Ergodic directional switching in mobile insect groups

We obtain a Fokker-Planck equation describing experimental data on the collective motion of locusts. The noise is of internal origin and due to the discrete character and finite number of constituents of the swarm. The stationary probability distribution shows a rich phenomenology including nonmonotonic behavior of several order and disorder transition indicators in noise intensity. This complex behavior arises naturally as a result of the randomness in the system. Its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models. Our results suggest that the collective switches of the group’s direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation.

Figure 1

(Color online) Panels (a) and (b) show the profile of the SPD, $P_s\left(u\right)$, from Eq. (6) plotted against the normalized mean velocity $u$ for a varying noise intensity ${\beta }_2/\left(N{\alpha }_2\right)$. In panel (a) the solid red line represents ${\beta }_2/\left(N{\alpha }_2\right)=0.05$, the dashed green line represents ${\beta }_2/\left(N{\alpha }_2\right)=0.15$ and the dotted blue line represents ${\beta }_2/\left(N{\alpha }_2\right)=1.5$. In panel (b) the solid red line represents ${\beta }_2/\left(N{\alpha }_2\right)=1.5$, the dashed green line represents ${\beta }_2/\left(N{\alpha }_2\right)=3.0$ and the dotted blue line represents ${\beta }_2/\left(N{\alpha }_2\right)=5.5$. Panel (c) displays the second moment of the revised model, centered at the origin, $S_0$, versus noise strength ${\beta }_2/\left(N{\alpha }_2\right)$ and panel (d) shows the second moment at a maximum, $S_m$, versus noise strength ${\beta }_2/\left(N{\alpha }_2\right)$. The minima are attained for ${\beta }_2/\left(N{\alpha }_2\right)\approx 0.27\left(S_0\right)$ and for ${\beta }_2/\left(N{\alpha }_2\right)\approx 0.12\left(S_m\right)$.