Large-Scale Patterns in a Minimal Cognitive Flocking Model: Incidental Leaders, Nematic Patterns, and Aggregates

We study a minimal cognitive flocking model, which assumes that the moving entities navigate using the available instantaneous visual information exclusively. The model consists of active particles, with no memory, that interact by a short-ranged, position-based, attractive force, which acts inside a vision cone (VC), and lack velocity-velocity alignment. We show that this active system can exhibit—due to the VC that breaks Newton’s third law—various complex, large-scale, self-organized patterns. Depending on parameter values, we observe the emergence of aggregates or millinglike patterns, the formation of moving—locally polar—files with particles at the front of these structures acting as effective leaders, and the self-organization of particles into macroscopic nematic structures leading to long-ranged nematic order. Combining simulations and nonlinear field equations, we show that position-based active models, as the one analyzed here, represent a new class of active systems fundamentally different from other active systems, including velocity-alignment-based flocking systems. The reported results are of prime importance in the study, interpretation, and modeling of collective motion patterns in living and nonliving active systems.

Figure 1

Depending on vision cone size $\beta$ (insets), we can observe the formation of locally polar structures, which we call “worms,” panel (a)—$\beta =0.8$, or the emergence of aggregates or millinglike patterns as shown in panel (b) for $\beta =2.4$, and panel (c) for $\beta \sim \pi$. Parameters (a)–(c): $\sqrt{2D_{\theta }}=0.12$, $L=100$, $N=10^4$. See [45] for movies.

Figure 2

Phase diagram: vision angle $\beta$ vs angular noise intensity $D_{\theta }$. The color code indicates the value of the average cluster size $m^{*}$ in simulations with $N=10^4$. Vertical rectangles refer to cuts of the phase diagram shown in Figs. 3 and 4.

Figure 3

(a) Normalized number of clusters $M^{*}$ (normalized average cluster size $m^{*}$, inset) as function of time. Worms coagulate at a much faster rate than aggregates. (b) The (normalized) diffusion coefficient $D_{\mathrm{eff}}/D_{\mathrm{NAP}}$ and polar order $S_1$ as a function of $\beta$ (see text). (c) Temporal evolution of $⟨{\mathbf{x}}^2⟩/(4t)$. (d) Velocities and positions of particles that form a worm: particles copy the behavior of the particle at the front, which we refer to as the incidental leader. See [45] for a movie. Parameters: $N=10^4$, $L=100$, $\sqrt{2D_{\theta }}=0.12$.

Figure 4

Nematic bands. (a) The nematic order parameter $S_2$ as a function of $\beta$ for $\sqrt{2D_{\theta }}=0.84$. The inset shows $S_2$ as a function of $N$, with error bars obtained using 50 realizations, $\beta =1.9$. (b) and (c) display simulation snapshots at the steady state for $N=10^4$, $L=100$, $\sqrt{2D_{\theta }}=0.84$, and $\beta =1.9$. Insets in (b) and (c) correspond to magnified views of the bands. See [45] for movies.