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.
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.
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.