Flocking by stopping: a novel mechanism of emergent order in collective movement

TL;DR

This work introduces a minimal, one-dimensional model of collective motion in which individuals can be in three states ($X_+$, $X_-$, $X_0$) and interact via spontaneous switching, copy interactions, and a novel halting interaction that stops motion when encountering oppositely moving neighbors. By deriving mean-field ordinary differential equations and Itô stochastic differential equations, the authors show that the combination of a stopped state and halting interactions can produce robust, large-scale flocking (nonzero $m$) even with only pairwise interactions, contrasting with conventional constant-speed models. The main contributions include a clear phase transition condition, the demonstration that halting interactions enable order for large $N$, and the characterization of finite-size effects where noise can induce or amplify order in small groups. The results offer a new mechanism—'flocking by stopping'—that highlights speed variability as a potential driver of collective order and provides a framework for exploring extensions to higher dimensions and more realistic motion patterns.

Abstract

Collective movement is observed widely in nature, where individuals interact locally to produce globally ordered, coherent motion. In typical models of collective motion, each individual takes the average direction of multiple neighbors, resulting in ordered movement. In small flocks, noise induced order can also emerge with individuals copying only a randomly chosen single neighbor at a time. We propose a new model of collective movement, inspired by how real animals move, where individuals can move in two directions or remain stationary. We demonstrate that when individuals interact with a single neighbor through a novel form of halting interaction -- where an individual may stop upon encountering an oppositely moving neighbor rather than instantly aligning -- persistent collective order can emerge even in large populations. This represents a fundamentally different mechanism from conventional averaging-based or noise-induced ordering. Using deterministic and stochastic mean-field approximations, we characterize the conditions under which such ``flocking by stopping'' behavior can occur, and confirm the mean-field predictions using individual-based simulations. Our results highlight how incorporating a stopped state and halting interactions can generate new routes to order in collective movement.

Figures (5)

• Figure 1: A schematic illustration of the model.Left: Each individuals can be in one of three states + (moving clockwise), - (moving counterclockwise), or 0 (stopped). Right: The individuals can switch between these states either spontaneously or by interacting with a random neighbor, through the different types of interactions as shown.
• Figure 2: Collective order via stopped state and halting interactions.Top: Without halting interactions ($h = 0$), the disordered state is the stable-state. Bottom: With halting interactions ($h = 7$), the system can show ordered dynamics. (a) and (c): Example trajectories of order parameters $m, v$, with and without halting interactions respectively. (b) and (d): Phase plane of the mean-field model with the background heatmap showing steady state histograms, computed from Gillespie simulations for $N=500$, with and without halting interactions respectively. (Parameter values: $s_S = s_M = s_C = c_S = c_C = 0.2, h \in \{0,7\}$ and $c_M = 2$.)
• Figure 3: Role of copy interactions in collective order. The ordered state is a deterministic stable state for only sufficiently high values of $c_M$. (a): As $c_M$ increases, the order $m$ undergoes a bifurcation. (b): A similar bifurcation occurs for speed $v$.(Parameter values: $s_S = s_M = s_C = c_S = c_C = 0.2, h = 7$ and $c_M \in [0,5]$)
• Figure 4: Stochastic dynamics and noise effects in small flocks. Histograms of $m$ and $v$ obtained from stochastic simulations of the SDE model, as a function of $c_M$, with $N=10$ (top row, panels (a) and (b)) and $N=100$ (bottom row, panels (c) and (d)). The bifurcation plots, representing the mean-field predictions, are overlaid on the normalized histograms (heatmaps). Top row: For $N=10$, we identify three ranges of copying interactions $c_M$, A, B and C. In region A (representing the region of disorder), the histograms match the expectations of mean-field theory. However, in region B, the histograms show a finite order, suggesting a noise-induced order, which deviates from the mean-field stable state of disorder. In the region C, the obesrved order is quantitatively higher than the mean-field predictions, indicating a case of noise facilitating the order. Bottom row: For $N=100$, we do not observe the noise-induced or noise-facilitated order, consistent with mean-field theory.
• Figure 5: Stochastic dynamics of the model. The SDE approximation captures the actual dynamics of the system over a range of parameters.