Figure 1

Modeling boundary following in an experimental study of ants Messor sancta forming cemeteries [59]. (a) Trajectory. Typical trajectory of an ant in the vicinity of a conspecific's corpse. (b) Angular distribution. Considering criterion 1 (defined in the main text) we have calculated the normalized cumulative $I\left(\theta \right)$ of ${\tilde{f}}_{\mathrm{stat}}(d,\theta )|cos\left(\theta \right)|$ for $\theta \in [0,\pi ]$ where we used the circular symmetry around the corpse to write the stationary distribution function $f_{\mathrm{stat}}(r,\omega )\equiv {\tilde{f}}_{\mathrm{stat}}(d,\theta )$ with $d$ being the distance to the corpse (approximated by a disk of radius 2 mm) and $\theta$ being the absolute angle between $\omega$ and radial vector $e_r$. The plot shows $I\left(\theta \right)$ as a function of $I\left(\theta \right)$ calculated for an isotropic distribution. In the case of isotropy (dashed curve), this corresponds to a straight line of unit slope, whereas in the case of boundary alignment [solid curve, Eq. (2) with $\varphi (r,\omega )=10$ if $\parallel \omega ·e_r\parallel <0.3$ and $\varphi (r,\omega )=1$ else, note that $\alpha \left(r\right)$ and $K\left(r\right)>0$ do not impact $I\left(\theta \right)$], the curve deviates from the straight line. The experimental ant distribution (circles, $\mathrm{mean}\pm \mathrm{SEM}$ for $n=625$ trajectories, distance $d=5$ mm) is isotropic. This conclusion has also been confirmed for other distances $d$ (from 0 to 8 mm). (c) Residence time. Considering criterion 2, we have calculated the boundary-following criterion $\mathcal{C}=\frac{\langle T\rangle }{{\langle T\rangle }_{\mathrm{ref}}}$ (defined in the main text). An ant is tracked as long as its distance to the corpse is shorter than 8 mm, and the boundary-following time $T$ is defined as the time spent within the band of width $d$ around the corpse during this track [shaded zone and thick parts of trajectory in (a)]. The presented experimental curve $\mathcal{C}\left(d\right)$ (circles, $\mathrm{mean}\pm \mathrm{SEM}$, 507 trajectories) indicates boundary-following behavior ($\mathcal{C}\in [1.2,1.4]$ compared to 1 expected for free-field behavior) and can be fitted with our model for $\frac{\widehat{\alpha }}{1-g}=46e_r{\mathrm{m}}^{-1}$ [solid curve, see Eq. (2)]. (d) Spatial distribution. Alternatively, criterion 2 can be considered (without any path tracking) by looking at the ant density profile around a corpse. Here $\widehat{\eta }\left(u\right)$ is the surfacic ant density at the stationary state, averaged over the contour corresponding to the distance $u$ to the corpse. The figure shows the cumulative ant density $\frac{c}{{\Phi }_{\delta \Omega ,\mathrm{stat}}}{\int }_0^d\widehat{\eta }\left(u\right)2\pi udu$, normalized by the incoming ant flux ${\widehat{\varphi }}_{\delta \Omega ,\mathrm{stat}}$ at $d=8\mathrm{mm}$ and by the mean ant speed $c$. Thus $\widehat{\eta }\left(u\right)$ is proportional to the derivatives of the curves. The observed ant density (circles, $\mathrm{mean}\pm \mathrm{SEM}$) is about 30% higher than expected for free-field behavior (dashed curve), thus demonstrating boundary following and is again in close agreement with our model fit (solid curve).