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.



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 Inner Working of Parallel Processing

The concept of scale-free animal space use becomes increasingly difficult to avoid in modeling and statistical analysis of data. The empirical support for power law distributions continue to pile up, whether the pattern appears in GPS fixes of black bear movement or in the spatial dispersion of a population of sycamore aphids. What is the general class of mechanism, if any? In my approach into this challenging and often frustrating field of research on complex systems, one particular conjecture – parallel processing (PP) – percolates the model architecture. PP requires a non-mechanistic kind of dynamics. Sounding like a contradiction in terms? To illustrate PP in a simple graph, let’s roll dice!

Please note: the following description represents novel details of the PP concept, still awaiting journal publication. Thus, if you are inspired by this extended theory of statistical mechanics to the extent that it percolates into your own work, please give credit by referring to this blog post (or my book). Thank you.

The basic challenge regards how to model a process that consists of a mixture of short term tactics and longer time (coarser scale) strategic goals. Consider that the concept of “now” for a tactical response regards a temporally finer-grained event than the time scale for executing a more strategic event, which consequently takes place within a more “stretched” time frame relative to the tactical scale.

“Strategy” is defined in a hierarchy theoretical manner; coarser scale strategy consequently invokes a constraint on finer scaled events (references in my book). For example, while an individual executes a strategic change of state like starting a relatively large-distance displacement (towards a goal), finer-scaled events during this execution (consider shorter time goals) are processed freely but within within the top-down constraint that they should not hinder the execution of the coarser goals. Hence, the degrees of process freedom increases with the scale distance between a given fine-scaled goal and a coarser-scaled goal.

To illustrate such a a PP-compliant scale range from tactics to strategy within an extended statistical-mechanical system, consider the two-dimensional graph to the right. The x-axis represents a sequence of unidirectional classic time and the y-axis represents a log2-scaled* expression of time’s orthogonal axis, “elacs” (ε) along this sequence.

The continuous x-y plane has been discretized for simpler conceptualization, and each (x,y) pair shows a die. This die represents a potential change of state of the given process at the given point in time and at the given temporal scale. An actual change of state at a given (t,ε) location is marked by a yellow die, while a white die describes an event still in process at this scale. The respective number of eyes on each die could represent a set of available states for a given system variable at this scale. To illustrate complex dynamics (over-)simplistically in terms of concepts from quantum mechanics, consider each magnitude of ε at the y-axis to represent a wave length in a kind of “complex system” wave function and each yellow die represents a “collapse” of this probability wave into a specific execution of the given event at a given point of unit time this time scale.

As the system is viewed towards coarser time scales (larger ε), the average frequency of change of state vanishes proportionally with 1/ε = 1/bz, where b is the logarithmic base and increasing z describes increasing scale level of bz. In other words, the larger the z, the more “strategic” a given event at this scale. In short, consider that each die on scale level 1 [log(b0)=1] is rolled at each time increment t=1, t=2, …, t=8; each die at level 2 [log(b1)=2] is on average rolled each second time increment, an so on.

In the illustrative example above, no events have taken place during the eight time increments at the two coarsest scales bz where z=7 (ε=128) and z=8 (ε=256). A substantial increase of the observation period would be needed to increase the probability of actually observing such coarse-scaled change of system state.

More strategic events are executed more rarely. Strategic events at a given scale bare initiated in a stochastic manner when observed from a finer time scale (smaller z), but increasingly deterministic when observed from coarser time scales. At finer scales such a strategic event may be inexplicable (thus appearing unexpectedly at a given point in time), while the causal relationship of the given process is established (visible) when the process is observed at the properly coarsened time scale. However, at each time scale there is an element of surprise factor, due influence from even coarser scale constraints and even lower frequency change of state of the system at these coarser scales. 

The unit time scale, log(b0)=1, captures the standard time axis, which is one-dimensional as long as the system can be described as non-complex. In other words, the y-axis’ dynamics do not occur, and – consequently – it makes no sense to talk about a parallel process in progress**. In this standard scale-specific framework, time is one-dimensional and describes scale-specific processes realistically. This includes the vast theories of low order Markovian processes (“mechanistic” modeling), the  mathematical theory of differential equations (calculus), and standard statistical mechanics.

For a deeper argument why a PP kind of fundamental system expansion seems necessary for a realistic description of system complexity, read my book and my previous blog posts. By the way, it should of course be considered pieces of a theoretical framework in progress.

The ε-concept was introduced in my book to allow for complex dynamics within a non-Markovian physical architecture. In other words, to allow for a proper description of parallel processing the concept of time as we know it in standard modeling in my view needs to be heuristically expanded to a two-dimensional description of dynamics.

The bottom line: it works! In particular, it seems to survive the acid tests when applied on empirical data, both with respect to individual space use and population dispersion.

Environment is hereby expanded with a two-dimensional representation of dynamical time. This implies that an individual’s environment not only consists of its three-dimensional surroundings at a given point in time but also its temporal “surroundings” due to the log compliant (scale-free) scale-stretching of time. In this manner an implementation of parallel processing turns the common Markovian, mechanistically modeled framework into a special case. According to the special case of standard mechanistic dynamics a given process may be realistically represented either by a scale-specific process at a given (unit) scale or a trivial linear superposition of such processes (e.g., a composite random walk toggling between different magnitudes of the diffusion parameter for each “layer”). On the other hand, complexity arises when such a description that is based on one-dimensional time is not sufficient to reproduce the system realistically.

Observe that in a PP-system several events (change of system state) may be executed in parallel! In the illustration above, see for example the situation for t=5 where events at three time scales by chance are initiated simultaneously but at different time scales as defined by ε. Such a kind of dynamics represents a paradox within the constraint of a Markovian (mechanistic) system.

An earlier illustration of the PP framework was given here. For other examples, search this blog for “parallel processing” or read my book.

Various aspects of scaling in animal space use; from power law scaling of displacement lengths (Lévy like distribution), fractal dispersion of GPS fixes (the home range ghost model) and scale free distribution of populations (Taylor’s power law and the Zoomer model) may be natural outcomes of systems that obey the PP conjecture.


*) The base, b, of the logarithm does not matter. Any positive integer introduces scaling of the ε-axis.

**) in a standard, mechanistic process an event describes a change of system state at a given point in space at a given point it time. No “time stretching” takes place.


Scrutinizing the MRW Model: Random Returns

The devil is in the details. The Multi-scaled Random Walk (MRW) model is merging scale-free habitat exploration with occasional returns to previously visited locations. Both components are by default expressed by stochastic rules. How can random returns be justified as realistic in a heterogeneous environment – where some localities are expected to experience a higher return frequency than others?

MRW regards a mixture of scale-free exploratory steps and targeted return events. The former kind of movement is modeled by a standard Lévy walk equation for random displacements with no directional bias. On the other hand, return targets are modeled as randomly chosen points in time where the animal revisits a previous location rather than performing yet another exploratory Lévy step. In other words, the animal is in this parsimonious (ground level) model formulation executing a mixture of scale-free moves and occasionally non-random self-crossing of its previous path.

Little stint, Calidris minuta, taking a nap while considering what to do next. Photo: AOG.

By default such self-crossings – strategic returns – are in the basic simulations picked randomly and at a chosen average frequency among the entire series of previous locations. Consequently, the only bias towards site preference that gradually build up is the effect from self-reinforcing space use: if a site by chance has been targeted twice in the past, it has a twice as large probability to receive yet another visit, relative to another location with only one visit. Before considering the concept of “randomness”, let’s focus on this question: how can this model design, which intrinsically reflects movement in a homogeneous environment, be considered realistic, given that the animal’s environment typically is strongly heterogeneous in both time and space?

Randomly picked return targets are easy to imagine in a homogeneous environment. However, simple statistical logic may justify such random picking also in a more realistic scenario with heterogeneous distribution of various resource patches and no-go zones. In particular, two home ranges in a homogeneous versus a heterogeneous environment may show a similar average characteristic scale of space use (CSSU) when the full set of relocations (e.g., GPS fixes) are considered, given that the average environmental conditions are similar for the two sets, respectively. In short, in this case the two data sets is expected to show similar magnitudes of the parameters in the Home range ghost equation, I(N) = cNz, where c expresses the CSSU and 1/c is the intensity of space use*.  The heterogeneity with respect to locally varying intensity of space use within a home range will surface only if one studies local or more short term conditions; for example, by splitting the data into spatial or temporal sub-sets. Influence of locally or temporally variable influence on habitat selection is then expected to be reflected in the respective subsets’ variation of 1/c and z of the Home range ghost equation at the chosen intra-home range scale of analysis.

In this simulation scenario the intra-home range CSSU varies from large (I) to small (IV), reflecting a more intense space use in the latter section. Despite this intra-home range heterogenity of space use the average CSSU for the four sub-sets I-IV equals the CSSU that is found in the pooled set of relocations; i.e., at the home range scale. Note that calculation of CSSU from the home range ghost formula makes it independent of the number of actual relocations in each section.

Hence, even if returns during the path sampling time are skewed towards some localities (on expense of other localities) due to preference based on habitat heterogeneity, this effect will be “averaged out” when considering the path as a whole. In other words, the basic model may reflect the over-all space use condition realistically, as seen from the scale of the home range. This aspect is thoroughly verified by simulations of MRW in homogeneous vs. heterogeneous environment (Gautestad, unpublished).

First, the property of random moves is simply a reflection of a sufficiently deep “hidden layer” to allow for a statistical-mechanical representation of movement and space use. Second, by zooming into subsections (space or time) of this over-all space use picture, the environmental heterogeneity may be revealed. Thus, the basic version of the MRW model for homogeneous habitat conditions my be feasible even in a heterogeneous habitat when studying the over-all conditions.

Similarly, if one finds difference in CSSU between two sets of home range data, this reflects difference at the home range scale of space use between these two sets. Finer-grained heterogeneity is hidden from the analysis (coarse-grained away) for the sake of studying inter-home range differences rather than intra-home range heterogeneity.


*) When calculating CSSU at different spatial scales, some trivial statistical-mechanical “tweaking” (rescaling) is performed to adjust for this difference.

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.


*) 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).


Slow Motion in Books on Animal Movement

Over the last years we have seen a range of interesting and important books appearing in the field of animal movement and space use. In this post I mention four of them. Unfortunately, only two of these books (disregarding my own contribution) presents any reference to animals’ capacity for spatial reorientation beyond the individual’s current perceptual field. However, all that is offered in these two titles – covering hundreds of pages with deep theory – is a couple of sentences or paragraphs. Why such a slow implementation in mainstream models with respect to this key aspect of behavioural ecology? Why such stubbornness to bridge theory to empirical knowledge by including spatial memory as an important factor that influences how animals use their habitat?

First, I was happy to discover that Turchin’s classic book “Quantitative analysis of animal movement” (Turchin 1998) has now appeared in new print, dated 2015. However, like the original version you will search in vain for any reference to animals that utilize spatial memory. Since this is not a re-write, Dr Turchin is excused. However, I had expected some update on models on movement as they have developed over the last 20 years, in particular with respect to scale-free space use (e.g., Lévy walk) and spatially explicit returns, where the latter explain an individual’s home range as an emergent property rather than some kind of mystic and pre-set constraint on area use.

Second, I have read “Stochastic foundations in movement ecology” (Mendéz et al. 2014) with great interest. However the downturn also here was the lack of any reference to animals utilizing space in a strategic manner; i.e., relating to its environment beyond the perceptual field around the present location. Several sections are included to cover advanced space use in the context of scale free movement and even continuous time random walk models (CTRW). However, again I missed descriptions of animals with spatial memory utilization. Before this field of research reach a full integration between scale-free movement and strategic, memory-influenced displacements the theory is a half-told story.

Third, I was thrilled by the title of the latest book in this field of research, “Animal Movement: Statistical Models for Telemetry Data” (Hooten et al. 2017). This book is written by researchers with a proper empirical experience with modern data sets on space use. Still, only two paragraphs over the 306 pages were dedicated to the spatial memory aspect, only referring to papers where the reader may find some research on spatial memory.

Fourth, “The Physics of Foraging” (Viswanathan et al 2011)  was very promising and I really looked forward to get a copy, since we here finally got a book that was explicitly dedicated to the interface between physics (e.g., statistical mechanics-related) and behavioural ecology. It does in fact include 12 index references to memory, but – disappointingly – all but two of them regard temporal memory (CTRW, correlated random walk and other variants) rather than spatially explicit memory as expressed by utilization of a memory map. Anyway, I appreciate that this important book does at least mention references to our own work on spatial memory!

To conclude, so far my book Animal space use is still alone on the arena what regards focusing on the integration of scale free space use and spatial map utilization. I hope it soon gets competition.

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.