Parallel Processing: From Metaphysics Towards Biophysics, Part II

When I returned to the University of Oslo in 1990 to explore alternative pathways towards complex dispersion of populations it was natural to start out by orbiting around the Department of biology’s division that focused on population dynamical modelling. However, as become increasingly obvious was a tension that grew up between my choice of off-piste approaches, the introduction of rather unorthodox concepts and on the other side meeting a culture that focused on the classical mathematical and statistical toolbox. I simply could not find satisfactory local support for working on scale-free dispersal processes under these terms, despite what I observed as thoroughly and broadly documented instances of such paradigm-breaking behaviour in the hundreds of papers surrounding confirmation of – for example – Taylor’s power law and fractal-patterned population dispersion. The theoretical culture was stubbornly shying away from exploring a series of paradoxes that in my view were crying for an entirely novel approach for a solution. Thus, luckily in 1992 I knocked on Ivar Mysterud’s door further down the same corridor.

Ivar (now Professor emeritus) was mentor of my Master’s degree on a population of tawny owl Strix aluco, Now I sat in his office again, advocating my ideas for a PhD on animal space use in broader terms and how I felt it was necessary to drill into behavioural ecology at the individual level to seek solutions to the paradoxical dispersion processes in real populations. Characteristically for Ivar, he was immediately positive to such an endeavor that was aiming at some rather unfamiliar theoretical terrain. The more off-piste, the better! He already had concluded from his own field experience that completely new approaches were probably needed. But where to find them? Not in standard text books. Neither in contemporary papers that were influencing animal movement and habitat selection those days.

Ivar Mysterud (left) and myself, displaying a novel equation that links home range utilization to to its fractal properties. From a 2010 article in the Norwegian resarch magazine Apollon. Photo: Francesco Saggio.

With Ivar’s very solid background as field ecologist in wildlife behaviour and management, he had already struggled in practical terms with analyzing animal space use complexity, in particular related to the aspects of home range behaviour. He offered me both an office and a four year scholarship to work freely (!) on whatever successively emerged as the most promising path forward. Best of all, Ivar’s and his students’ telemetry data on habitat use by free-ranging sheep Ovis aries were made available. He also ensured contact with his good friend Mike Pelton, professor at the University of Tennessee, where we were allowed to work on his lab’s extensive material on black bear Ursus americanus movement.

Within a year following the toggle from population kinetics to individual space use the first “metaphysical” property of the sheep data emerged from our analyses under the new terms (Gautestad and Mysterud 1993), including frequent and inspirational rounds on the office’s too small blackboard.

A catchy new concept emerged, “the Home range ghost” (Gautestad and Mysterud 1995).

For the initial 1993 paper the minimum convex polygon method* was used to study home range size as a function of sample size of telemetry fixes, n, including all ‘outliers’ available (more proper methods in follow-up works confirmed the same “ghost” aspects). The data were found to be satisfactorily non-auto-correlated at the given sampling intervals of several hours between successive relocations (fixes).

Then, what was the “metaphysical” Home range ghost property? For a start, consider demarcating space use from a total sample of N fixes using two protocols; (a) samples of n fixes (nmin < n <= N) that are drawn uniformly from the pool N, and (b) samples of n fixes that represent continuous series (time-close segments) from the total path of N fixes.

Ordinary theory and common sense predicted that the A(n) curves for home range area A from both methods should be quite overlapping, given that the fixes were temporally non-auto-correlated. In other words, n fixes from uniform sampling were expected to show similar A(n) as continuous sampling series. Otherwise, the latter should show smaller A for a given n. Further – again from the home range paradigm – the A(n) should be expected to flatten out towards an area asymptote for larger A.

The sheep data told us another story.

First, the asymptote of A(n) was not found (Figure 5a above). In fact, with log-log scaled axes the function satisfied a straight line with slope z about 0.5; i.e., a scale-free power law log(A) = log(c) + z*log(n). Area grew proportionally with square root of n rather than towards an asymptote, despite large N. Second, despite the non-asymptotic function, the two sampling methods “uniform over N” and “sections within N” overlapped!

A similarly strange power law pattern with z=0.5 was also found in Mike’s and his students material on black bear (Gautestad et al. 1998), which I recently also re-confirmed using latest methods of the Home range ghost theory (see this blog post). In a meta-analysis of A(N) data from many species, taxa and ecological conditions, the overall result also confirmed the same power law with z≈0.5 (Gautestad and Mysterud 1994).

In short, we were able to show scale-free space use by applying the relatively new concepts of statistical fractals (Mandelbrot 1983, Feder 1988). But what was most explosive in our results were a property of space use that resonated well with the strange, apparently “metaphysical” property of sycamore aphids with respect to “time-independent” and “scale range dependent” pattern, like seen in 1/f noise spectra and in a subsection of Taylor’s power law (see Part I of this 2-part post). This time from analysis of individual space use.

NOTE

*) The MCP method, which was quite dominant in home range analyses at the time, was replaced by more robust statistical procedures in the follow-up work.

REFERENCES

Gautestad, A. O. and I. Mysterud. 1993. Physical and biological mechanisms in animal movement processes. J. Appl. Ecol. 30:523-535.

Gautestad, A. O. and I. Mysterud. 1994. Fractal analysis of population ranges: methodological problems and challenges. Oikos 69:154-157.

Gautestad, A. O. and I. Mysterud. 1995. The home range ghost. Oikos 74:195-204

Gautestad, A. O., I. Mysterud, and M. R. Pelton. 1998. Complex movement and scale-free habitat use: testing the multi-scaled home range model on black bear telemetry data. Ursus 10:219-234.

Non-Mechanistic Dynamics: a Simple Illustration

In my previous post I summarized my critique of mechanistic modelling when applied on animal movement. Simply stated, the Markovian design on which mechanistic models depend is in my view incompatible with a realistic representation of memory-influenced and scale-free space use. Below I illustrate the alternative approach, non-mechanistic dynamics, by a simple Figure. As conceptually described by the Scaling cube, an extra system dimension based on relative scale (“hierarchical scaling” of the dynamics), resolves the apparent paradox of non-mechanistic dynamics.

I cite from my first post on the Scaling cube (December 25, 2015):

The scaling cube brings these directions of research together under a coherent biophysics framework. It also forces upon us a need to differentiate between mechanistic dynamics (the M-floor) and non-mechanistic dynamics (the PP-ceiling).

As a supplement to my book presentation I have published a series of posts on this theme, where its unfamiliar nature has been revealed in a piece-wise manner. In particular, you got a “rolling dice” perception of how non-mechanistic dynamics pays out as a special universality class:

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 “now” at 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 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.
From “The Inner Working of Parallel Processing” (blog post, February 8, 2019).

 

How to visualize non-mechanistic dynamics? Consider the output from a simple representation of a simulated animal path during a given time interval. The movement rules are constant and deterministic. Then consider repeating the simulation four times (marked by numbers in the image to the right). In compliance with classic design principles all repeat runs of the dynamics should be expected to show identical path progression over the habitat during the given interval. However, under non-mechanistic dynamics even a deterministic progression is expected to show occasional “surprise” moves (red line in run no 4).

Simply stated, what appears to be a surprising and random move from one time resolution (fine-scale “now”) may appear deterministic and quite rational from a coarser time resolution, in compliance with the rules for the simulation at these scales (coarser scale “now”).

Such unfamiliar and unconventional kind of model behaviour is expected to appear due to the fact that the path from the simulation is logged at a specific temporal scale (unit size time increments) while the dynamics are executed over a scale range, including coarser scales that the logged scale. The dynamics from parallel processing at coarser temporal scales will by necessity appear as stochastic surprise moves from the perspective of finer time scales.

For example, with reference to the “throwing dice” analogy (where number of eyes determine a specific state at the respective scale levels), most moves are executed at unit scale while some are executed at coarser scales. The latter may be formulated as deterministic moves at respective temporal resolutions, but will appear randomly; i.e., as a surprise, when observed from finer scales. The dice represent discrete-scale description of levels, which of course should be considered a continuous range in a real system.

The crucial difference between mechanistic and non-mechanistic kind of stochastic moves is thus buried in the process itself; is the move a result of a Markovian rule that involves some influence of randomness in the decision making about what to do next, or is the move the result of a more strategic decision? To clarify this key issue one has to apply specific statistical methods that have the potential to test for parallel processing. The acid test of parallel processing is performed by comparing some basic statistics from simulations of non-mechanistic dynamics with the similar statistics from real data. My papers, book and blog provide many results of such tests, spanning many aspects of the space use behaviour.

On the Paradox of Mechanistic Movement Models

Mechanistic design is still dominating animal space use modelling. As the readers of my papers, book and blog have understood I’m very critical to this framework. In particular, because both mechanics and statistical mechanics – due to their dependence on Markovian dynamics – under their present formulations seem to be unable to implement memory-influenced movement in a realistic manner. Thus, paradoxes abound. Unfortunately most theoreticians in movement ecology either don’t care or don’t know how to approach this issue. In this post I seek to pinpoint the most basic challenge, and how it may be potentially resolved by exploring a qualitatively new direction of modelling.

Consider the standard, simplified illustration of animal foraging, which takes up much of an individual’s focus during a day. At each time increment the behaviour adheres to rules under the mathematical framework of a low order Markovian process. A similar diagram could have been shown for other behavioural modes; like looking for a mate, seeking a shelter for resting, and so on. In short, mechanistic behaviour describes rules, which may be executed deterministically, stochastically or as a mixture. The key point is that the process in model terms is described at a specific temporal resolution. In other words, each “Start” in the illustration to the right regards execution of behaviour during the current time increment; i.e., at the current moment at the given time scale.

A Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. In probability theory and related fields, a Markov process, named after the Russian mathematician Andrey Markov, is a stochastic process that satisfies the Markov property (sometimes characterized as “memorylessness”). Roughly speaking, a process satisfies the Markov property if one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process’s full history, hence independently from such history, that is, conditional on the present state of the system, its future and past states are independent.
https://en.wikipedia.org/wiki/Markov_chain

Animals including at least mammals, birds, reptiles, amphibians and fish (apparently also many species of invertebrates), have a cognitive capacity both to orient themselves in space and to relate to a larger or lesser degree to past experiences from visited space. For example, a home range is an emergent property of this capacity, where the animal under specific biological and ecological circumstances prefers to revisit some of its previously visited locations more frequently than by chance.

There have been many proposals for modelling spatial memory, based on the standard principles of the mechanistic (Markovian) framework. For example, the animal could cognitively store its experience from successive locations along its path with respect to local food attributes and the location of these experiences (memory map utilization). Along a trailing time window; i.e., assuming a high order rather than low order Markovian design, older experiences beyond a given memory capacity is lost by computational necessity.

However, as I’ve documented over a large range of simulation examples where theory and empirical results are compared, the space use pattern as seen in large sets of individual relocations (e.g., in telemetry/GPS-based studies) do not seem to comply with the basic statistical properties of mechanistic/Markov behaviour. Animals seem to have a capacity for long term and spatially explicit memory utilization* well beyond the time scale of any high order Markov model. Simple statistical analyses from spin-offs under the alternative statistical-mechanical model designs reveal that these animals also relate to their environment in a scale-free manner, rather than the intrinsically scale-specific constraint of processing  that by necessity follows from the standard Markovian designs.

The alternative approach to modelling animal space use – MRW and the Zoomer model – is advocated my papers, my book and on this blog. These contributions also provide several simple methods to test model compliance – Markovian vs. non-Markovian processing – using your own data.

It’s of course up to each ecologist to ignore these warning signs for the realism of standard dynamic modelling of animal movement and space use, but be aware that an increasing number of physicists are now drawn towards this field of research. In an upcoming post I’ll summarize some of these recent developments, which in large part seem to have been initially inspired by my MRW simulations that began appearing in papers almost 15 years ago**.

NOTES

*) Biological and ecological conditions may constrain the utilization to a more narrow range, for example when or where the environment is relatively unpredictable. In that case the value of old information i rapidly diminishing as a function of time.

**) In fact, the first simulations of MRW were published already in my PhD thesis back in 1998.

 

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 “now” at 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 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 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.

NOTE

*) 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.

 

Temporally Constrained Space Use, Part III: Critique of Common Models

There is no doubt among field ecologists that animals from a broad range of taxa and over wide range of ecological conditions utilize their environment in a spatial memory-influenced manner. Spatial map utilization have now been verified also well beyond vertebrates, like dragonflies and some solitary wasps. To me at least it is thus a mystery why theoretical models that are void of influence from a memory map; for example ARS, Lévy walk and CTRW (see Part I, II), are still dominating ecological research with mostly no critical questions asked about their feasibility.

It is a fact that the memory-less mainstream models all have a premise that the data should not be influenced by map-dependent site fidelity. In other words, applying ARS, Lévy walk and CTRW models as stochastic representation of space use also implies accepting that the animal’s path is self-crossing by chance only, and not influenced by targeted returns. Such returns can be expected to seriously disrupt results on – for example – habitat selection, since self-reinforcing patch utilization (positive feedback) obviously becomes a serious issue for methods that are based on memory-less space use where revisits are statistically independent events.

Despite performing hypothesis tests on data that obviously contradicts this hidden assumption about lack of spatial memory influence, for example movement in a home range context (where the home range is an emergent property from such returns), memory-less models are applied by cultural instinct or a misconception that alternatives do not exist. “Everybody else is using these standard models, so why not me?”

This attitude obviously hinders space use-related ecological research on its path towards becoming hard science at the level we are used to find in physics, chemistry and geology; i.e., models with strong predictive power. The laid-back excuse that animal ecology is not only more complicated but also basically more complex does not hold anymore. Biophysical research, for example based on inspiration from – or developed in compliance with – my parsimonious MRW model (Song et al. 2010; Boyer et al. 2012; Boyer and Solis-Salas 2014; Mercado-Vásquez and Boyer 2018), show how even complex space use systems may now be treated analytically with success.

So far, there still exists only one book (Gautestad 2015) that is dedicated to criticizing the sloppy culture of model selection in ecological research. The statistical errors that follow from ignoring the frequently violated assumption about memory-less space use are percolating both my book and my blog*.

MRW implements a combination of scale-free space use with memory-dependent, occasional returns to previous sites in accordance to the parallel processing conjecture. The average return interval tret to a previously visited location relative to the sampling interval tobsρ = tret/tobs, will lead to different analytical results a a function of ρ.

This important ratio defines how the observed distribution of step lengths is a function of  memory-influenced movement that complies with the MRW formulation: a mixture of scale-free exploratory steps and occasional returns to a previous location. I cite from Part II:

If the animal in question is utilizing spatial memory a lot of confusion, paradoxes and controversy may thus appear if the same data are analyzed on the basis of erroneously applying memory-less models within different regimes of ρ!

For example, an decreasing tret for a given tobs implies stronger site fidelity. The variable observer effect that is expressed by tobs becomes apparent within a quite wide transition range around tobs ≈ tret. For example, a Brownian motion-like form of the step length distribution may erroneously be found if ρ << 1, and a power law form can be expected when ρ >> 1, with truncated power law to be observed in-between. However, power law compliance may arise both in scale-free but spatially memory-less behaviour (Lévy walk) and MRW when ρ >> 1. Recall that MRW implies a combination of spatially memory-influenced and Lévy walk-like kind of movement in statistical terms.

The step length distributions to the right (Gautestad and I. Mysterud 2005)  illustrates from MRW-simulated data the effect on changing the ratio ρ >> 1 towards ρ < 1 apparently makes the step length distribution shape-shifting from a power law (apparently Lévy) to a negative exponential (apparently Brownian). This paradoxical pattern appears simply from changing sampling frequency of a given series of successive relocations.  As observation frequency becomes larger than the return frequency the paradox appears from comparing the expectation from erroneous of model based on the memory-less space use assumption; i.e., Brownian motion vs. Lévy walk. 

The Figure to the right (Gautestad and A. Mysterud 2013) illustrates the same transition more graphically. The hump (blue colour)  that is observed for for ρ = 10 towards the extreme tail of the distribution, leading to a hump-like “hockey stick” pattern, becomes almost invisible at ρ = 100 Appendix 1 in Gautestad and A. Mysterud, 2013; see also Gautestad 2012). This gradual appearance/disappearance of the hockey stick as a function of ρ >> 1 illustrates the pseudo-LW aspect of MRW. By the way, such a “hump” on the tail part of a power law distribution has in fact been found and commented in several analyses of empirical data. Citing from Gautestad and A. Mysterud (2013):

It is interesting that one of the main issues raised in this  respect regards the “problematic” occasional over-representation of very long step lengths even relative to an ideal Lévy walk distribution, invoking the term “Lévy walk-like” search (Sims and Humphries 2012; Sims et al. 2012). This “hump” in the long tail part of the distribution has been hypothesized to emerge from some kind of environmental forcing (Sims and Humphries 2012). However, here we have shown (Figure 3) that a similar hump – called a hockey stick – is in fact expected by default if MRW-compliant data are analysed within a specific range of the ratio between return events and observation interval.
Gautestad and Mysterud 2013, p14.

The take-home message from these two examples is stressing the importance of testing for spatial memory before choosing which statistical model(s) to apply for a specific analysis.

NOTE

*) In my research I also criticize memory-implementing models where spatial utilization beyond the individual’s current perceptual field builds on a mechanistic (Markov-compliant) kind of information processing. See, for example, this post. Consequently, in the Scaling cube, these Markov models are located in the lower right corner (MemRW), in contrast to the “parallel processing”-based MRW, which you find in the upper right corner. In Gautestad et al. (2013) we tested these alternative model classes on red deer Cervus elaphus, and found strong support for the MRW framework. The red deer moved both in compliance with a scale-free space utilization, in parallel with site fidelity from targeted returns in a manner which supported parallel processing. Additional research has also given support to to MRW lately; for example see Merkle et al. (2014), who tested a set of contemporary hypotheses on memory-influenced movement in free-ranging bison Bison bison and found support for a central premise of MRW in the summer ranges of this species.

REFERENCES

Boyer, D., M. C. Crofoot, and P. D. Walsh. 2012. Non-random walks in monkeys and humans. Journal of the Royal Society Interface 9:842-847.

Boyer, D. and C. Solis-Salas. 2014. Random walks with preferential relocations to places visited in the past and their application to biology. arXiv 1403.6069v1:1-5.

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.

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

Mercado-Vásquez, G. and D. Boyer. 2018. Lotka-Volterra systems with stochastic resetting. arXiv:cond-mat.stat-mech:1809.03975v03971.

Merkle, J. A., D. Fortin, and J. M. Morales. 2014. A memory-based foraging tactic reveals an adaptive mechanism for restricted space use. Ecology Letters Doi: 10.1111/ele.12294.

Sims, D. W. and N. E. Humphries. 2012. Lévy flight search patterns of marine predators not questioned: a reply to Edwards et al. ArXiv 1210.2288: [q-bio.PE].

Sims D. W, N. E. Humphries, R. W. Bradford and B. D. Bruce. 2012. Lévy flight and Brownian search patterns of a free-ranging predator reflect different prey field characteristics. Journal of Animal Ecology 81:432-442.

Song, C., T. Koren, P. Wang, and A.-L. Barabási. 2010. Modelling the scaling properties of human mobility. Nature Physics 6:818-823.

Temporally Constrained Space Use, Part II: Approaching the Memory Challenge

In Part I three models for temporally constrained space use were summarized. Here in Part II I put them more explicitly into the context of ecology with focus on some key assumptions for the respective models. Area restricted search (ARS), Lévy walk (LW) and Continuous time random walk (CTRW) are statistical representations of disparate classes of temporally constrained space use without explicit consideration of spatial memory effects. Hence, below I reflect on a fourth model, Multi-scaled Random Walk (MRW), where site fidelity gets a different definition relative to its spatially memory-less counterparts.

A cattle egret Bubulcus ibis is foraging within a wide perimeter surrounding its breeding site. Spatial memory is utilized not only to be able to return to the nest but also to revisit favored foraging locations during a bout, based on a memory map of past experience. Photo AOG.

First, ARS is typically formulated as a composite random walk-like behaviour in statistical terms, which could be suitable for situations where a Markovian compliant (“mechanistic”) behaviour is either verified or can be reasonably assumed (memory-less  and scale-specific movement in both time and space). In this scenario the diffusion exponent can be estimated for movement bouts in different habitats and time intervals, and the result can be interpreted behavioural-ecologically. For example, the diffusion rate can be expected to be smaller i optimal patches than elsewhere. In other words, the local staying time increases due to a more jagged path.

Second, Lévy walk is a special kind of random walk. Most steps are relatively short but others may be extremely long. Sequences of short steps in-between the long ones make the overall space use appear locally constrained during these periods*). Lévy walk is characterized by a spatially memory-less statistical representation of scale-free (“hierarchical”) movement within a given spatial scale range. Beyond this range the distribution of step lengths will show increased compliance with a non-scaling, truncated Lévy walk; i.e., a composite model with exponential tail rather than a power law for the extreme part of the step length distribution. By analyzing the step length distribution within the scale-free (power law) regime using different sampling intervals one should be able to verify model compliance from stationary power exponent. A Lévy walk is statistically self-similar in space, and thus the power exponent is expected to be relatively unaffected by the sampling scheme; see Reynolds (2008). Calculating the difference in the median step length for a given sampling interval when studying subsets of the movement data under different environmental conditions brings the model into the realm of ecology [see a practical method in; for example, Gautestad (2012)].

Third, Continuous Time Random Walk (CTRW) is suitable where the animal is found to occasionally stop moving. The temporal distribution of the duration of such resting episodes can then be fitted to statistical models; for example, a power law, a negative exponential, or a mixture as in the distribution for truncated power law. The spatial distribution of step lengths is in CTRW fitted independently of the temporal distribution. Bartumeus et al. (2010) applied the CTRW framework to study “intensive versus extensive searching” (scale-free sub-diffusive versus super-diffusive search) in foraging of Balearic shearwaters Puffinus mauretanicus and Cory’s shearwaters Calonectris diomedea along the coast of Spain (Bartumeus et al. 2010). The authors  interpreted the results ecologically with weight on difference between presence and absence of local trawling activity. See Part I, where I gave a brief summary.

However, is CTRW a proper framework for these seabirds? At the end of each foraging bout they obviously utilized spatial memory to successfully return to their breeding location. CTRW assumes consistently random crossing of the movement path due to the model’s lack of spatial memory description. To me it seems illogical to assume that these birds should toggle between memory-dependent and goal-oriented returns at the end (and possibly at the start) of each trip and memory-less Brownian motion (ARS-like?) during foraging when moving in the proximity of trawlers. The same argument about conditional memory switch-off may be raised for scale-free (Lévy-like) search in the absence of trawlers.

In the context of memory-less statistical modelling of movement (the three models above), site fidelity is defined by the strength of “slow motion”, and how the distribution of local staying times is expected to vary with ecological conditions. Compare this with the alternative model Multi-scaled Random Walk, where site fidelity is defined as the strength (frequency) of targeted returns to a previous location on a path. This return frequency may be interpreted as a function of ecological conditions. Hence, MRW explicitly invokes both spatial memory and its relative strength:

Three time scales are defined: the implicit interval between successive displacements in simulations (t), the average return interval to a previous location (tret), and the observation interval on the movement path (tobs). The latter represents GPS locations in real data, and is applied to study the effect from varying ρ = tret/tobs (relative strength of site fidelity for a given tobs).
Gautestad and Mysterud (2013), p4

Note that an increasing tret for a given tobs implies weakened site fidelity, and the functional form of the step length distribution is influenced by the ρ = tret/tobs ratio. For example, a Brownian motion-like form may be found if ρ << 1, and a power law form can be expected when ρ >> 1, with truncated power law (Lévy-like) to be observed in-between. See Figure 3 in Gautestad and I. Mysterud (2005) and Figure 3 in Gautestad and A. Mysterud (2013).

If the animal in question is utilizing spatial memory a lot of confusion, paradoxes and controversy may thus appear if the same data are analyzed on the basis of erroneously applying memory-less models within different regimes of ρ!

The MRW model may thus offer interesting aspects with a potential for alternative interpretation of the results of space use analyses when put into the context of – for example – foraging shearwaters. Thanks to the three times scales for MRW as above – where the third variable, t, represents the unit (t≡1) spatiotemporal scales for exploratory moves – it should be possible to test for example Lévy walk or CTRW against MRW using real movement data. 

More on this in Part III.

NOTE

*) While temporally constrained space use in ARS regards difference in environmental forcing, the occurrence of short-step intervals of random occurrence in a Lévy walk is by default due to intrinsic behaviour.

REFERENCES

Bartumeus, F., L. Giuggioli, M. Louzao, V. Bretagnolle, D. Oro, and S. A. Levin. 2010. Fishery discards impact on seabird movement patterns at regional scales. Current Biology 20:215-222.

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

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

Reynolds, A. 2008. How many animals really do the Lévy walk? Comment. Ecology 89:2347-2351.