**Please observe**: there is an unfortunate typo at page 2 of my book. Correct © should read 2015, not 2016. Thus, please refer to the book as Gautestad (2015), which is in compliance with ISBN.

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**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 awayfrom a preferred depth, not finding an improvement in conditions elsewhere and soreturningto the original depth.Such behaviour sits outsidethe Lévy search paradigmwhere 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.

Hayset al. (2012), p471

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 t_{ret} and the sampling interval, t_{obs} of the animal’s path; *ρ* = t_{ret}/t_{obs}, is key to understanding the statistical pattern if movement is memory-influenced:

… by analysing the data with different t

_{obs}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 t_{obs}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 t_{obs}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 t_{obs} and different t_{ret}), 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

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).

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.

**In this post I present a guideline that summarizes how a memory-based model with an increasing pile of empirical verification covering many species – the Multi-scaled Random Walk model (MRW) – may be applied in ecological research. The methods are in part based on published papers and in part based on some of the novel methods which you find scattered throughout this blog. **

In the following, let us assume for a given data set that we have verified MRW compliance (using the standard memory-less models or alternative memory-implementing models as null hypotheses) by performing the various tests that have already been proposed in my papers, blog posts and book. Typically, a standard procedure should be to verify (a) site fidelity;* i.e.*, presence of a home range, (b) scale-free space use by studying the step length distribution from high frequency sampling, and (c) the fractal dimension D≈1 of the spatial scatter of relocations in the resolution range between the dilution effect (very small pixels) and the space fill effect (very large pixels).

The obvious next step is to explore specific ecological hypotheses using the MRW as the model for space use. Here follows a quick tutorial:

- Study the Home range ghost model,
*I*(N) =*c*N, to estimate^{z}*c*, the individual’s characteristic scale of space use (CSSU). Search my blog for methods how to optimize grid resolution, and in particular also consider the recent breakthrough 11 months ago where I show how to estimate CSSU from auto-correlated data. Variations in CSSU quantifies difference in intensity of space use, which logically illuminates aspects of habitat selection. - After proper estimation of CSSU study the power exponent,
*z*, of the Home range ghost model. If your data lands on*z≈*0.5 – the default condition – you have verified that the individual has not only utilized its environment in a scale-free manner but also has put “equal weight” into relating to its habitat across the spatial scale range within its home range. On the other hand, finding 0.2<*z<*0.3 indicates that a model for an alternative movement class, the Markov-compliant MemRW, may be more suitable for your data. 0 <*z*< 0.3 indicates that the individual has concentrated its space utilization primarily towards finer resolutions, like you would expect from a Markov-compliant kind of cognitive processing. More detailed procedures should be applied to select model framework, since MemRW and MRW describes qualitatively disparate classes. - Is the individual’s space use stationary, or is the home range drifting over time? Spatial autocorrelation in your series of fixes typically has two causes; high-frequency sampling of fixes from space use relative to a slower return frequency (
*ρ*>>1; see my previous post) or high- or medium-frequency fix sampling under the condition of a drifting home range. Split the data into several subsets of magnitude N_{s}where the number of fixes (N) in each set is constant. Then study the overlap pattern of incidence*I*(N) at spatial resolution of CSSU (see method here). Low degree of overlap between successive subsets implies a non-stationary kind of home range. By comparing non-adjacent subsets in time one may even quantify the degree of non-stationarity (the speed by which the space use is drifting). These results can then be interpreted ecologically. - What about the fractal dimension of the total set of fixes, for example by applying the box counting method? By default one expects D≈1 when
*z*=0.5. Deviations from D=1 over specific spatial resolutions can be interpreted ecologically. For example, 1.5<D<2 at the coarsest resolutions may indicate that space use is constrained by some kind of borders. However, it could also appear from missing outlier fixes in the set (Gautestad and Mysterud 2012) or a simple statistical artifact (the space fill effect). On the other end, 1<<D can be hypothesized to emerge where the animal has concentrated its space use among a set of fine-scale patches rather than scattering is optimization more smoothly (in a statistical sense) over a wider range of scales. In Gautestad (2011) I simulated central place foraging, where i found 0.7<D<1. More sophisticated but logically simple methods can contribute to various system properties and statistical artifacts that contribute to deviation from D≈1, for example by varying the sample size of fixes as illustrated in the Figure to the right (copied from the link above).

The MRW theory also offers several other methods to study ecological and biological aspects of space use. For example, the data may reveal whether the temporal memory horizon has been constrained or unlimited (infinite memory, or remembering previous visits only over a limited, trailing time window). Temporally constrained memory will be shown by example in my next post. For more theoretical or technical details of the methods above please search this blog for the actual term, or find references in the subject index of my book.

REFERENCES

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. and I. Mysterud. 2012. The Dilution Effect and the Space Fill Effect: Seeking to Offset Statistical Artifacts When Analyzing Animal Space Use from Telemetry Fixes. Ecological Complexity 9:33-42.

**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 t_{ret} to a previously visited location relative to the sampling interval t_{obs}, *ρ* = t_{ret}/t_{obs}, 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 t_{ret} for a given t_{obs} implies stronger site fidelity. The variable observer effect that is expressed by t_{obs} becomes apparent within a quite wide transition range around t_{obs} ≈ t_{ret}. 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.

**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.**

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 (t

_{ret}), and the observation interval on the movement path (t_{obs}). The latter represents GPS locations in real data, and is applied to study the effect from varying ρ = t_{ret}/t_{obs}(relative strength of site fidelity for a given t_{obs}).

Gautestad and Mysterud (2013), p4

Note that an increasing t_{ret} for a given t_{obs} implies weakened site fidelity, and the functional form of the step length distribution is influenced by the *ρ* = t_{ret}/t_{obs} 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.

**Temporally constrained space use is a key property of animal movement. With respect to vertebrates three main statistical representations are particularly popular among modelers, based on disparate theoretical foundations. Which one should one use for analysis of a particular data set? As always in ecological research, one needs some simple protocol to distinguish between alternative model assumptions.**

Animals paths are neither straight lines nor a dense dot of positions from juggling back and forth at the same spot. Typically we see a complicated combination of these two extreme patterns; some quite straightforward moves occasionally abrupted by more jagged movement. In order to infer behavioural and ecological results from space use one needs to study the data in the context of a realistic theoretical framework.

Outside the realm of temporal site fidelity; *e.g.*, a drifting home range (Doncaster and Macdonald 1991), ecological textbooks typically explain the mixture of straight and convoluted movement bouts as Area restricted Search (ARS).

Area-restricted search.A foraging pattern in which a consumer responds to an intake of food by slowing down its movement and remaining longer in the vicinity of the most recently located food item. This behaviour causes consumers to remain longer in areas where the density of food items is high than in areas where it is low.

A Dictionary of Ecology. Encyclopedia.com. 29 Jul. 2018.

In terms of statistical models, this rather qualitative description of behaviour may be formulated in many ways. A popular one is to combine classic or correlated random walk with two distinct parameter values for intrinsic step length distribution (the *λ* value) in F(r) ∝ e^{-λr}. In this manner, movement varies with the jaggedness of the path (number of turns pr. period of time) rather than the movement speed. A larger *λ* implies smaller step lengths on average, which tend to increase local staying time during intervals when this movement mode is active. By fine-tuning respective *λ*_{1} and *λ*_{2} and in such a superposition of two “randomly toggling modes” this so-called composite random walk can even be made to mimic the second main approach to model complicated paths, the Lévy flight model (Benhamou 2007; but see Gautestad 2013):

Lévy flightsare, by construction, Markov processes. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a stable distribution due to the generalized central limit theorem, enabling many processes to be modeled using Lévy flights.

https://en.wikipedia.org/wiki/Lévy_flight.

Lévy flights (and walks) are typically thought of as producing “fat tailed” step length distributions. However, within an often observed parameter range of the distribution (Pr(*U*>*u* = *O*(*u*)^{-k} with 1<*k*<2) in real animals, one should not forget that half of the displacements in the distribution are in fact relatively short! In fact, the dominating step length bin is ultra-short moves, leading to a path that is conceptually (albeit not statistical mechanically) similar to the slow-down effect of more jagged moves during composite random walk.

Such “knots” of ultra-short moves of a Lévy path brings us to the third class of movement models, **Continuous time Random Walk (CTRW)**. In this case the movement may be arrested for a shorter or longer period:

The step length distribution and a waiting time distribution (“resting” between steps) describe mutually independent random variables. This independence between jump lengths and waiting time to perform the next step makes the difference between CTRW and ordinary random walk (including Brownian motion).

Page 91 in: Gautestad, A. O. 2015, Animal Space Use: Memory Effects, Scaling Complexity, and Biophysical Model Coherence Indianapolis, Dog Ear Publishing.

Again, by fine-tuning the intrinsic model formulations and parameter conditions of CTRW one may in fact produce a Lévy flight-like movement pattern (Reynolds 2010; Schulz *et al.* 2013). Towards another extreme, one may achieve a classic random walk look-alike pattern. Conceptually, CTRW thus belongs on the left wall of the Scaling cube (x=0; y and z varies in accordance to model variant and parameter settings).

CTWR (which is closely connected to the concept of fractional diffusion) is typically invoked to explain a kind of switching – “intensive versus extensive searching” (scale-free sub-diffusive versus super-diffusive search) – that is now being found in real animal data like foraging of Balearic shearwaters *Puffinus mauretanicus* and Cory’s shearwaters *Calonectris diomedea* along the coast of Spain (Bartumeus et al. 2010).

In particular, they found that the birds in the presence of fisheries were more “restless” when taking advantage of fishery discards than in absence of trawlers, implying a higher probability of leaving a localized area pr. unit time during trawling activity. Specifically, during period of fishery discard utilization tended to show a smaller temporal scaling exponent for staying time (the temporal aspect of CTRW; larger *β* in their site fidelity function S(t) ∝ t^{–β}, implying fewer events with a particularly prolonged staying time). In the spatial aspect of flight length distributions, when fisheries discard was present the birds tended to show good compliance with a negative exponential function*.

On the other hand, in the absence of trawlers a compliance with Lévy walk (truncated power law) was found. Space and time brought together, they found that the birds tended towards sub-diffusive foraging in the presence of fisheries discard (despite – somewhat counter-intuitively – a smaller local staying time within a given patch of a given spatial resolution), and super-diffusive foraging under natural conditions.

**Despite the mathematical and numerical attractiveness of composite random walk, Lévy walk and CTRW and their well-explored statistical properties, they unfortunately all lack what may be a crucial component of foraging behaviour: spatial memory; i.e., the condition x>0 of the scaling cube. **

Without spatial memory, self-crossing of an individual’s path happens by chance only, not intentionally by returns to a previous location (site fidelity, whether we consider short term or long term time scales).

- May a model that implements spatial memory offer an alternative interpretation of the results presented by Bartumeus
*et al.*2010? - May this alternative hypothesis even offer a logical explanation for the apparent paradox that the birds were more restless locally when the movement simultaneously was more spatially constrained in overall terms?

The memory aspect of temporally constrained space use will be explored in Part II.

NOTE

*) Somewhat confusingly relative to common practice Bartumeus *et al.* (2010) use the exponential formula variant F(r) ∝ e^{-r/λ}, which makes average step length proportional with *λ *rather than 1/*λ.*

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.

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

Doncaster, C. P., and D. W. Macdonald. 1991. Drifting territoriality in the red fox *Vulpes vulpes*. Journal of Animal Ecology 60:423-439.

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.

Reynolds, A. 2010. Bridging the gulf between correlated random walks and Lévy walks: autocorrelation as a source of Lévy walk movement patterns. J. R. Soc. Interface 7: 1753–1758.

Schulz, J. H. P., A. V. Chechkin, and R. Metzler. 2013. Correlated continuous time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics. J. Phys. A: Math. Theor. 46:1-22.

**Examples of positive feedback loops in population dynamics abound. Even if the majority of models are focusing on negative feedback, like the logistic growth function, non-equilibrium “boom and bust” kind of model designs have also been developed. In this post I elaborate on the particular kind of positive feedback loop that emerges from cross-scale dual-direction flow of individuals that is based on the parallel processing conjecture.**

The image to the right illustrates – in simplistic terms – a spatially extended population model of standard kind (e.g., a coupled map lattice design) where each virtually demarcated local population *j* at spatial resolution *i* and at a given point in time *t* contains N* _{ij}* individuals. No borders for local migration are assumed; i.e., the environment is open both internally and externally towards neighbouring sites.Typically, these individuals are set to be subject to a locally

As previously explained in other posts, this kind of model framework depends on a premise of Markov-compliant processes at the individual level (mechanistic system), and thus also at the population level (local or global compliance with the mean field principle). In this framework intrinsic dynamics may be density dependent or not, but from the perspective of a given N* _{ij}*, extrinsic influence – like immigration of individuals – is always stochastic and thus density independent with respect to N

**To implement cross-location and ***dual-direction*** deterministic dynamics, multi-scaled behaviour and spatial memory needs to be introduced. My **parallel processing** conjecture; which spins off various ***testable*** hypotheses, creates turmoil in this standard system design for population dynamics because it explicitly introduces such system complexity. For example, positive feedback loops may emerge. Positive feedback as described below may effectively also counteracting the paradoxical Allée effect, which all “standard” population models are confronted with at the border zone of a population in an open environment**. **

The dynamic driver of the complexity is the introduction of spatial memory in combination with a scale-free kind of dynamics along both the spatial and the temporal dimensions. In statistical-mechanical terms, parallel processing is incompatible with a mechanistic system. Thus, a kind of extended statistical mechanics is needed. I refer to the post where I describe the scale-extended description of a metapopulation system.

For the most extensive individual-level test of the parallel processing conjecture until now (indirectly also verifying positive feedback of space use), see our paper on statistical analysis of space use by red deer *Cervus elaphus *(Gautestad *et al.* 2013; Gautestad and Mysterud 2013). In my blog I have also provided several anecdotal examples of third party research potentially supporting the parallel processing conjecture. For the sake of system coherence, i*f parallel processing is verified for individual space use of a given species and under given ecological conditions, this behaviour should also be reflected in the complementary population dynamical modelling of the given species and conditions*.

**Extending the standard population model**. As explained in a range of blog posts, my Zoomer model represents a population level system design that is coherent with the individual-level space use process (in parsimonious terms), as formulated by the Multi-scaled random walk model. In my previous post I described the latter in the context of positive feedback from individual-level site fidelity. Below I illustrate positive feedback also at the population level, where site fidelity get boosted by conspecific attraction. In other words, conspecifics become part of the individuals’ resource mapping at coarser scales, as it is allowed for by spatial memory. Consequently, a potential for dual-direction deterministic flow of individuals is introduced (see above). Conspecific attraction is assumed to be gradually developed by individual experience of conspecifics’ whereabouts during exploratory moves.

First, consider the zooming process, whereby a given rate, *z*, of individuals (for example, *z*=5% on average at a chosen time resolution Δt) at a “unit” reference scale (*k*=*i*) are redistributing themselves over a scale range beyond this unit scale***. During a given Δt consider that 100 individuals become zoomers from the specific location marked by the white circle. In parallel with the zooming out-process the model describes a zooming in-process with a similar strength. *The latter redistributes the zoomers in accordance to scale-free immigration of individuals under conspecific attraction*.Thus, number of individuals (N) at this location *j* at scale *i*, marked as N* _{ij}*, will at the next time t+1 either embed N-100 individuals if they all leave location

In the ideal model variant of zooming we are thus assuming a scale-free redistribution of individuals during zooming, with zooming to a neighbourhood at scale *k*_{i+x} takes place with probability 1/*k*_{i+x} (Gautestad and Mysterud 2005). Under this condition, zoomers to successively coarser scales become “diluted” over proportionally larger neighbourhood area, the maximum number of immigrants in this example is 100 + N’, where N’ is the average number of zoomers pr. location at unit scale *k*=*i* within the coarsest defined system scale *k*=*i*(max) for zooming surrounding location *j* at scale *i*.

**As a consequence of this kind of scale-free emigration of zoomers, the population system demonstrates zooming with equal weight of individual redistribution from scale to scale over the defined scale range (Lévy-like in this respect, with scaling exponent β≈2; see Gautestad and Mysterud 2005). By studying the distribution of step lengths, this “equal weight” hypothesis may be tested, when combinded with othe rstatistical fingerprints (in particular, verifying memory-dependent site fidelity; see Gautestad and Mysterud 2013).**

Putting this parsimonious Zoomer model with its system variables and parameters into a specific ecological context implies a huge and basically unexplored potential for ecological inference under condition of scale-free space use in combination with site fidelity.

**Positive feedback in the Zoomer model**. As shown in my series of simulations of the Zoomer model a few posts ago, a positive feedback loop emerges from locations with relatively high abundance of individuals having a relatively larger chance of received a net influx of zoomers during the next increment, and vice versa for locations with low abundance. The positive feedback emerges from the conspecific attraction process, linking the dynamics at different scales together in a parallel processing manner.

This positive feedback loop from conspecific attraction also counteracts extinction from a potential Allée effect (see this post and this post), which have traditionally been understood and formulated from the standard population paradigm. The Zoomer model represents an alternative description of a process that effectively counteracts this effect.

NOTES

*) The migration rates connects the local population to surrounding populations. Immigration is – by necessity from the standard model design – density *independent* with respect to the dynamics in N* _{ij}*.

**) Since the process is assumed to obey a Markovian and the mean field principles (standard, mechanistic process), the arena and population system must either be assumed to be infinitely large or the total set of local populations has to be assumed to be demarcated by some kind of physical border. Otherwise, net emigration and increased death rate in the border zone will tend to drive N towards zero in open environments (extinction from standard diffusion in combination with local N drifting below critical density where Allée kicks in). Individuals will “leak” from an open border zone to the surroundings where N is lower.

***) The unit temporal scale for a population system should be considered coarser than the unit scale at the individual level, since the actual scale range under scrutiny typically is larger for population systems. In particular, to find the temporal scale where for example 5% of the local population can be expected to be moving past the inter-cell borders of a given unit spatial grid resolution *k _{i}=*1, one should be expected to find Δt substantially larger than Δt at the individual level.

Consider that the difference in Δt is a function of the difference of the area of short-range versus long range displacements under the step length curve for individual displacements, where the ∼5% long-step tail of this area represents the relative unit time in comparison to the rest of the distribution (thereby defined as intra-cell moves). Since this area is a fraction of the area for the remaining 95% of the displacements, the difference in Δt should scale accordingly.

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