Waiting times.

<p>Waiting times are an ubiquitous pattern of animal movement behaviour and they may follow power-law scaling, as is the case of <i>Cornitermes cumulans</i> 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 (straight line slope, calculated with a MLE procedure <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0111183#pone.0111183-Clauset1" target="_blank">[36]</a>.)</p

Scaling exponents of four representative examples of termite walking.

<p>The values of exponents and are related by theoretical results <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0111183#pone.0111183-Solomon1" target="_blank">[16]</a>, and are also compatible with the results of a Lévy exponent via a Kolmogorov structure functions analysis.</p><p>Scaling exponents of four representative examples of termite walking.</p

Power spectrum.

<p>Power spectrum of a <i>Cornitermes cumulans</i> 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.</p

Long-range correlations.

<p>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 .</p

Examples of time-series containing traveled distances by four different <i>Cornitermes cumulans</i> individuals.

<p>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.</p

Anomalous diffusion.

<p><i>Cornitermes cumulans</i> termites exhibit anomalous diffusion () in their walking patterns because the mean squared displacement grows faster than it does in the normal diffusion of a Brownian particle (black), where . MSD superdiffusive scaling exponent values of four termite workers are (purple), (green), (blue) and (red). Notice that the termite MSD scaling separate away from a power-law at values of and beyond, this is common and correspond to the typical diffusive behaviour of truncated motion in confined environments. is the diffusion coefficient of each individual termite.</p

Picture of the observation set.

<p>(A) One <i>Cornitermes cumulans</i> 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.</p

Turning angle distribution.

<p>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.</p

IFS algorithm.

<p>(A) A <i>Cornitermes cumulans</i> 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 where is a normally distributed random variable with zero mean and unit variance and 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 , where is the frequency in a Fourier transformed space. is the scaling exponent and when , the process is uncorrelated white noise (C), is correlated pink noise (D) and 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 . For details on how the IFS algorithm operates, see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0111183#pone.0111183-Miramontes3" target="_blank">[30]</a>.</p

Sampling rates.

<p>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.</p