecology

Data collection of individual movement like a series of GPS fixes provides a potential for a physical – a statistical-mechanical – interpretation of animal space use. Such material represents indirect studies of behaviour in contrast to direct observation and interpretation. The GPS pattern of dots on the map provides a coarse-grained image of how the individual in overall terms relocated itself during the period of sampling. It is fascinating that this “out of focus” image may in fact not only be scrutinized with respect to verifying many similar behavioural traits as traditionally studied by ethological methods, but  also allows for interpretation of specific relationships that are difficult or outright impossible to test from the classic methods in behavioural ecology.  In this post I’m focusing on one of these space use properties, scale-free habitat utilization.

First, what is “scale-free movement”? Statistically, this property apparently should be easy to verify (or falsify) by studying the binned distribution of displacements from one GPS location (fix) to the next in a large collection of such displacements. Since a scale-free distribution implies a power law (see below), it is expected to adhere to a linear relationship between log(F) and log(L), where F is the number of displacements of a given length L: log [F(L)] = αβ*log(L), where β ≈ -2 in many data sets [β is the power exponent and the parameter α sets the intercept with the log(F) axis].

However, as everybody working in the context of animal space use and movement ecology has experienced, in practice it is not that simple to verify scale-free movement. In fact, as time has gone by since this concept began to appear in journals in the 1990s (Viswantathan 1996) it has been repeatedly underscored that the step length distribution is extremely sensitive to a lot of conditions. For example, as verified by simple simulations a change of the sampling frequency when collecting fixes may turn an apparent scale-free distribution scale-specific (Gautestad and Mysterud 2013) and vice versa (Gautestad 2013).

Obviously, regardless of the level of sophistication of the statistical method and whether the distribution or some alternative approach is applied (e.g., MLE), to understand the issues and controversies progress depends on a deeper understanding of the actual space use process behind the statistical pattern. In other words, in the context of fix sampling a statistical mechanical approach is needed to interpret the pattern! From this approach the above referred sensitivity to sampling frequency is no longer an issue but what to be expected from specific classes of space use – in statistical-mechanical terms. Ignoring the respective biophysical models’ statistical expectations will create both confusion and controversy and thus hamper progress in ecological analysis of animal space use.

When analyzing large data sets it is easy to understand how relocations will tend to show successive displacement vectors that distribute themselves uniformly over 0-360 degrees. Further, well known statistical mechanical theory can explain why a scale-specific kind of space use will tend to show a step length distribution that confirms a negative exponential function (number of steps falling in range Li+1 = Li+d is p percent less frequent than number of steps in bin Li, where d is bin width). However, it is still quite murky how a power law pattern of steps is emerging from GPS fixes. Lévy walk is just one of many candidate models for the underlying process, and it has come under stronger scrutiny lately.

Scale-free movement is closely linked to the concept of more or less “equally distributed” space use over a range of scales (see for example this post). For example, for one particular slope β of the log [F(L)] distribution (β=-2), the animal is expected to distribute spatial displacements from short-, medium, and longer-term temporal goals relatively evenly over a scale range. By “evenly” it is understood that execution of a k times larger strategic event during exploratory movement is expected with frequency 1/k² (i.e., β=-2). In other words, the product of frequency of occurrence if a given step length and length scale L is constant over the scale range for scale-free space use (observe the equality of the two red triangles in the illustration below). In particular, when the data analysis reveals β=-2 the animal has in over-all terms shown inversely proportional use of the landscape at large spatial resolutions in two-dimensional space terms (L2) relative to finer resolutions (Gautestad 2005).

Figure1A

The illustration shows the property of a Lévy walk (LW), which is one of several classes of a scale-free movement, where a given increment of log(L) is expected to show a given decrease in log(occurrence) regardless of scale L. In the smaller inset – with standard arithmetic axes – the expectation from a scale-specific process, Brownian motion (BM), is also included for comparison to LW (see  details in Gautestad and Mysterud 2013). Observe that the slope of the log-log distribution of step lengths become steeper with increasing L as shown by the blue triangles, verifying a scale-specific kind of space use (typically, a negative exponential distribution rather than a power law).

For ecology the main take-home message from the concept of scale-specific versus scale-free space use is

  • a statistical-mechanical approach allows for a proper analysis of the dualism between a statistical pattern and the physical movement process
  • such a process-oriented approach initially introduced the very concept of scale-free space use to behavioural ecology
  • studying the ecological conditions under which individuals and populations adhere to a scale-free or -specific kind of space use raises important hypotheses with practical relevance also for wildlife management.

REFERENCES

Gautestad, A. O., and I. Mysterud. 2005. Intrinsic scaling complexity in animal dispersion and abundance. The American Naturalist 165:44-55.

Gautestad, A. O. 2013. Lévy meets Poisson: a statistical artifact may lead to erroneous re-categorization of Lévy walk as Brownian motion. The American Naturalist 181:440-450.

Gautestad, A. O., and A. Mysterud. 2013. The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion. Movement Ecology 1:1-18.

Viswanathan, G. M., V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, and H. E. Stanley. 1996. Lévy flight search patterns of wandering albatrosses. Nature 381:413-415.

 

Parallel Processing – a Simplistic Introduction

The parallel processing concept (PP) is a core postulate of the Multiscaled random walk model. PP provides the backbone of an attempt to understand in a statistical-mechanically consistent manner why animal movement generally tend to show scale-free distribution of displacement lengths at a given frequency of GPS position sampling (Lévy walk-like movement). Contrary to standard Lévy walk theory, PP-based movement seems to offer a plausible explanation for why superlong displacements – the “long tail” part of the Lévy walk-like step length distribution – may appear even in environments with frequent direction-perturbing events. From standard theory such events should tend to “shorten the tail” by prematurely terminate long steps, and simultaneously inflate the frequency of shorter displacements. Thus, the standard Lévy model is – in my view – lacking some essential aspects of many animals’ cognitive computation of environmental conditions and the individual’s internal state. In my book I have elaborated on PP from many angles, both from the individual and the population perspective. In this post I present a new attempt to squeeze the PP concept into a simple visual image.

parallel-processing-scaling

First, consider the lower dotted line, representing a timeline covering the past, present,and the future. Let the shaded area surrounding the timeline demarcate the potential movement distance from one moment to the next during the high-frequency time resolution at this “Level 0”. The series of arrows along the timeline represents in a simplistic manner movement to the left (upwards-pointing arrow) or to the right. It is trivial to expand space to higher dimensions, but let’s keep it simple. It is also trivial to expand the system with forward-pointing arrows for no movement during the given increment.

Next, consider that this series of left-right move distances during respective time increments are the result of completely deterministic behaviour, where the following move is the resultant decision based on present time conditions (a cognitive computation of factors from environmental input and internal state). In this manner, the vertical line marked “Present” slides towards the right as time goes by. As mentioned, the arrows along the timeline show the potential movement distance during a given increment in time. Since all arrows at the given timeline are equally long, it means that the series for simplicity either illustrates the average step length or the maximum length at this scale (constrained by limits on movement speed).

In a large series of steps that is explicitly executed in space by successively adding the arrows to create a path (in this example, back and forth on a one-dimensional line as the arrows on the Level 0 timeline are added as spatial vectors) this process will comply with statistics of a standard random walk – similar to Brownian motion – whether each movement decision is successively stochastic or deterministic or something in-between. Such statistics could for example be described by the step length distribution (net movement distance after N steps) or other classical aspects of such a path’s diffusion properties. The reason for Brownian motion compliance for the series in total, and thus a transition from a deterministic process at unit time scale to a stochastic process at the scale of several increments, is the fact that we defined the successive steps to be represented by independent movement decisions from one increment to the next; i.e., a Markovian (mechanistic) kind of dynamics.

In short, the illustration’s Level 0 when seen in isolation describes a Markovian; i.e., a mechanisic movement process. This implies that the next step’s direction – shown by a question mark – will depend on the animal’s evaluation of the present conditions at the specific time resolution for this level. At the next increment a new evaluation is taking place, and so on.

A Markovian/mechanistic model may include several time resolutions, but not more than one resolution at any given increment. For example, a so-called composite random walk may toggle stochastically or deterministically between shorter or longer periods of high- and long-frequency directional change of its path. In behavioural ecology this model is often called area-restricted search (ARS).

Expanding a mechanistic model to a non-mechanistic PP model implies that the cognitive process is postulated to involve a range of time resolutions, organized in a hierarchical design (whether animals actually adhere to this design is testable; see below). In the illustration above this temporal scale range is shown by several additional timelines, marked Level 1-4. The lines represent a geometric discretization of a continuous time range (logarithmic base 2 with levels 20, 21, 22, 23 and 24). Why log-linearity and why base 2? I have to refer to my book for these arguments. The similar scaling of the spatial dimensions (wider “grey area” surrounding the higher-level timelines) is explained in this post.

The main effect from expanding the mechanistic model to a non-mechanistic PP model may – for example – be seen in the statistics of movement path that emerges from considering the steps from all levels in the illustration above; i.e., not ignoring the Level 1-4 timelines (or even larger levels if the PP scale range is enlarged). Steps at higher levels are the expected outcome from temporally coarser scale goals. From the perspective of a lower level, higher level goals will appear more strategic. Due to the geometric scaling (statistically speaking) of the frequency of steps from respective levels under the PP postulate, in the long run one should expect twice as many Level 3 displacements for every Level 4 displacement; twice as many Level 2 displacements for every Level 3 displacement, and so on.

Thus, the distribution of step lengths from PP complies with a power law – similar in some respect to a Lévy walk. However, there are qualitative and testable differences between such scale-free space use by a Lévy walker and a PP walker. For example, the issue mentioned above with respect to Lévy walk and truncation of large displacements are under the PP model resolved for environments with frequent path disruptions from environmental conditions (see Figure 24 in my book).

As an extra twist of the non-standard statistical mechanical framework that accompanies PP, observe that in the illustration above I have widened the concept “Present” as we zoom from fine towards coarser time resolutions. Consequently, while mechanistic theory defines an event (like a movement decision) to take place at s given point in space at at given point in time, the PP framework defines the concept of “Now” i more broad terms. “Now” by necessity implies a coarser time resolution at higher levels. This aspect is qualitatively incompatible with standard statistical mechanics. A PP-influenced kind of space use is in fact mathematically non-differentiable both from the Lagrangian and the Eulerian angle. Hence, paths and spatial utilization distributions comply with statistical fractals.

What is potentially achieved by switching away from the mechanistic framework?

  • The PP concept generalizes the standard modeling framework for ecological research from – by default and tradition – being strictly Markovian/mechanistic towards allowing cognitive processing to be executed over a range of temporal resolutions in parallel. In other words, decisions taking place at a higher level in the PP hierarchy is conjectured not to be a simple superposition (sum) of the myriad of decisions that are executed at finer temporal resolutions. The mechanistic principle is thus qualitatively violated, but may be regained if the PP scale range is suppressed towards a single scale (like a non-linear functional response being squeezed into a linear one).
  • The PP theory describes several process-specific properties that deviate from the mechanistic counterpart.  These properties may be described and tested both in simulations and on real data. In my book, my papers and here at my blog I have provided several anecdotal empirical examples supporting the PP framework (for example, the lab rat data). In more stringent settings than these anecdotal examples model variants from the mechanistic framework have represented the null hypotheses, which have been falsified in every test so far (for example, Gautestad et al. 2013).
  • The transition between deterministic and stochastic movement decisions gets an unexpected theoretical twist. For example, a deterministic decision at Level 4 to move to a neighbour area during the next couple of hours will probably not influence a randomly picked finer-grained decision among the set of decisions that are embedded in the similar-sized time period at Level 1 (with logarithmic base 2 for the scaling of the timelines, Level 1 regards time resolution 120 min/8 = 15 min). During one or more of the eight embedded Level 1-events during the actual Level 4-event, the goal in progress at Level 4 is actually expected to be executed as a surprise factor (i.e., stochastically) from the perspective of Level 1. In other words, the deterministic aspect of this event can only be revealed by considering the goal at its proper level 4. Thus, from Level 1’s perspective, the “moving to the neighbour area” appears in a manner that cannot be properly resolved by expanding the model with more detailed movement rules at Level 1. The degree of surprise events from the perspective of the finer-resolved timeline increases with the scale distance to the  higher levels where the actual goals reside.
  • From an evolutionary perspective it is interesting that a study of animal space use from a statistical-mechanical approach (e.g., based on a series of GPS fixes) may provide strong empirical support for complex behaviour, involving a hierarchical and geometrically scaling execution of decisions. A priori, one should expect an animal with such cognitive capacity to achieve a higher fitness than an animal lacking it.
  • From an ecological perspective the PP framework raises novel questions for research and methods how to answer them. The PP postulate also provides a potential to resolve some long-lasting confusion and even controversies, both in the field of movement ecology and in population dynamics (Gautestad 2013; Gautestad and Mysterud 2013).
  • The scaling cube visualizes how temporal memory (one of the premises for PP) may be expanded with spatial memory to create a model for multi-scaled space use that includes site fidelity (PP allows for emergence of a home range from non-mechanistic principles).

REFERENCES

Gautestad, A. O. 2013. Lévy meets Poisson: a statistical artifact may lead to erroneous re-categorization of Lévy walk as Brownian motion. The American Naturalist 181:440-450.

Gautestad, A. O., and A. Mysterud. 2013. The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion. Movement Ecology 1:1-18.

Gautestad, A. O., L. E. Loe, A. Mysterud. 2013. Inferring spatial memory and spatiotemporal scaling from GPS data: comparing red deer Cervus elaphus movements with simulation models. Journal of Animal Ecology 82: 572-586.