Time delay can facilitate coherence in self-driven interacting-particle systems

Directional switching in a self-propelled particle model with delayed interactions is investigated. It is shown that the average switching time is an increasing function of time delay. The presented results are applied to studying collective animal behavior. It is argued that self-propelled particle models with time delays can explain the state-dependent diffusion coefficient measured in experiments with locust groups. The theory is further generalized to heterogeneous groups where each individual can respond to its environment with a different time delay.

Figure 1

(a) Average velocity $U\left(t\right)$ of $N=20$ individuals which are modeled by Eqs. (2) and (3) where we use $\tau =0$ for times $0\le t<100$ min, $\tau =0.5$ s for times $100\le t<200$ min, and $\tau =1$ s for times $t\ge 200$ min. (b) Mean switching time as a function of $N$ for $\tau =0$ (blue circles), $\tau =0.5$ s (red squares), and $\tau =1$ s (green triangles). Note the log scale on the $y$ axis. (c) Mean switching time as a function of $\tau$ for model of Eqs. (2) and (3) (blue circles) and generalized model of Eqs. (16) and (17) (red squares). Parameter values used are $L=251.3$ cm, $R=13.9$ cm, $\nu =1.9$ cm/s, ${\eta }_1=1$ ${\mathrm{s}}^{-1}$ and ${\eta }_2=2$ ${\mathrm{s}}^{-1/2}$.

Figure 2

The approximation of stationary distribution given by Eq. (7) for parameters $N=20,$ ${\eta }_1=1$ ${\mathrm{s}}^{-1},{\eta }_2=2$ ${\mathrm{s}}^{-1/2}$, and $\tau =1$ s (green dash-dotted line), $\tau =2$ s (red dotted line), and $\tau =3$ s (blue solid line).

Figure 3

Diffusion coefficient estimated using Eq. (13) for the model of Eqs. (2) and (3) where $N=20,$ $L=251.3$ cm, $R=13.9$ cm, $\nu =1.9$ cm/s, ${\eta }_1=1$ ${\mathrm{s}}^{-1}$, and ${\eta }_2=2$ ${\mathrm{s}}^{-1/2}$. We use $\delta t=0.1$ s to estimate $D\left(U\right)$ using Eq. (13).

Figure 4

The stationary distribution obtained by Eq. (14) (solid red line) is compared with the histogram computed using the long time simulation of model in Eqs. (2) and (3). The parameter values are $N=20,L=251.3$ cm, $R=13.9$ cm, $\nu =1.9$ cm/s, ${\eta }_1=1$ ${\mathrm{s}}^{-1},{\eta }_2=2$ ${\mathrm{s}}^{-1/2}$, and $\tau =1$ s.

Figure 5

The mean switching time for delays that are normally distributed with $c=1$ (red triangles), normally distributed with $c=2$ (magenta squares), and exponentially distributed (green stars) as a function of the mean time delay. The results computed with constant delays are plotted as blue circles. We use $N=100,$ $L=251.3$ cm, $R=13.9$ cm, $\nu =1.9$ cm/s, ${\eta }_1=1$ ${\mathrm{s}}^{-1}$, and ${\eta }_2=4$ ${\mathrm{s}}^{-1/2}$.