Emergent Strategies for Shepherding a Flock

TL;DR

The paper tackles how to shepherd a cohesive flock to a target using minimal interaction rules. It develops two complementary models—a Reynolds-Vicsek-inspired agent-based model (ABM) and a coarse-grained ordinary differential equation (ODE) for an elliptical herd—together with a discrete gradient optimization of a cost function that trades off herd cohesion, proximity to the target, and line-of-sight of the shepherd. Three emergent strategies—droving, mustering, and driving—arise as the optimal regime in distinct regions of the phase space, characterized by the scaled herd size $\sqrt{N} l_a / l_s$ and the scaled shepherd speed $v_a / v_s$, with a phase diagram and analytic scaling for herd-area oscillations. The study also shows robustness to inertia and moving targets, and discusses practical implications for autonomous shepherding and understanding collective navigation in active matter.

Abstract

We investigate how a shepherd should move to effectively herd a flock towards a target. Using an agent-based (ABM) and a coarse-grained (ODE) model for the flock, we pose and solve for the optimal strategy of a shepherd that must keep the flock cohesive and coerce it towards a target. Three distinct strategies emerge naturally as a function of the scaled herd size {and} the scaled shepherd speed: (i) mustering, where the shepherd circles the herd to ensure compactness, (ii) droving, where the shepherd chases the herd in a desired direction while sweeping back and forth, and (iii) driving, where the flock surrounds a shepherd that drives it from within. A minimal dynamical model for the size, shape, and position of the herd captures the effective behavior of the ABM and further allows us to characterize the different herding strategies in terms of the behavior of the shepherd that librates (mustering), oscillates (droving), or moves steadily (driving).

Figures (13)

• Figure 1: Herding and models for it. (A) A herd of sheep, cattle, and ducks. (B) In our agent-based formulation, the position and orientation of individual herd members (blue arrows) respond to each other and the location of a nearby dog (red) that moves so as to optimally transport the herd center (black dot) to the target location (red cross). (C) In our mean-field ODE formulation, the herd is modeled as an ellipse with area $A$ and aspect ratio $Q$ that evolve in response to the distance $R$ to the dog. Cattle from Montana images courtesy of Jordan Kennedy. Ducks image by Cao Ky Nhan (https://mindthegraph.com/blog/photography-contest/). Free roaming sheep image from CBS (https://www.youtube.com/watch?v=0yMmc2xmE8I). Cattle with fence image from BBC (https://www.bbc.com/reel/video/p07t7zbv/turning-old-pastures-new).
• Figure 1: Fig. \ref{['SI_fig:area_and_fft']}A illustrates how the herd area fluctuates as a function of time during the mustering phase. Fig. \ref{['SI_fig:area_and_fft']}B illustrates the fourier-transform of the area shown in Fig. \ref{['SI_fig:area_and_fft']}A. Solid blue line depicts the power spectrum of the herd;s area fluctuation; dashed vertical line shows the peak (corresponding to the primary oscillation frequency).
• Figure 2: Shepherd and herd trajectories across three regimes derived from the ABM (eq.(1-2)) and ODE (eq.(3-7)) models. Trajectories correspond to movies S1 (A.) and S2 (B.) respectively (params in SI). In each of the six plots in the left column, the mean path of the flock (blue) over an interval is shown as it is driven by a shepherd on a separate path (red) towards a target (green square). Columns 2-4 show snapshots from column 1, with trajectories indicated in black, where fading indicates history. From left to right, snapshots represent the flock at later timesteps.
• Figure 2: Illustration of the different herd shapes that can be achieved with our equation for the herd boundary, ranging from circular to lunate, to ring-like. The reader can play with the model if they so choose using the following link: https://www.desmos.com/calculator/v7l33lt5k0.
• Figure 3: (A) Shepherding behavior (droving, driving, mustering) in herd orientation-angular velocity ($\psi$-$\dot \psi$) space, with data extracted from steady-state periods of the ABM (eq. (1-2)) simulations (Movie S1), and correspond to the different states of a simple pendulum (see text for details).(B) Plot of the shepherd-repulsion length scale ($l_s$) versus the herd's area compression rate ($\omega_{\rm area}$); dots correspond to mustering and droving, respectively, and lines represent the theoretical fit to the scaling law $\omega_{\rm area} = c \frac{v_s}{\sqrt{N}l_a + d l_s}$, with $c=0.416, d=0.328$ for droving; and $c=0.046, d=0.706$ for mustering.