Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer

Abstract

The study of animal foraging behaviour is of practical ecological importance 1 , and exemplifies the wider scientific problem of optimizing search strategies 2 .Lévy flights are random walks whose step lengths come from probability distributions with heavy power-law tails 3, 4 , such that clusters of short steps are connected by rare long steps. flight durations (time intervals between landing on the ocean) were then calculated as consecutive hours for which a bird remained dry, to a resolution of 1 h. It was assumed that birds landed on the water solely to feed, and that flight durations were thus indicative of distances between prey.Time series for 19 separate foraging trips 7 were pooled to give a total of 363 3 flights. The resulting log-log histogram of flight durations gave a straight line with a slope of approximately 2, and is reproduced in Supplementary Fig. 1 from the original raw data. The crux of the conclusion that the albatrosses were performing Lévy flights was that the slope of 2 implied the probability density function (pdf) of flight durations t (in hours), was 7, 10for t ≥ 1 h (leaving out the normalization constant). This is consistent with the Lévy flight definition that the tail of the pdf is of the power-law form t −µ , where 1 < µ ≤ 3 (though technically this is a Lévy walk 4,7,22 We first analyze a newer, larger, and higher resolution data set of albatross flight durations to test for Lévy flights. In 2004, 20 wandering albatrosses on BirdIsland were each fitted with a salt-water logger and a GPS device. The GPS data were too infrequent (at most one location h −1 ) to give distances between landings, but were needed to estimate each bird's departure time from Bird Island, in order to calculate the duration of the initial flight before first landing on the water (we calculated return flights similarly). The resulting data set of flight records was 4 pooled, as in ref. 7, yielding a total of 1416 flights to a resolution of 10 s (Fig. 1).The flights ≥ 1 h are clearly inconsistent with coming from the power law t −2 ascertained 7 for the 1992 data. Furthermore, data from a power law of any exponent (not just 2) would yield a straight line 23 , and this is clearly not the case.In fact, the flight durations t (in h) are consistent with coming from the shifted gamma distribution given by the pdfwhere y = t − 1/120 accounts for the assumed 30 s period before the bird searches for new food sources (see Methods), s = 0.31 is the shape parameter, r = 0.41 h −1 is the rate parameter, and Γ(·) is the gamma function. Equation (2) is valid for flights >30 s; for shorter flights we have f (t) = 0. The exponential term of (2) dominates for large t, implying Poisson behaviour, such that for long enough flights the birds essentially encounter prey randomly with a constant low probability.A Brownian random walker's displacement increases as t H where H = 1/2.If H > 1/2, we have "superdiffusion" as originally inferred in Fig. 2a The gamma distribution (2) has µ = 1 − s = 0.69. This is such a slow powerlaw ...

Changing measurements or changing movements? Sampling scale and movement model identifiability across generations of biologging technology

“…One is to take a “naive” maximum likelihood approach, that is, ignore the observational process, and instead assume that the observed data are drawn without noise from the underlying distribution. Another approach is to use a multinomial maximum likelihood approach that explicitly models the observational process (Edwards et al., 2007). In this case, the log‐likelihood of the parameters θ , given a record r (a set of observations, see Table 1), takes the general form $\ell false(\mathrm{bold\theta }false|\mathrm{boldr}false)=\sum _{j=1}^J{d}_{j}logfalse[p(j|\mathbf{\theta })false],$where $d_j$ is the number of recorded flights of length $j\in [1,J]$, and $p(j|\mathbf{\theta })$ is the probability of observing a flight of length j given the underlying flight‐time distribution and observation process.…”

“…Following Edwards et al. (2007); Reynolds et al. (2016) and others, we use a lower limit of flight duration (part of an overall trip) of 30 s, on the biological assumption that this is not likely to be a flight to a different food patch.…”

“…Researchers use satellite transmitters or data loggers to collect location information; these provide data at intervals which are often constrained by battery power or memory capacity (Edwards et al., 2007; Fedak, Lovell, McConnell, & Hunter, 2002; Shillinger et al., 2012). In large birds and mammals, particularly in recent years, the deployment of GPS loggers, and, in marine animals, of saltwater immersion or temperature‐depth loggers, have generated a wealth of tracking data at high temporal resolution and at relatively low cost (Block et al., 2011; Mackley et al., 2010; Scales et al., 2016).…”

“…For instance, although the initial study on albatrosses indicated a Lévy pattern of foraging (Viswanathan et al., 1996), after correcting and augmenting the original data, and utilizing improved statistical methods, a later study by Edwards et al. (2007) showed that the Lévy flight model was not supported; instead, flight times were more likely to be gamma‐distributed. Subsequent studies claim new evidence for Lévy‐like behavior in certain marine predators (Focardi & Cecere, 2014; Hays et al., 2012; Humphries et al., 2010; Reynolds, Paiva, Cecere, & Focardi, 2016; Sims, Humphries, Bradford, & Bruce, 2012; Sims et al., 2008), and that humans exhibit more complex behaviors (González, Hidalgo, & Barabási, 2008).…”

“…Our analyses focus on a particular type of data from loggers which detect and record saltwater immersion, providing information on wet and dry periods (so called immersion loggers; Edwards et al., 2007; Mackley et al., 2010). Although a geographic location was not available in some of the earlier deployments, PTT or GPS data have been collected concurrently with the immersion data in the past 20 years, providing improved insights into movements and habitat use (Carneiro et al., 2016; Mackley et al., 2010; Scales et al., 2016).…”