Jellyfish behavior: LFF or MRW?

Scale-free distribution of displacement lengths is often found in animal data, both vertebrates and invertebrates. In marine species this pattern has often been interpreted in the context of the Lévy flight foraging hypothesis (LFF), where optimal search is predicting a scale-free power law compliant movement when prey patches are scarce and unpredictably distributed and a more classic and scale-specific Brownian motion-like motion when such patches are encountered (Viswanathan et al. 1999). In a study on the jellyfish Rhizostoma octopus such an apparent toggling between two foraging modes were found, but critical questions were also raised by the authors (Hays et al. 2012). Here I come the authors “to the rescue” by suggesting that an alternative model – the Multi-scaled Random Walk (MRW) – could be included when testing statistical classes of foraging behaviour.

I cite from their Discussion (with my underscores):

In some periods (when integrated vertical movement was low), vertical excursions were followed by a vertical return to the depth occupied prior to the excursion. This pattern of ‘bounce’ movements has also been seen in some fish [ref.] and presumably represents an animal prospecting away from a preferred depth, not finding an improvement in conditions elsewhere and so returning to the original depth. Such behaviour sits outside the Lévy search paradigm where it is assumed that a prey patch is not purposefully revisited once deserted. Again this finding of ‘vertical return’ behaviour, points to jellyfish movements, at least on occasions, being fine-tuned to prey resources.
Hays et al. (2012), p471

Another jellyfish species. Photo by Pawel Kalisinski from Pexels

Such a space use mixture of “prospecting away” in combination with targeted returns, and where the former complies with a scale-free step distribution (as now shown in jellyfish), is in fact MRW in a nutshell. As repeatedly underscored in previous posts the LFF hypothesis rests on a premise that individuals do not have a cognitive capacity to return non-randomly to a previous location, while MRW includes this capacity (Gautestad 2012; Gautestad and Mysterud 2013).

When search behaviour is studied using a spatially memory-less model framework that contrasts behavioural toggling between Lévy and Brownian motion, the standard statistical method (MLE) typically explores the continuum from a pure power law to a pure exponential, with a so-called truncated Lévy flight in-between. In addition to Hays et al. (2012), also Ugland et al. (2014) documented this transition, with Lévy pattern during night time swimming of another large jellyfish, Periphylla periphylla.

I cite from one of my papers, where the ratio between the average return interval tret and the sampling interval, tobs of the animal’s path; ρ = tret/tobs, is key to understanding the statistical pattern if movement is memory-influenced:

… by analysing the data with different tobs relative to system-specific boundary conditions, two observers may reach very different conclusions with respect to step-length compliance with a negative exponential or a power law. Both may in fact be right! In particular, if the animal in question has used its habitat under the influence of long-term memory, then the observed pattern at temporal level tobs may shape-shift from power law, through a hockey stick pattern, to a truncated power law pattern (figure 1c), and ultimately to a negative exponential (BM compliance) if tobs is chosen large enough. Hence, this paradox may to some (testable) extent be rooted in a relative difference in observational scale between the respective studies.
Gautestad 2012, p8.

A jellyfish has a very rudimentary nervous system. It doesn’t have a brain or central nervous system, only a very basic set of nerves at the base of their tentacles. These nerves detect touch, temperature, salinity etc. and the animal reflexively respond to these stimuli. For example, the jellyfish can orient to olfactory cues from prey (Arai 1991). Hence, the movement is expected to include targeted returns in a very rudimentary and environmental field-dependent manner. On the other hand, Kaartvedt et al. (2015) have demonstrated the ability of a jellyfish species, P.  periphylla, to locate and team up with each other in a surprisingly “individualistic” manner. That fact raises an interesting (and speculative) hypothesis; could jellyfish movement along the Lévy-Brownian gradient be explained as temporally difference in targeted return frequency (same tobs and different tret), whether returns go to a previous depth or as a means to keep contact with conspecifics? The MRW framework – including its parallel processing postulate for cognitive tactics/strategy complexity – provides a tool to test this hypothesis.

In short, do these returns in different context for these two jellyfish species embed tactical and Markovian-like behaviour only (for example, simply following an olfactory gradient on a moment-to-moment basis) or is a jellyfish capable of returning more strategically by initiating a return without such a specific taxis-response within its current perceptual field?

What is extremely interesting in Hays et al. (2012) is that the jellyfish apparently shows a capacity both to long distance prospecting and long distance returns. According to MRW the returns should emerge from a capacity for spatial mapping of previously encountered foraging patches, without necessarily following an olfactory gradient towards this target! Hence, the test to differentiate between these classes of spatially explicit behaviour is to study if the animal is capable of targeted returns in absence of – or even disobeying (!) – a simple “following the gradient” (taxis) kind of return.

Hays et al. (2012) documented “occasional sallies” (prospecting) in the foraging behaviour of jellyfish. This behaviour obviously implies moving away from the current foraging patch and thus “down” the hypothetical olfactory gradient. Returning may then either imply swimming “up” the gradient or targeting a previous location per se; hypothetically as a function of spatial memory rather than getting moment-to-moment guidance from an environmental, chemical field. For a conceptual illustration of complex movement spanning the tactics-strategy gradient (parallel processing), see this post.

A long and speculative shot, I agree, to suggest that jellyfish nervous system may express MRW behaviour. However, perhaps the cognitive capacity of animals with simple nerve systems like jellyfish are more powerful than traditionally anticipated, and that statistical analyses of their movement paths from the perspective of (memory-extended) statistical mechanics may contribute to studying this capacity?

For example, due to Hays et al.‘s (2012) documentation of the combined capacity to explore its environment in a scale-free manner within a given scale range and occasional returns to a previous location (which may take several minutes; i.e., “strategic moves”) the jellyfish behaviour may cast light on evolutionary initial steps towards a more sophisticated kind of spatial behaviour, as it is found in animals with developed brain structures.

Such a potential for rudimentary MRW behaviour could, for example, imply a capacity to perform targeted returns to a recent part of the individual’s path but not further back as in the default MRW. Such a constrained variant of parallel processing may be tested statistically, by comparing simulations under this condition with true paths. In fact, I’ve already done introductory simulation studies (Gautestad 2011; A. O. Gautestad, unpublished).

MRW is simulated in 2-dimensional space with return steps at frequency 1:100 of original series (tret=100 in relative terms) to a trailing time window of 1000 last steps; i.e., a short memory horizon. Left: spatial pattern from 9000 observed fixes at frequency 1:1000 of original series (tobs. = 1000). Middle: box counting method shows fractal dimension D = 1.06 over a mid-range of spatial resolution, k. A larger number of fixes, N, would have increased this range. Right: Studying incidence, I, as a function of N shows a positive log–log slope of 0.96 and 1.01 for grid resolutions k = 1:64 and 1:300, respectively. This example illustrates that MRW under  the condition of temporally constrained memory still shows a statistical fractal of spatial fixes. However, the limited capacity for targeted returns makes incidence increase proportionally with N (log-log slope of 1) rather than with square root of N (log-log slope of 0.5), as when memory is infinite and ρ = tret/tobs << 1. From Gautestad (2011).

Reynolds (2014) explored the results in Hays et al. (2012) by simulating an alternative model for jellyfish search, called Fast simulated annealing (FSA). In the present context this algorithm is scanning the environment to find and select optimal food patches. This post is too short to describe and discuss this very interesting approach, so I may return to it later. However, as a preliminary comment to those familiar with FSA I suggest that it may be very promising to combine principles from MRW and FSA. In particular, FSA implies patch selection that on one hand is based on a Cauchy-distributed step length distribution during searching; i.e., very long tailed next-location selection, and on the other hand occasional “escape” steps to avoid local trapping in patches that are potentially only locally optimal but not globally. Long steps at scales beyond the animal’s perceptual field will logically require a cognitive capacity for some kind of directed returns to more optimal patches after “prospecting”; i.e., spatial memory may be required. By default, FSA does not include spatial memory. In other words, the perceptual field is assumed to span the entire search arena. This capacity is obviously not a feasible premise in the jellyfish case, so what remains to sufficiently extend the individual’s overview of its environment is a cognitive utilization of a spatial map?

On the other hand, combining MRW and FSA will have to bridge two system representations, which may require a novel mathematical formulation of FSA. While MRW requires a sufficiently deep hidden layer to ensure compliance with a statistical-mechanical system description, the FSA in current formulations describes a mechanistic and Markovian kind of dynamics on a fine-grained temporal scale; i.e., a very shallow hidden layer. Further, FSA describes a tactical search algorithm, while MRW is based on a gradient from tactical to strategic time scales in a non-trivial kind of superposition (the parallel processing conjecture).

To conclude, the experimental outline for studying optimal foraging needs to include a test for strategic space use beyond a purely tactical/Markovian kind of displacements.


Arai, M. 1991. Attraction of Aurelia and Aequorea to prey. Hydrobiologia 216:363–366.

Gautestad, A. O. 2011. Memory matters: Influence from a cognitive map on animal space use. Journal of Theoretical Biology 287:26-36.

Gautestad, A. O. 2012. Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion. Journal of the Royal Society Interface 9:2332-2340.

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.

Hays, G. C., T. Bastian, T. K. Doyle, S. Fossette, A. C. Gleiss, M. B. Gravenor, V. J. Hobson, N. E. Humphries, M. K. S. Lilley, N. G. Pade, and D. W. Sims. 2012. High activity and Lévy searches: jellyfish can search the water column like a fish. Proc. R. Soc. B 279:465-473.

Kaartvedt, S., K. I. Ugland, T. A. Klevjer, A. Røstad, J. Titelman, and I. Solberg. 2015. Social behaviour in mesopelagic jellyfish. Scientific Reports 5:1-8.

Reynolds, A. M. 2014. Signatures of active and passive optimized Lévy searching in jellyfish. Journal of the Royal Society Interface 11:20140665.

Ugland, K. I., D. L. Aksnes, T. A. Klevjer, J. Titelman, and S. Kaartvedt. 2014. Lévy night flights by the jellyfish Periphylla periphylla. Mar. Ecol. Prog. ser. 513:121-130.

Viswanathan, G. M., S. V. Buldyrev, S. Havlin, M. G. E. d. Luz, E. P. Raposo, and H. E. Stanley. 1999. Optimizing the success of random searches. Nature 401:911-914.

The Lesser Kestrel: Natal Dispersal In Compliance With The MRW Model

The Multi-scaled random walk (MRW) model defines a specific dispersal kernel for animal movement; a power law, which is qualitatively different from standard theory (a negative exponential function). Alcaide et al. (2009) analyzed long-term ringing programmes of the lesser kestrel Falco naumanni in Western Europe, and showed results from re-encounters of 1308 marked individuals in Spain. They found that most first-time breeders settled within 10 km from their natal colony (i.e., a strong philopatric tendency), with a negative association between natal dispersal and geographical distance. While Alcaide et al. (2009) were mainly concerned with gene flow and population effects, here I take a deeper look at their dispersal data and find strong support for MRW-compliant behaviour in the natal dispersal data. Indirectly, this pattern at the individual level also supports the MRW-analogue at the population level, the Zoomer model (Gautestad 2015).

I allow myself to copy their Figure 1, showing the natal dispersal distances:

Fig. 1. Frequency distribution of natal dispersal distances of lesser kestrels in the Guadalquivir Valley (SW Spain, N = 321 individuals, black bars; Negro et al. 1997) and in the Ebro Valley (NE Spain, N = 961, white bars; Serrano et al. 2003).


To visualize the difference between the expected dispersal kernel from MRW and from standard theory I here present the data above with log-scaled axes:


Under this transformation, compliance with a power law should resemble a straight regression line, with a slope that is defined by the power exponent. Such log-log linearity of a power law contrasts with a log-log transformed negative exponential function, which becomes convex. Interestingly, the two subsets of natal dispersal distances show strong compliance with a power law (R2=0.90 and R2=0.96, respectively), while the best-fitting negative exponential does not match the pattern that well (R2=0.60; dotted line).

Quite remarkably, even the power exponents (β=-2.02 and β=-2.00) show up very close to the standard MRW expectancy of β=-2  (Footnote 1). This particular magnitude of β is – according to MRW theory – expected from scale-free space use where the individual on average during the sampling period has put equal effort into utilizing its environment over the given scale range (in this case, from a spatial grain resolution of 10 km to an extent resolution of 440 km).

The discovery of natal dispersal data as summarized by Alcaide et al. (2009) allows me – for the first time – to study empirical model compliance in a species at relatively coarse temporal scales; i.e., over the interval from birth to first breeding the following year or two. Previous resolutions for MRW tests have typically been at temporal resolution of a few hours (GPS relocation data). Simultaneously, the good fit to power exponent β=-2, even at this coarse temporal scale, translates to β’=-1 in area terms rather than distances (Gautestad and Mysterud 2005). I recycle an illustration of this population kinetic aspect, which was also shown in this post and in my book:


The grey-shaded inset represents the classic dispersal kernel, expected from standard random walk at the individual level and diffusion at the population level; i.e., a negative exponential. The other elements in the illustration regard MRW (scale free power law, see also Footnote 2).

In particular, observe for the F(L) movement kernel that the coloured rectangle area of each log-scaled interval (bin) for squared distance, L2; representing “effort” by the individual to relate to respective spatial resolutions of their environment, is of similar magnitude when F=(L2)-1 = 1/L2. The area of each of the rectangles is the same. In other words; in a two-dimensional arena, an individual is then utilizing a k times larger landscape resolution 1/k times as frequently. In a population context (the Zoomer model, switching from a Lagrangian to the complementary Eulerian system perspective) – since a k times larger arena is expected to embed k times more individuals in average terms – when β=-2 the population is utilizing the landscape with equal intensity over the given scale range (Gautestad 2015, p122-132).

Footnote 1: what about the Lévy flight/walk model, which also predicts a scale-free and thus a log-log linear dispersal kernel? With respect to the lesser kestrel, as well as all other bird species, spatial memory is part of their cognitive capacity. A home range, which requires directed returns to previous locations, is exemplifying this utilization. MRW regards a combination of scale-free space use and site fidelity. Lévy flight only regards the former.

Footnote 2: With respect to lesser kestrel’s natal dispersal, the data represents the displacement distribution of many individuals (called an ensemble in statistical mechanics) rather than the distribution of a set of displacements for a given individual. Thus, the power law curve reflects these individuals’ pooled tendency for scale free space use during natal dispersal. When establishing their respective home ranges with centre of activity at the chosen breeding site, it would have been interesting to see whether the median displacement length (and β) for the following 1-2 year period deviated from natal dispersal at the same temporal resolution.


Alcaide, M., D. Serrano, J. L. Tella and J. J. Negro. 2009. Strong philopatry derived from capture–recapture records does not lead to fine-scale genetic differentiation in lesser kestrels. Journal of Animal Ecology 78:468–475.

Gautestad, A. O. 2015. Modelling parallel processing. pp114-148 inAnimal Space Use: Memory Effects, Scaling Complexity, and Biophysical Model Coherence. Dog Ear Publishing, Indianapolis. 298pp.

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

Making the Science of Animal Space Use Less Soft

A provocative headline is a double-edged sword. Why do I indicate that one of the most rapidly developing fields of animal ecology should still be regarded as a soft science? When it comes to individual space use rest assured that I’m thrilled by the substantial leaps forward in some parts of the theory of animal whereabouts. On the other hand, I also have critical comments. In my view there is still too strong disconnection between some general properties of movement-related animal behaviour and theoretical representations of this behaviour in models.

Patch and resource sharing – butterfly Aglais io and bumblebee. Photo: AOG.

Both in my book and in previous blog posts I have repeatedly pointed out the unfortunate fact that contemporary models in the field often referred to as “movement ecology” have matured into two quite distinct premise foundations. On one hand  we see a broadened recognition of scale-free movement as a quite general property and on the other hand also a broadened recognition that spatial memory is influencing movement under common ecological conditions. The former insight is often referred to a “Lévy walk-like movement”, while the latter focus regards strong and recent developments of the classic theory related to home range behaviour, philopatric dispersal, and similar memory map-dependent properties. Unfortunately, these two directions of theoretical development – scaling and memory aspects – still tend to be progressing quite independently of each-other. Lévy walk is inherently void of spatial memory influence, and site fidelity models are generally building on a scale-specific framework with spatial memory influence as  a model extension of this framework. Hence, a scientific field consisting of disparate theories when it comes to the other camp’s main system premises has to fail. Thus, I dare to call animal movement theory unnecessarily soft – until a unification is reached.

To summarize, from a bird’s view of the current state I dare to categorize the science of animal movement as soft science due to lack of coherence between on one hand modelling the sliding scale from scale-specific to scale-free space use, and on the other hand modelling the variable strength of directed returns to familiar patches – on the basis of a common theoretical framework.

When I during the early 90’s started locking my focus on these two aspects of animal behaviour the interplay between multi-scaled space use and the simultaneous expression of site fidelity, both fields of research were in respective camps surrounded by confusion and controversies.  Conferences, workshops and books on animal space use were typically surrounded by a myriad of ideas and concepts in the emergent field of landscape ecology. Thus, as far as I recall the scene back then, nobody else seemed to be tempted to make the science of animal space use even messier by studying model coherence between scaling and spatial memory. The general attitude was that “scale-free movement” – if it was at all recognized – might be interesting as a concept, but the theory had a far way to go before it could reach general acceptance with consequences for statistical and dynamical modelling and ecological analysis.On the other hand, “the Lévy camp” typically disregarded the site fidelity aspect of movement all together, except for occasional and brief reference to the concept as a challenge for the future.

Progress in respective camps have obviously been strongly propelled by large databases of animal movement that has been collected from modern GPS technology. Thus, theory and empirical data have been brought closer together. Wildlife ecology, computer modelling and advanced statistical analysis is thriving together in many strong research groups. Still, I’m waiting for bolder steps towards stronger unification between scaling and memory.

Wildlife ecologists have much to look forward to from such a leap towards better model realism. Consider all the interesting aspects that could be more easily studied and tested when the respective hypotheses are based on a more coherent theory of animal space use. Respective quantification of parameters connected to scaling and memory will then be founded on models with stronger predictive power. It is tempting lo start listing a long string of examples of potential theory applications at this point, but first things first. Conferences and workshops on the scaling/memory issue lay in the cards!

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).


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.


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.


Lévy Walking or Not Lévy Walking – That’s the Question

As summarized and reviewed by many, one of the hot topics in the field of movement ecology regards to what extent – and under which ecological conditions – an animal performs scale-free movement. As readers of my book and my blog definitely have observed, I advocate a distinction between several kinds of scale-free movement (summarized by the Scaling cube); in particular, (a) the standard battleground of Lévy walk vs. Brownian motion (scale-free vs. scale-specific movement) (b) Lévy walk vs. composite random walk (a “Lévy walk look-alike”, i.e., pseudo-scale-free movement), and (c) Multi-scaled random walk (MRW; scale-free movement with site fidelity). The third variant has so far not received much attention in this debate. This is in my view unfortunate, because this approach apparently has the potential to resolve much of the controversy!

It is a fact that empirical research on scale-free vs. scale-specific movement generally has ignored the effect from animal site fidelity. This important and very universal property of animal space use seems to have got lost in convoluted and heated discussions on statistical methods and statistical interpretation of results. Why is this unfortunate? Simply because an inclusion of site fidelity when modeling space use allows a given movement pattern to appear scale-free or scale-specific, simply as a function of sampling frequency.

Figure3AResearch in this field tends to focus on the Lagrangian aspect of scale-free vs. scale-specific movement; i.e., the distribution F(L) of inter-fix displacement lengths L in a set of relocations from sampling at interval t along the true path. Based on our simulation results, a visual inspection of F(L) may apparently give a good clue about the actual generating process, as illustrated conceptually by the three classic candidate models in the left-hand part of the illustration above (the Figure is copied from Gautestad and Mysterud, 2013). For example, with log-transformed axes, linearity over a range of 10-100 length units is needed to make the distribution acceptable as statistically “scale-free” and thus Lévy walk compliant over this range, based on a first visual inspection (more formal MLE statistics will then typically confirm this). Such linearity implies a power law distribution, with log-log slope of magnitude –β and 1<β<3.

As the slope becomes steeper than -3 in log-log plot, the data is most likely reflecting scale-specific movement – apparently! Why “apparently”? Because the standard set of candidate models all define this specific transition as a critical magnitude of β to distinguish scale-free from scale-specific movement, without considering the influence from the sampling frequency 1/t; i.e., without considering a site fidelity effect on observed β.

The traditional set of models are implicitly based on an assumption that the animal’s path is not influenced by site fidelity. In other words, the animal is assumed to self-cross its historic path by chance only. Very unlikely, under a broad range of ecological conditions! MRW, which in my view should be included in the set of candidate models in this melting pot, will force upon the analysis a modification of how this scale-specific/scale-free transition at β≈3 is interpreted. In the MRW model, scale-free space use is combined with site fidelity. At present, MRW is the only model that provides this combination! 

Site fidelity implies a sensitivity to sampling frequency. If 1/t is substantially larger (interval smaller) than the animal’s average return frequency towards previously visited locations, we do not expect much effect from site fidelity on the distribution F(L). There is only a minute chance for a return event taking place during a given sampling interval. The MRW process will at these temporal resolutions appear scale-free and Lévy walk compliant. However, if the ratio ρ between the animal’s average return interval and the researcher’s choice of sampling interval is approaching unit size from above – and perhaps even passing over to ρ<1 (high probability of at least one return event during t) – the distribution F(L) will no longer look scale-free. Under this condition the effect from spatial memory on space use comes into focus and will tend to blur any log-log linearity in the F(L) distribution.

The site fidelity effect is illustrated in the right-hand part of the image above. At high frequency sampling relative to the individual’s average return interval, the effect from return events – the site fidelity effect – is basically concentrated in the outermost part of the tail of the distribution (blue area). In the MRW model, this “hump” pattern is reflecting that return events, whenever they have happened during a sampling interval, will tend to produce a quite large displacement at the next relocation in the series, relative to the median step length from sampled displacements that are void of such returns. Next, consider that ρ is decreased from e.g. 100 towards a smaller ratio using a larger t. For example, in the range 1<ρ<10 simulations show that the slope may be “artificially” steepened by the site fidelity effect, making the functional form increasingly similar to a truncated Levy walk as ρ →1. A statistical analysis may be expected to show a somewhat reduced β in the small-L range and an increased β in the large-L range. Under this condition the influence from return events become more broadly distributed over the F(L) range (red area). When ρ<1, F(L) may give the impression of Brownian motion compliance due to a concentration of return steps in the left-hand part of the distribution (green area).

It seems like a distributional mixture in the zone between truncated Lévy walk and Brownian motion is the typical pattern observed in real F(L) distributions, making it appear like an animal that my be mode-shifting between truncated Lévy walk and Brownian motion. However, considering the alternative MRW model introduces another interpretation: the movement could potentially be scale-free under influence of site fidelity, which makes the functional form of F(L) observer-dependent. The latter implies that F(L) becomes a function of ρ (where the observer defines the denominator). Based on the typically observed intermittent appearance of F(L) in real data, the discussion continues circling back and forth, year after year: is the pattern complying best with truncated Lévy walk, or a scale-specific Brownian motion? Adding MRW to the set of candidate models for Bayesian analysis might perhaps resolve the gridlock and bring us closer to consensus?

t should be easy enough to test MRW’s potential strength in this regard, given a sufficiently large data set, collected over a sufficiently high sampling frequency. Studying the effect on F(L) from different ρ then becomes a matter of sub-sampling the series.

Both truncated Lévy walk and MRW will show a steepened slope upon such sub-sampling, but these two alternative hypotheses may be easily distinguished by supplementing the F(L) analysis with fractal analysis of the spatial fix pattern (MRW will maintain D≈1, while truncated LW will tend towards larger D). One could also look for the home range ghost (incidence increasing proportionally with √N for MRW and incidence increasing proportionally with N for LW).

Appendix Fig (large rho)When ρ>≈10, simulations of MRW show that return events tend to inflate the occurrence of very long steps, giving the impression of a “hump”, for example as shown above for at ρ≈10. Several researchers in the field of Lévy research have in fact anecdotically observed such a hump, without having a plausible behavioural explanation for it within the context of the traditional candidate models. However, if ρ>>10, the hump may in the MRW scenario almost disappear, making F(L) even more Lévy-like. The illustration to the right shows the simulation result for ρ=100 (Gautestad and Mysterud, 2013).

Already in Gautestad and Mysterud (2005) we illustrated the ρ effect on the “problematic”  F(L) distribution. In Gautestad and Mysterud (2013) we elaborated further on this aspect. However, so far it looks like nobody has grasped this approach to see to what extent this “process-oriented” MRW framework might contribute to bringing the controversy out of the quagmire. In the meantime, the Lévy controversy rolls on, based on models void of spatial memory effects on F(L).

The MRW approach might also contribute to resolve other hot topics in movement ecology. For example, according to the Lévy walk/flight hypothesis for optimal foraging (a more specific aspect of the issue outlined above), Brownian motion is expected in an environment with a relatively productive environment with a relatively predictable resource distribution. In contrast, the MRW working hypothesis would be: in an environment with a relatively productive environment the individual’s home range is expected to be smaller, due to higher return frequency (stronger site fidelity, and thus a smaller ρ for a given sampling frequency). Hence, based on MRW one may predict that the site fidelity effect on F(L) will be more pronounced under these circumstances. If observation frequency, 1/t, is not increased accordingly, one may erroneously find support for the Levy flight foraging hypothesis. Supplementary test should be performed before concluding.


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

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.

Snapshot From the Lévy Controversy

Are animals moving in a scale-specific or a scale-free manner? Are they toggling between these modes under different environmental conditions (basically represented by the Lévy flight foraging hypothesis)? This theme has emerged as a hot potato, which does not seem to cool down anytime soon.

For example, Andy Reynolds’ recently replied to no less than nine (!) responses to his recent paper on this topic. The following reference provides an overview:

Reynolds, A. 2015. Venturing beyond the Lévy flight foraging hypothesis: Reply to comments on “Liberating Lévy walk research from the shackles of optimal foraging”. Physics of Life Reviews 14:115–119.

In my book I summarize elements of the debate on scale-free space use, and put it in context of the parallel processing (PP) conjecture. PP is conceptualized as representing the ceiling in the Scaling Cube, which I briefly described in another post.

In the present post, however, I want to draw the attention to another interesting paper:

Breed et al. 2014. Apparent power-law distributions in animal movements can arise from intraspecific interactions. J. R. Soc. Interface 12: 20140927.

Baltimore_Checkerspot. TaylorsCheckerspot

Females of the two butterfly species Baltimore checkerspot and Taylor’s checkerspot were tracked during foraging and other activities, and their locations were collected every 15-20 seconds. The authors provide a very thorough statistical analysis of the respective distributions of step lengths (fix-to-fix lengths during the constant sampling interval). They conclude that these individuals are not performing scale-free (i.e., Lévy-like) movement but a mixture of scale-specific movement bouts with different parameter characteristics. Such “composite walks” imply in this case a superposition of two exponential functions for the pooled step length distribution with different mean step lengths. Consider area-restricted search (the classic ARS model) as an analogy, often referred to as toggling between intensive and extensive moves. However, in the present scenario the more extended type of moves were frequently triggered by harassment by males. The total distribution of both types of movement may then give the wrong impression of a “fat tail”, which is considered an indication of scale-free (power law) movement.

This work is a very nice and well executed study, which will probably ignite some responses. The potato will be kept warm. Here are my own preliminary thoughts on the results and conclusions in Breed et al. (2014):

  • The authors have chosen to analyze the entire distribution of step lengths, including the relatively high frequency of very small ones. They argue sensibly for their choice: “…short steps were not attributable to observation error and they are numerically important. Thus, we could not justify ignoring some or all of these short steps—often fitting procedures have excluded the shorter steps and only fit the distribution tails.” (p4). My comment to this choice: butterflies can be expected to have spent a considerable amount of time sucking nectar during short displacements (calculated as the distance between successive moves at 15-20 sec. intervals). Such episodes are obviously less dominant during medium and longer range displacements. In other words, the animals may on average have spent a disproportionally smaller part of available time on actual locomotion during intervals with small displacements. Consequently, the number of short steps may then tend to be inflated during periods when the individual is moving slowly, relative to when it is moving faster, when measured at a fixed time scale. This may lead to artificially increased statistical weight of the smallest step lengths and thus to a stronger support for a two-phased kind of scale-specific moves (composite random walk) relative to – for example – power law (Lévy-like) movement.
  • This effect could perhaps easily be adjusted for? If – for example – the butterflies on average spent x% of the time nectar-feeding during displacements belonging in a given distribution bin, the number of steps in this bin could artificially be reduced by x% (random removal) prior to performing the statistical analysis. If x is substantially larger for small-step bins than for larger bins such an adjustment may be critically important.
  • A long-lasting challenge with respect to differentiating between a composite random walk and a true scale-free walk is the fact that the former may easily appear scale-free in statistical tests if the exponential function parameters and the respective components’ frequency of occurrence are “tuned” towards fitting a scale-free distribution of steps (Benhamou, S. 2007. How many animals really do the Lévy walk? Ecology 88: 1962-1969). However, in my book I summarize two very simple methods to distinguish a composite walk  – “a Lévy walk in disguise” – from a true scale-free walk; the coarse-graining method (Gautestad, A. O. 2013. Animal space use: Distinguishing a two-level superposition of scale-specific walks from scale-free Lévy walk. Oikos 122: 612-620) and the parallel shift method (Gautestad, A. O. 2012. Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion. Journal of the Royal Society Interface 9: 2332-2340). More statistically sophisticated methods have also been proposed (Auger-Methe et al. 2015. Differentiating the Lévy walk from a composite correlated random walk. Methods in Ecology and Evolution 6: 1179–118). However, a prerequisite for applying these tests is a large data set, particularly since the coarse-graining method and the parallel shift method depends on studying the distribution of step lengths as the sampling interval is varied (in practice, the original set of relocations is sub-sampled). This requirement probably exclude the application of these tests on the present material, where each series is relatively short (still impressive from a field ecologist’s perspective, I may add).