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Lévy Flights and Self-Similar Exploratory Behaviour of Termite Workers: Beyond Model Fitting

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Lévy Flights and Self-Similar Exploratory Behaviour of Termite Workers: Beyond Model Fitting

Octavio Miramontes et al. PLoS One.

Abstract

Animal movements have been related to optimal foraging strategies where self-similar trajectories are central. Most of the experimental studies done so far have focused mainly on fitting statistical models to data in order to test for movement patterns described by power-laws. Here we show by analyzing over half a million movement displacements that isolated termite workers actually exhibit a range of very interesting dynamical properties--including Lévy flights--in their exploratory behaviour. Going beyond the current trend of statistical model fitting alone, our study analyses anomalous diffusion and structure functions to estimate values of the scaling exponents describing displacement statistics. We evince the fractal nature of the movement patterns and show how the scaling exponents describing termite space exploration intriguingly comply with mathematical relations found in the physics of transport phenomena. By doing this, we rescue a rich variety of physical and biological phenomenology that can be potentially important and meaningful for the study of complex animal behavior and, in particular, for the study of how patterns of exploratory behaviour of individual social insects may impact not only their feeding demands but also nestmate encounter patterns and, hence, their dynamics at the social scale.

Conflict of interest statement

Competing Interests: The authors have declared that no competing interests exist.

Figures

Figure 1
Figure 1. Picture of the observation set.
(A) One Cornitermes cumulans worker with a painted abdomen was allocated into a circular glass arena (205 mm inner dia.). When detected by the video recording system, the individual appears as an image measuring 5×5 pixels representing 4.7 mm2 aprox. In (B) the termite worker is the small black dot at the top of the circular area. A single total trajectory is drawn in (C) showing the typical entangled pattern of individual steps. This particular example contained 35,000 points sampled at 0.5 seconds intervals. Notice that most of the trajectory occurs near the arena border, however inner exploratory excursions are also frequent.
Figure 2
Figure 2. A termite walking trajectory segment as reconstructed in 3D, with a time axis added.
Three walking behaviours are visible. (C) is the individual walking termite following the circular geometry of the container border, (B) is a straight long nearly-ballistic displacement from one container side to the opposite, and (W) is a waiting time.
Figure 3
Figure 3. Examples of time-series containing traveled distances by four different Cornitermes cumulans individuals.
The time-series totaled 35,000 data points in (A) and 43,000 in (B, C and D), but a window of 18,000 points is shown for each. Sample rate was one point at every 0.5 seconds.
Figure 4
Figure 4. Sampling rates.
Different sampling rates give very different results when the length of the steps are plotted as a histogram of their frequency (log-binned). In the plot, three different sampling rates (sr) were exemplified as the time series as captured by the video recording device at each 0.5, 2.0, and 5.0 sec. Note that a region resembling a power-law scaling is only obvious at 2-sec sampling rate.
Figure 5
Figure 5. Anomalous diffusion.
Cornitermes cumulans termites exhibit anomalous diffusion (formula image) in their walking patterns because the mean squared displacement grows faster than it does in the normal diffusion of a Brownian particle (black), where formula image. MSD superdiffusive scaling exponent values of four termite workers are formula image (purple), formula image (green), formula image (blue) and formula image (red). Notice that the termite MSD scaling separate away from a power-law at values of formula image and beyond, this is common and correspond to the typical diffusive behaviour of truncated motion in confined environments. formula image is the diffusion coefficient of each individual termite.
Figure 6
Figure 6. Kolmogorov -functions.
Plots shown at the left column depict four examples of Kolmogorov q-functions and their power-law scaling (red lines). First eight values of the exponent formula image were calculated but only four are shown for the sake of clarity. The column at the right depicts the linear scaling of formula image (red lines), resulting in four formula image slope values: 0.93, 0.87, 0.85 and 0.8. These correspond to Lévy exponents of values formula image, formula image, formula image and formula image, respectively.
Figure 7
Figure 7. Power spectrum.
Power spectrum of a Cornitermes cumulans termite walking time-series (A) and an artificially generated one (B). Both time series contained 4096 points and were transformed with a Fast Fourier Transform (FFT) algorithm.
Figure 8
Figure 8. IFS algorithm.
(A) A Cornitermes cumulans termite walking time-series as seen with a IFS algorithm. Notice the subtle details of a self-similar structure. (B) An artificially correlated time-series generated with a relaxation return map formula image where formula image is a normally distributed random variable with zero mean and unit variance and formula image is real valued parameter whose value determines the colour of the resulting time-series scaling. Colour in this context means the classification of a noise mode formula image, where formula image is the frequency in a Fourier transformed space. formula image is the scaling exponent and when formula image, the process is uncorrelated white noise (C), formula image is correlated pink noise (D) and formula image is a correlated brown noise (E). Termite walking (A) lies in between a pink (D) and brown noise (E) scaling, being compatible with the fact that the termite scaling exponent in the FFT is formula image. For details on how the IFS algorithm operates, see .
Figure 9
Figure 9. Long-range correlations.
Termite walking exhibit power-law decaying long-range correlations as measured by a correlation function along the walk time-series (blue). An artificial correlated time-series, as explained in Fig. 8 was used also to compare a correlated decaying process (red). The black line is a power-law with a scaling exponent formula image.
Figure 10
Figure 10. Waiting times.
Waiting times are an ubiquitous pattern of animal movement behaviour and they may follow power-law scaling, as is the case of Cornitermes cumulans workers when performing exploratory behaviour. The graph at the left (A) depicts a typical example of termite spatial distribution of accumulated waiting times over the circular arena (squared root axis for enhancing visualization). The plot in (B) is the waiting-time bouts histogram showing a power-law with a scaling exponent value of formula image (straight line slope, calculated with a MLE procedure .)
Figure 11
Figure 11. Turning angle distribution.
Turning angle distribution in termite walking. Four examples are depicted exhibiting a bell shaped distribution centered at 0 degrees. No preferential angles were identified apart from the persistence of moving forwards.

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Grant support

OM thanks DGAPA-PAPIIT Grant IN101712 and the Brazilian Ciência Sem Fronteiras program (CSF-CAPES) 0148/2012. ODS is supported by CNPq-Brasil, fellowship 305736/2013-2. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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