Self-organization of collective escape in pigeon flocks

Abstract

Bird flocks under predation demonstrate complex patterns of collective escape. These patterns may emerge by self-organization from local interactions among group-members. Computational models have been shown to be valuable for identifying what behavioral rules may govern such interactions among individuals during collective motion. However, our knowledge of such rules for collective escape is limited by the lack of quantitative data on bird flocks under predation in the field. In the present study, we analyze the first GPS trajectories of pigeons in airborne flocks attacked by a robotic falcon in order to build a species-specific model of collective escape. We use our model to examine a recently identified distance-dependent pattern of collective behavior: the closer the prey is to the predator, the higher the frequency with which flock members turn away from it. We first extract from the empirical data of pigeon flocks the characteristics of their shape and internal structure (bearing angle and distance to nearest neighbors). Combining these with information on their coordination from the literature, we build an agent-based model adjusted to pigeons' collective escape. We show that the pattern of turning away from the predator with increased frequency when the predator is closer arises without prey prioritizing escape when the predator is near. Instead, it emerges through self-organization from a behavioral rule to avoid the predator independently of their distance to it. During this self-organization process, we show how flock members increase their consensus over which direction to escape and turn collectively as the predator gets closer. Our results suggest that coordination among flock members, combined with simple escape rules, reduces the cognitive costs of tracking the predator while flocking. Such escape rules that are independent of the distance to the predator can now be investigated in other species. Our study showcases the important role of computational models in the interpretation of empirical findings of collective behavior.

Fig 1. Collective motion schematic in the computational model HoPE.

The colored areas represent the field of view of a focal individual i with heading ${\widehat{h}}_i$, split into a ‘front’ (light blue, ${\theta }_{FoV}^f$) and ‘side’ area (blue). Based on its total field of view (θ_FoV), i interacts with its topological neighbors n, j, m, and k. Agents g and q are in the blind angle of i and are thus ignored. All pseudo-forces (ψ) acting on i are represented by colored arrows. Alignment (ψ^a) is the average vector of topological neighbors’ headings (indicated by blue doted arrows). Centroid attraction (ψ^ct) is the vector from i’s position to the center of the topological neighborhood (c). Accelerating attraction (ψ^ca) is a vector in the direction of motion (aligned with heading ${\widehat{h}}_i$), depending on the distances of i to the neighbors in its front field of view, namely j, m and k (with distances d_ij, d_im and d_ik respectively). If there are no agents in the front field of view, then ψ^ca is negative. Separation (ψ^s) is the vector away from the position of the nearest neighbor n. Avoidance of the predator (ψ^p) is a vector perpendicular to the heading of the pigeon-agents, in the direction away from the predator’s heading (${\widehat{h}}_p$). In other words, forcing a turn away from the predator’s heading clockwise or anticlockwise (left or right) relative to the agent’s heading. Vector ψ^f represents the flight-control force that drags i towards its preferred speed. The ψ^w vector is the force to create random-error in the orientation of the agents.

Fig 2. Definition of turning direction.

(A) Prey individuals i and j are members of a flock with average heading ρ_α and face different conditions to escape the predator p. The headings (unit vectors) of the individuals are represented by $\widehat{h}$. The red angles θ_p are the turns away from the heading of the predator and the blue angles θ_a, the turns towards the heading of the flock. Based on the relative heading of individuals i and j to the headings of the flock and the predator, j needs to escape while aligning with its flockmates, while i is in-conflict, since it needs to turn away from the flock in order to escape. (B) The change of heading of an individual between consecutive time steps (dt) is represented by Δθ. Its sign shows whether the individual turned towards the flock (same sign with θ_a), away from the predator (same sign with θ_p), towards neither or both (depending on its escape conditions).

Fig 3. Comparison of real (top row) and simulated (bottom row) flocks of 8 pigeons.

The shown data are representative of the majority of distributions; for a detailed comparison across more flocks see S1 Table. (A-B) The vertical dotted lines show the mean of each distribution. (A) The distributions of individual speed throughout a flight of real pigeons (A1) and a simulation (A2). (B) The distributions of nearest-neighbor distance throughout a flight (B1) and a simulation (B2). (C) Density of the nearest-neighbor positions in the coordinate system of each focal individual, based on the bearing angle and distance to the nearest neighbor (m), estimated for a real flight (C1) and a simulation (C2). The area of highest density is expected to be around the mode of the distribution of nearest neighbor distance. The triangle represents the position of the focal individual heading to the north direction.

Fig 4. Turning direction frequencies of flock members.

The percentage of turns towards the four turning directions at consecutive time-steps (for individuals under ‘conflict’ and ‘non-conflict’ scenarios) as a function of distance between them and the predator, across the empirical data and two simulation experiments. (A) Empirical data of Sankey et al. (2021) [24]. (B) Simulated data with predator avoidance that is independent of the distance between the predator and the prey individuals (modeled as in Fig 1). (C) Data from control simulations where the prey does not react to the predator.

Fig 5. Distance-dependency in simulated flocks with and without predator avoidance.

(A) The angle of the flock members’ headings relative to the predator’s at each distance-cluster. The boxes include the 50% of the distribution and the horizontal line shows its median. When pigeon-agents react to the predator, the measurement increases showing that the flock is turning away from the predator. (B) The frequency that the escape-direction of each flock-member is changing direction (Eq 7). The height of each bar shows the mean value of all individuals per distance cluster and the error bar shows one standard deviation above this mean. The escape direction remains more stable when the flock is turning away from the predator. (C) Consensus in escape direction across a flock at each sampling point (Eq 8). More flock members have the same escape direction closer to the predator in simulations with predator-avoidance.

Fig 6. Progression of a collective escape.

(A) The tracks of a simulated flock of 10 pigeons under attack. For simplicity, we present the part of the track when the predator is within 40 meters distance from the flock (excluding larger distances shown in previous plots). The points represent the position of single pigeons per time step of 0.2 seconds. The filled black rhombi show the position of the predator at 7 discrete time points. The numbers represent the link between the position of the predator and prey in time. (B) Positions of the pigeon-agents at time points 1 to 6 (0 to 9.8 seconds). Their color shows the distance to the predator of each individual (according to the clusters of Fig 4). Arrows represent the heading of each agent. The shadow of the arrows shows the escape direction of each agent at that time point. (C) The effect of centroid-attraction and alignment forces across the flock during the respective time points (density map of $F_f^{\psi }\left(t\right)$, Eq 9). The point represents the position of all individuals in their local reference frame and the dotted line shows the average heading of the flock at that time point.

Fig 7. Effect of coordination forces on ‘in-conflict’ flock members within 30 meters to the predator in HoPE.

The density of turning-attraction (low center) and alignment (high center) forces (Eq 9) acting on the coordinate system of pigeon-agents that are in-conflict during the pursuit sequence shown in Fig 6. The triangle represents the position of each focal individual and the dotted line and arrow its heading. The predator sign represents the turning direction of predator avoidance. Alignment and centroid-attraction have opposite effects on turning direction relative to the agents’ headings and predator-avoidance is mostly in accordance with the centroid-attraction direction.