Parallel Processing – How to Verify It

In my previous post I contrasted the qualitative difference between animal space use under parallel processing (PP) and the standard, mechanistic approach. In this post I take the illustration one step further by illustrating how PP – in contrast to the mechanistic approach – allows for the simultaneous execution of responses and goals at different time scales. This architecture is substantially different from the traditional mechanistic models, which are locked into a serial processing kind of dynamics. This crucial difference in modelling dynamics allows for a simple statistical test to differentiate between true scale-free movement and look-alike variants; for example, composite random walk that is fine-tuned towards producing apparently scale-free movement.

First, recall that I make a clear distinction between a mechanistic model and a dynamic model. The former is a special case of a dynamic model, which is broader in scope by including true scale-free processing; i.e., PP. In my previous post I rolled dice to explain the difference.

In the traditional framework there is no need to distinguish between a mechanistic and a dynamic evolution, simply because in this special case of dynamics time pr. definition is one-dimensional. On the other hand, in the PP framework time is generally two-dimensional to allow for parallel execution of a process (for example, movement) at different scales at any moment in time.

Ignoring this biophysical distinction has over the years produced a lot of unnecessary confusion and misinterpretation with respect to the Multi-scaled random walk model (MRW), which is dynamic but non-mechanistic. The distinction apparently sounds paradoxical in the standard modelling world, but not in the PP world. I say it again: MRW is non-mechanistic, non-mechanistic, non-mechanistic – but still dynamic!

First, consider multi-scale movement in the comfort zone of mechanistic models. You may also call it serial processing, or Markov compliant. In the image to the right we see a (one-dimensional) time progression over a time span t=1,….,8 of a series where unit time scale per definition equals one (ε = b0=1; see my previous post). Some sequences are processed at a coarser scale than unit scale; for example, during the interval from t=2 to t=5 the animal “related to” its environment in a particularly coarse-scaled manner relative to unit time. Consider an area-restricted search (ARS) scenario, where the unit-scale moves (light blue events) regard temporally more high-frequency search within a local food patch and more coarse-scaled moves regard temporally toggling into a mode of more inter-patch movement. Consider that the animal during this time temporarily switched to a behavioural mode whereby environmental input is less direction- and speed-influencing (as seen from the unit scale) than during intra-patch search.

Within a mechanistic framework, processing at different scales (temporal resolutions) cannot take place simultaneously. The process needs to toggle (Gautestad 2011).

Mechanistically, the ARS scenario is often parsimoniously modelled by a composite, correlated random walk. By fine-tuning the model parameters an the relative frequencies of toggling it has been shown how such a pattern my even produce approximately scale-free distribution of displacements; i.e, Lévy-like movement (Benhamou 2007). Such statistical similarity between two distinct dynamical classes has produced much fuzz in the field of animal movement research.

Next, contrast the Lévy look-alike model above with a true scale-free process to the right. Due to the dynamics being executed over a continuum of temporal scales, we get a hierarchical structure of events. Thanks to the extra ε axis, there is no intrinsic paradox – as in a mechanistic system – due to a mixture of simultaneous events at different resolutions. Again, I refer to my previous “rolling dice” description. Despite a potential for fine-tuning the composite random walk model to look statistically scale-free, this mechanistic variant and the dynamically scale-free Lévy walk belongs to different corners of the Scaling cube.

Finally, how to distinguish a PP compliant kind of scale-free dynamics from the look-alike process? Coarse-grain the time series and see if the scale-free property persists or not (Gautestad 2013)!

Simulation of a two-level Brownian motion model was performed under four conditions of ratio lambda between the scale parameter of the respective levels, lambda2/lambda1, where frequency of execution t2/t1 = 10 under all conditions. For each condition of lambda the simulated series were sampled at three time scales (lags, tobs); every step, sampling 1:10 and sampling 1:100. Original series lengths were increased proportionally in order to maintain the same sample size under each sampling scheme (20 000 steps). A double-log scatter plot (logarithmic base 2) of step length frequency, log(F), as a function of binned step length, log(L), was then made for each of the four parameter conditions and each of the three sampling schemes. (a) The result from lambda = 4 shows a linear regression slope and thus power law compliance over some part of the tail part of the distribution, with slope b = 2.9; i.e. the transition zone between Lévy walk (1 < b < 3) and Brownian motion (b >= 3 and increasing with increasing L, leading to steeper slope). At coarser time scales tobs = 10 and tobs = 100 the pattern is transformed to a generic-looking Brownian motion with exponential tail, which becomes linear in a semi-log plot: the inset shows the pattern from tobs = 100. (b) The results from lambda = 8.

Both the step length distribution (above) and the visual inspection of the path at different temporal scales reveal the true nature of the model: a look-alike scale-free and pseudo-Lévy pattern when the data are studied at unit scale where the fine-tuning of the parameters were performed, but shape-shifting towards the standard random walk at coarser scales. A true PP-compliant process would have maintained the Lévy pattern even at different sampling scales (Gautestad 2012).

Simulated paths of two-scale Brownian motion where 1000 steps are collected at time intervals 1:1, 1:10 and 1:100 relative to unit scale for the simulation, with lambda2/lambda1 = 15. The pattern shifts gradually from Lévy walk-like towards Brownian motion-like with increasing temporal scale relative to the execution scale (t = 1) for the simulations. Since the number of observations is kept constant the spatial extent of the path is increasing with increasing interval.

By the way, the PP conjecture also extends to the MRW-complementary population dynamical expression of animal space use, the Zoomer model. This property can be clearly seen in the Zoomer model’s mathematical expression.

 

REFERENCES

Benhamou, S. 2007. How many animals really do the Lévy walk? Ecology 88:1962-1969.

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. 2013. Animal space use: Distinguishing a two-level superposition of scale-specific walks from scale-free Lévy walk. Oikos 122:612-620.

 

The Limited Scope of Lévy Walk and the LFF Model

The Lévy flight foraging (LFF) hypothesis describes a toggling between classic Lévy flight/walk (LW) and classic Brownian motion (BM) as a function of the individual’s current resource field properties (its “environment”). Both states of motion are statistical by nature – and explicitly defined as such. The LFF describes movement as random walk in two disparate modes; scale-free LW versus scale-specific BM. However, the LFF premise of animals moving like drunken LW/BM walkers logically does not make sense unless the animal in question does not possess a capacity for spatial memory utilization or because the environment is so volatile that returning to a previous location has no fitness value with respect to optimal foraging. Under these premises of value-less spatial map utlilization the LFF hypothesis should be expected to make sense, otherwise one should expect to find better compliance with other movement-related models and hypotheses.

The theoretical model developments surrounding both the LW concept in general and the LFF hypothesis in particular have ignited new life into the interface between biology (animal movement, behavioural ecology) and physics (Lagrangian aspects of statistical mechanics, biophysics of animal space use). At last, the classic random walk – Brownian motion (BM) – got justified competition, but recall that both LW and BM are variants of random walk; respectively scale-free and scale-specific. Hundreds of papers have emerged at an increasing pace over the last 30 years, exploring optimal foraging theory and other behavioral-ecological aspects using this statistical walk approach. However, despite the increasing pile of research that has verified LW-like movement in data the results have often spurred both confusion and controversy, including alternative explanations for LW-like power law compliance of step length distribution.

Setting the LW versus pseudo-LW discussion aside, my own critique has on one side focused on premise that the two movement classes contained in LFF regard toggling between behavioural modes, and not just realizations of basically deterministic behaviour that is viewed statistical-mechanically; through “a hidden layer” (path sampling rather than continuous observation). On the other hand I have also criticized LFF from the empirically supported perspective that LW is intrinsically scale-free with respect to processing time but in practical terms scale-specific with respect to processing space.

First, when a path is sampled rather than observing the movement behaviour directly, the statistical compliance with LW, BM or other classes is an emergent property of the statistical-mechanical and intrinsic property of the actual process, regardless of the degree of deterministic or stochastic movement mode at the real-time scale.

Second, how can a process that generates a scale-free distribution of step lengths be scale-specific in the spatial domain? The simple answer is the effect of environmental influence when moving in heterogeneous space. For example, a potentially long distance displacement in progress is under the LW model expected to be easily terminated (truncated) by movement-influencing events and local conditions that successively appear within the trailing perceptual field – the interrupt distance that idealizes a specific scale – as the animal moves along. Consequently, the result is a deflation of super-long step lengths relative to expectation from absence of such interrupts. Such “truncated power law” distributions of step lengths in a non-homogeneous environment dominates the empirical results, where scale-free movement is consequently constrained to medium-range displacements*. Further down this continuum towards increased frequency of large-step interrupts we meet the BM kind of movement – the patch utilization condition of LFF – where the scale-free aspect of movement has vaporized altogether.

According to the LFF hypothesis, next time the wagtail Motacilla alba revisits this patch it is always by chance. A strategic future return; i.e., a non-random crossing of the bird’s movement path that is based on a decision outside the perceptual field surrounding this patch, requires an element of spatial memory. Photo: AOG.

However, many animals have been shown to perform scale-free movement also in heterogeneous space beyond the expected scale of potential truncation from local step length interrupts. Even during foraging such LFF breaking pattern has frequently been documented, as I have described with various references to empirical data in several posts in this blog. Some vital aspect of animal movement is obviously missing from the LW framework, and thereby also constraining the feasible applicability of the specific LFF hypothesis.

In my view there are two dis-harmonic properties of LW when confronted with empirical data of vertebrate movement; (a) the apparent ability of animals to be “back on track” after some kind of local event has interrupted a long-distance move in progress. This obviously violates LW’s intrinsic property of being a Markov compliant process. Further,  (b) according to LW properties the individual is expecting to re-visit a previous location by chance only, not as a result of a strategic return. This behavioural aspect – involving spatial memory and strategic moves – is absent from the Lévy approach. Lack of spatial memory utilization does not harmonize with the behavioural ecology of a very wide range of species and ecological conditions. Vertebrates and large groups of invertebrates have now been verified to possess a capacity for memory map utilization. Under a very broad range of environmental conditions it makes sense to logically expect a potential fitness value from returning non-accidentally to a previously utilized food patch, for example.

In particular, under the premise of spatially memory-less LW, the emergent property of “clumped” and statistical fractal-compliant aggregations of locations from self-reinforced space use (strategic returns) cannot appear. If one finds support for scale-free space use in the step length distribution in combination with support for memory utilization – “the home range ghost” (Ic√N) – it is not LW that has produced the pattern but Multi-scaled random walk (MRW). LW and MRW may under some conditions share a similar distribution of both displacement lengths and spatial distribution of relocations (power laws, with or without identical exponents), but only MRW embeds the expectation of a home range kind of space use. A home range is a verification of spatial memory utilization. Even under the condition where the ratio of returns vs. exploratory sallies under MRW is very small (leading to a larger home range for a given N; i.e., a larger c in the ghost equation), it still has a large statistical effect on how the animal is utilizing local patches with respect to self-reinforcing space use.

So far, whenever we have tested for LW vs. MRW it has been the latter that has been supported: a Lagrangian power law distribution of step lengths and an Eulerian power law compliant home range ghost equation. And the respective environments have been heterogeneous from fine towards coarse scales. Hence, a wide range of species seem to utilize their environment in a scale-free processing manner (due to the statistical power law compliance) over both time and space. This should have consequences for future tests of statistical patterns in animal movement, in particular with respect to model assumptions. New ecological methods should be implemented if memory is a factor.

I do not claim that the LW and the LFF hypothesis are erroneous – all I underscore is that these models’ assumptions with respect to absence of spatial memory influence should be scrutinized a priori and not taken as granted. I believe these models’ scope of realism will be accepted to be substantially limited if this introductory test is performed on a regular basis.

A statistical-mechanical modelling approach is needed when animal movement and space use is sampled; i.e., a hidden layer is invoked. One is studying not only the statistical aspects of the behaviour in the traditional sense. One is simultaneously also shifting the level of system abstraction from a temporally moment-to-moment analysis to a more abstract process level that appears from system coarse-graining of observational time scale (path sampling).

A statistical-mechanical theory for animal movement and space use should implement all variants of movement classes. In The scaling cube I have capture these classes by its eight corners, where the cube’s interior is expressing respective continua. From the Lagrangian perspective, spatio-temporally scale-free displacements are formulated by the MRW model’s step length distribution; from the Eulerian (spatially explicit) perspective the MRW model is expressed both by the Home range ghost formulation and by the complementary Zoomer model. The mathematical formulation of the parsimonious Zoomer model – a memory-extended statistical mechanics of ensembles of particles (individuals) under scale-free influence of conspecific attraction –  is found here.

NOTE

*) This pattern where the extreme section of the long tail part of the distribution is constrained does not per se lend support to the classic Lévy framework despite partly power law. Other models, for example the Multi-scaled random walk (MRW), also gives expectation of a truncated scale-free step length distribution over a wide range of conditions (see various posts on this blog).

 

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.

REFERENCES

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.

REFERENCES

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).

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.