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Biological movement and the Lévy flight hypothesis

How do animals move through their environment as they search for resources, or try to satisfy other natural goals? Over the past two decades, researchers have examined this question using real world data for organisms such as albatross, marine predators and bees, often finding a pattern of many short or mid-scale movements punctuated by occasional long flights or excursions. Some researchers have interpreted the pattern as evidence for so-called “Lévy flights” – a particular stochastic process which deviates from ordinary Brownian motion – and asserted that such searching is optimal under some environmental conditions.

And yet, this interpretation remains a matter of debate. One problem is the subtlety of the statistical analysis involved in actually backing a claim that some empirical data is evidence for a specific underlying stochastic process. Another is the confounding influence of the environment, as observed behavioural patterns may reflect the distribution of resources, rather than any characteristic of the organism in question. In a new book chapter, LML Fellow Rainer Klages attempts to clear up some of the confusion in the field by examining how researchers have used the term Lévy flight in many distinct ways. He asks if it is even possible to use data and mathematical modelling to understand how biological organisms move, and suggests the answer is yes – but only if biologists and mathematically sophisticated analysts from physics or applied mathematics learn to collaborate more effectively. Much previous work, he laments, has been naïve either from the mathematical or biological point of view – and sometimes from both.

One problem afflicting many empirical analyses is the difficulty of identifying true Lévy flight behaviour from empirical data. As Klages notes, Levy flights lack any characteristic length scale, which implies that a mean square displacement does not exist. Mathematically this invalidates this model for the interpretation of  empirical observations. However, this problem can be addressed with more sophisticated mathematical approaches based on the theory of stochastic processes. For example, the related concept of Levy walks involves random walks in which jumps are drawn randomly from a heavy-tailed Lévy distribution, but with each step taking place over a time interval in proportion to the length of the jump. This process belongs to the broad class of continuous time random walks, for which the mean square displacement does exist. Future studies taking a more appropriate mathematical perspective, Klages believes, could help settle controversy over the Lévy flight interpretation.

But previous work on movement has also suffered from biological naiveté. Many studies, Klages notes, have tracked the locations of organisms and analysed the recorded data, treating the organism more or less as a point particle lacking habits or behavioural predispositions. Yet biologists in recent years have turned to a different perspective, acknowledging that distinct segments of an organism’s movements often reflect different phases of behaviour serving different purposes [see Ran Nathan, PNAS 105, 19050-19051 (2008]. For example, a newly hatched bird may at the outset forage for food, then later shift to making excursions that help it to learn and gain experience of its environment, and then later still migrate to wintering grounds far away. It’s a mistake to see all parts of a time series of data as reflecting a single behavioural process.

By way of analogy, one might compare a movement time series to DNA. One might well analyse a DNA strand as a simple linear string of base pairs. Yet we know thousands of short segments with specific structures and meanings – genes. DNA analyses that ignore genes would be seriously impaired, and so too with the analysis of movement data. Biologists refer to “canonical activity modes” as key behavioural episodes – runs and tumbles for bacteria, or standing, walking and running for larger animals. A lifetime movement track may consist of a patterned sequence of such modes, forming higher-level structures that also carry meaning – migration to a mating territory, for example.

Klages chapter takes an optimistic view on the future of studies of organism behaviour and movement. Early work may indeed have been naïve in many ways. And yet the errors and misunderstanding were perhaps also necessary, helping the exploration of an unknown space by asking difficult questions. Now the field is ready to move into a second, more sophisticated phase:

“My general point of view, or `hypothesis’,” says Klages, “is that typically biological processes may not be simply `scale-free’ as predicted by the simplest type of Lévy dynamics. I see no reason why this should be the favourite type of motion in nature. To me it makes more sense if biological processes are highly adaptive, and exhibit different dynamics on different spatio-temporal scales. Lévy dynamics might describe one limited range of many scales. In my view this is the way to go for future analyses – and I believe nowadays many colleagues think like this. Looking for a simple `theory of everything’ like Brownian motion or Lévy flights might be outdated. Nature is more rich and complex than this.”

The book chapter is available at https://arxiv.org/abs/1804.03738

 

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