Jellyfish behavior: LFF or MRW?

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

I cite from their Discussion (with my underscores):

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

Another jellyfish species. Photo by Pawel Kalisinski from Pexels

Such a space use mixture of “prospecting away” in combination with targeted returns, and where the former complies with a scale-free step distribution, 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.

I cite from one of my papers, where the observational lag (the sampling interval, tobs, of the animal’s path) is key to understanding the statistical pattern if movement is memory-influenced:

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

A jellyfish has a very rudimentary nervous system. It doesn’t have a brain or central nervous system, only a very basic set of nerves at the base of their tentacles. These nerves detect touch, temperature, salinity etc. and the animal reflexively respond to these stimuli. For example, the jellyfish can orient to olfactory cues from prey (Arai 1991). Hence, the movement is expected to include targeted returns in a very rudimentary and environmental field-dependent manner.

However, do these returns embed tactical and Markovian-like behaviour only (for example, 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 swimming “up” the gradient or targeting a previous location; hypothetically as a function of spatial memory rather than getting moment-to-moment guidance from an environmental, chemical field. In the latter case, also the similarly long-distance return events following prospecting may be considered a kind of occasional sally. For a conceptual illustration of complex movement spanning the tactics-strategy gradient (parallel processing), see this post.

A long and speculative shot, I agree, to suggest that jellyfish nervous system may express MRW behaviour. However, perhaps the cognitive capacity of animals with simple nerve systems like jellyfish are more powerful that 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 the combined capacity to explore its environment in a scale-free manner (within a given scale range) and returns to a previous location 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 brain structures. Such pre-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 variant of parallel processing may be tested statistically, by comparing simulations under this condition with true paths. In fact, I’ve already done such studies (A. O. Gautestad, unpublished).

In more general terms 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. 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.

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.

Accepting spatial Memory: Some Alternative Ecological Methods

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:

  1. Study the Home range ghost model, I(N) = cNz, to estimate 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.
  2. 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.
  3. 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 Ns 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.
  4. 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).

For more theoretical or technical details 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.

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.