Scale-free distribution of displacement lengths is often found in animal data, both vertebrates and invertebrates. In marine species this pattern has often been interpreted in the context of the Lévy flight foraging hypothesis (LFF), where optimal search is predicting a scale-free power law compliant movement when prey patches are scarce and unpredictably distributed and a more classic and scale-specific Brownian motion-like motion when such patches are encountered (Viswanathan et al. 1999). In a study on the jellyfish Rhizostoma octopus such an apparent toggling between two foraging modes were found, but critical questions were also raised by the authors (Hays et al. 2012). Here I come the authors “to the rescue” by suggesting that an alternative model – the Multi-scaled Random Walk (MRW) – could be included when testing statistical classes of foraging behaviour.
I cite from their Discussion (with my underscores):
In some periods (when integrated vertical movement was low), vertical excursions were followed by a vertical return to the depth occupied prior to the excursion. This pattern of ‘bounce’ movements has also been seen in some fish [ref.] and presumably represents an animal prospecting away from a preferred depth, not finding an improvement in conditions elsewhere and so returning to the original depth. Such behaviour sits outside the Lévy search paradigm where it is assumed that a prey patch is not purposefully revisited once deserted. Again this finding of ‘vertical return’ behaviour, points to jellyfish movements, at least on occasions, being fine-tuned to prey resources.
Hays et al. (2012), p471
Another jellyfish species. Photo by Pawel Kalisinski from Pexels
Such a space use mixture of “prospecting away” in combination with targeted returns, and where the former complies with a scale-free step distribution (as now shown in jellyfish), is in fact MRW in a nutshell. As repeatedly underscored in previous posts the LFF hypothesis rests on a premise that individuals do not have a cognitive capacity to return non-randomly to a previous location, while MRW includes this capacity (Gautestad 2012; Gautestad and Mysterud 2013).
When search behaviour is studied using a spatially memory-less model framework that contrasts behavioural toggling between Lévy and Brownian motion, the standard statistical method (MLE) typically explores the continuum from a pure power law to a pure exponential, with a so-called truncated Lévy flight in-between. In addition to Hays et al. (2012), also Ugland et al. (2014) documented this transition, with Lévy pattern during night time swimming of another large jellyfish, Periphylla periphylla.
I cite from one of my papers, where the ratio between the average return interval tret and the sampling interval, tobs of the animal’s path; ρ = tret/tobs, is key to understanding the statistical pattern if movement is memory-influenced:
… by analysing the data with different tobs relative to system-specific boundary conditions, two observers may reach very different conclusions with respect to step-length compliance with a negative exponential or a power law. Both may in fact be right! In particular, if the animal in question has used its habitat under the influence of long-term memory, then the observed pattern at temporal level tobs may shape-shift from power law, through a hockey stick pattern, to a truncated power law pattern (figure 1c), and ultimately to a negative exponential (BM compliance) if tobs is chosen large enough. Hence, this paradox may to some (testable) extent be rooted in a relative difference in observational scale between the respective studies.
Gautestad 2012, p8.
A jellyfish has a very rudimentary nervous system. It doesn’t have a brain or central nervous system, only a very basic set of nerves at the base of their tentacles. These nerves detect touch, temperature, salinity etc. and the animal reflexively respond to these stimuli. For example, the jellyfish can orient to olfactory cues from prey (Arai 1991). Hence, the movement is expected to include targeted returns in a very rudimentary and environmental field-dependent manner. On the other hand, Kaartvedt et al. (2015) have demonstrated the ability of a jellyfish species, P. periphylla, to locate and team up with each other in a surprisingly “individualistic” manner. That fact raises an interesting (and speculative) hypothesis; could jellyfish movement along the Lévy-Brownian gradient be explained as temporally difference in targeted return frequency (same tobs and different tret), whether returns go to a previous depth or as a means to keep contact with conspecifics? The MRW framework – including its parallel processing postulate for cognitive tactics/strategy complexity – provides a tool to test this hypothesis.
In short, do these returns in different context for these two jellyfish species embed tactical and Markovian-like behaviour only (for example, simply following an olfactory gradient on a moment-to-moment basis) or is a jellyfish capable of returning more strategically by initiating a return without such a specific taxis-response within its current perceptual field?
What is extremely interesting in Hays et al. (2012) is that the jellyfish apparently shows a capacity both to long distance prospecting and long distance returns. According to MRW the returns should emerge from a capacity for spatial mapping of previously encountered foraging patches, without necessarily following an olfactory gradient towards this target! Hence, the test to differentiate between these classes of spatially explicit behaviour is to study if the animal is capable of targeted returns in absence of – or even disobeying (!) – a simple “following the gradient” (taxis) kind of return.
Hays et al. (2012) documented “occasional sallies” (prospecting) in the foraging behaviour of jellyfish. This behaviour obviously implies moving away from the current foraging patch and thus “down” the hypothetical olfactory gradient. Returning may then either imply swimming “up” the gradient or targeting a previous location per se; hypothetically as a function of spatial memory rather than getting moment-to-moment guidance from an environmental, chemical field. For a conceptual illustration of complex movement spanning the tactics-strategy gradient (parallel processing), see this post.
A long and speculative shot, I agree, to suggest that jellyfish nervous system may express MRW behaviour. However, perhaps the cognitive capacity of animals with simple nerve systems like jellyfish are more powerful than traditionally anticipated, and that statistical analyses of their movement paths from the perspective of (memory-extended) statistical mechanics may contribute to studying this capacity?
For example, due to Hays et al.‘s (2012) documentation of the combined capacity to explore its environment in a scale-free manner within a given scale range and occasional returns to a previous location (which may take several minutes; i.e., “strategic moves”) the jellyfish behaviour may cast light on evolutionary initial steps towards a more sophisticated kind of spatial behaviour, as it is found in animals with developed brain structures.
Such a potential for rudimentary MRW behaviour could, for example, imply a capacity to perform targeted returns to a recent part of the individual’s path but not further back as in the default MRW. Such a constrained variant of parallel processing may be tested statistically, by comparing simulations under this condition with true paths. In fact, I’ve already done introductory simulation studies (Gautestad 2011; A. O. Gautestad, unpublished).
MRW is simulated in 2-dimensional space with return steps at frequency 1:100 of original series (tret=100 in relative terms) to a trailing time window of 1000 last steps; i.e., a short memory horizon. Left: spatial pattern from 9000 observed fixes at frequency 1:1000 of original series (tobs. = 1000). Middle: box counting method shows fractal dimension D = 1.06 over a mid-range of spatial resolution, k. A larger number of fixes, N, would have increased this range. Right: Studying incidence, I, as a function of N shows a positive log–log slope of 0.96 and 1.01 for grid resolutions k = 1:64 and 1:300, respectively. This example illustrates that MRW under the condition of temporally constrained memory still shows a statistical fractal of spatial fixes. However, the limited capacity for targeted returns makes incidence increase proportionally with N (log-log slope of 1) rather than with square root of N (log-log slope of 0.5), as when memory is infinite and ρ = tret/tobs << 1. From Gautestad (2011).
Reynolds (2014) explored the results in Hays et al. (2012) by simulating an alternative model for jellyfish search, called Fast simulated annealing (FSA). In the present context this algorithm is scanning the environment to find and select optimal food patches. This post is too short to describe and discuss this very interesting approach, so I may return to it later. However, as a preliminary comment to those familiar with FSA I suggest that it may be very promising to combine principles from MRW and FSA. In particular, FSA implies patch selection that on one hand is based on a Cauchy-distributed step length distribution during searching; i.e., very long tailed next-location selection, and on the other hand occasional “escape” steps to avoid local trapping in patches that are potentially only locally optimal but not globally. Long steps at scales beyond the animal’s perceptual field will logically require a cognitive capacity for some kind of directed returns to more optimal patches after “prospecting”; i.e., spatial memory may be required. By default, FSA does not include spatial memory. In other words, the perceptual field is assumed to span the entire search arena. This capacity is obviously not a feasible premise in the jellyfish case, so what remains to sufficiently extend the individual’s overview of its environment is a cognitive utilization of a spatial map?
On the other hand, combining MRW and FSA will have to bridge two system representations, which may require a novel mathematical formulation of FSA. While MRW requires a sufficiently deep hidden layer to ensure compliance with a statistical-mechanical system description, the FSA in current formulations describes a mechanistic and Markovian kind of dynamics on a fine-grained temporal scale; i.e., a very shallow hidden layer. Further, FSA describes a tactical search algorithm, while MRW is based on a gradient from tactical to strategic time scales in a non-trivial kind of superposition (the parallel processing conjecture).
To conclude, the experimental outline for studying optimal foraging needs to include a test for strategic space use beyond a purely tactical/Markovian kind of displacements.
Arai, M. 1991. Attraction of Aurelia and Aequorea to prey. Hydrobiologia 216:363–366.
Gautestad, A. O. 2011. Memory matters: Influence from a cognitive map on animal space use. Journal of Theoretical Biology 287:26-36.
Gautestad, A. O. 2012. Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion. Journal of the Royal Society Interface 9:2332-2340.
Gautestad, A. O. and A. Mysterud. 2013. The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion. Movement Ecology 1:1-18.
Hays, G. C., T. Bastian, T. K. Doyle, S. Fossette, A. C. Gleiss, M. B. Gravenor, V. J. Hobson, N. E. Humphries, M. K. S. Lilley, N. G. Pade, and D. W. Sims. 2012. High activity and Lévy searches: jellyfish can search the water column like a fish. Proc. R. Soc. B 279:465-473.
Kaartvedt, S., K. I. Ugland, T. A. Klevjer, A. Røstad, J. Titelman, and I. Solberg. 2015. Social behaviour in mesopelagic jellyfish. Scientific Reports 5:1-8.
Reynolds, A. M. 2014. Signatures of active and passive optimized Lévy searching in jellyfish. Journal of the Royal Society Interface 11:20140665.
Ugland, K. I., D. L. Aksnes, T. A. Klevjer, J. Titelman, and S. Kaartvedt. 2014. Lévy night flights by the jellyfish Periphylla periphylla. Mar. Ecol. Prog. ser. 513:121-130.
Viswanathan, G. M., S. V. Buldyrev, S. Havlin, M. G. E. d. Luz, E. P. Raposo, and H. E. Stanley. 1999. Optimizing the success of random searches. Nature 401:911-914.