Modeling three-dimensional bait ball collective motion

Collective motion of animal groups such as fish schools and bird flocks in three-dimensional (3D) space are modeled by considering a topological (Voronoi) neighborhood. The tridimensionality of the group is quantified. Apart from the patterns of swarming, schooling, and milling, we identify a 3D bait ball around the phase transition boundary. More significantly, we find that by considering a blind angle in this topology based model, an individual interacts statistically with six to seven neighbors, consistent precisely with the previous field observations of the starling flocks. This model could be expected to enable more insightful investigation on realistic collective motion of shoals or flocks.

Figure 1

(a) Sketch of two interacting individuals (red), showing the position ${\mathbf{r}}_i$ and ${\mathbf{r}}_j$, the orientation ${\mathbf{e}}_i^{\left|\right|}$ and ${\mathbf{e}}_j^{\left|\right|}$, the distance vector ${\mathit{\rho }}_{ij}$ pointing from individual $i$ to its neighbor $j$, and the moving vector for individual $i$ from $t-1$ (pink) to $t$ (red). (b) Substep sketch of Eq.(4), namely the rotation matrix and the repulsive force.

Figure 2

The collective patterns exhibited by the model, with the snapshots of spatial individual distributions for (a) swarming at $I_{\left|\right|}=0.2$ and $I_n=1.8$, (b) milling at $I_{\left|\right|}=0.6$ and $I_n=0.2$, and (c) schooling at $I_{\left|\right|}=2.5$ and $I_n=0.5$. For all simulations, $\beta =1.0$, and the snapshots are taken at the end of 10 000 time units. Note that the data from five sequential time instants are superposed to represent the moving directions. For long-time dynamics, i.e., their evolution from randomly initialized positions and orientations in a $20\times 20\times 20$ cubic box, see movies 1–3 provided in the Supplemental Material [39].

Figure 3

Contour plots of $P$, $M$, $D_1,$ and $D_2$ in the ($I_n,I_{\left|\right|}$) parametric space: (a) the polarization $P$ with the dashed yellow line of $P=0.5$ marking the phase transition to a highly coordinated schooling pattern as $I_{\left|\right|}$ is high; (b) the milling $M$ identifying a milling pattern at the bottom left corner from the rest regimes, with $M=0.5$ (the dashed cyan line); (c) the dimension indicator $D_1$ gives a value of 0.1 for a planar spatial structure, while 1.0 for a sphere, and here, the superposition of the lines for $P=0.5$ and $M=0.5$ separates the swarming pattern to the bottom right of the space; (d) the distance $D_2$ indicates the spatial distance of the group (the average distance from individuals to the group center). A hundred simulations are run for each pair of $I_n$ and $I_{\left|\right|}$ with 10 000 time units, and the last 1000 time units are taken out for averaging.

Figure 4

Time histories of $P$, $M$, and $D_1$ at ($I_n,I_{\left|\right|})$=(0.0,0.7), corresponding to a milling pattern, with two different individual numbers of $N=100$ (blue lines) and $N=400$ (red lines), while keeping group density the same. It demonstrates the independence of individual number in this topological neighborhood based model.

Figure 5

Snapshots of two distinct milling patterns simulated with 400 individuals: (a) a flat milling structure at ($I_n,I_{\left|\right|})$=(0.2, 0.6), with $D_1=0.283$ and $D_2=5.02$; (b) a bait ball at ($I_n,I_{\left|\right|})$=(0.2, 0.9), with $D_1=0.585$ and $D_2=4.01$. Here, the snapshots are taken at the end of simulations, and $D_1$ and $D_2$ are averaged over the last 1000 time steps. Note that the views are adjusted according to their rotating axes for clearer presentation and more appropriate comparison. Please see movies four and five provided in the Supplemental Material [39] for the long-time dynamics of (a) and (b) respectively.

Figure 6

Sketch of 3D Voronoi neighbors for a schooling pattern, where (a) the red cell represents the focal individual, (b) the transparent gray cells represent its first shell of Voronoi neighbors, and (c) voronoi tessellation for the entire group.

Figure 7

The probability density of the average number of neighbors for different values of the parameters: (a) swarming for $(I_n,I_{\left|\right|})=(1,8,0.2)$; (b) milling for $(I_n,I_{\left|\right|})=(0.2,0.6)$; and (c) schooling for $(I_n,I_{\left|\right|})=(0.5,2.5)$. Each histogram is the result of an average over 100 different simulations in the same set of parameters.