Anisotropic interaction rules in circular motions of pigeon flocks: An empirical study based on sparse Bayesian learning

Coordination shall be deemed to the result of interindividual interaction among natural gregarious animal groups. However, revealing the underlying interaction rules and decision-making strategies governing highly coordinated motion in bird flocks is still a long-standing challenge. Based on analysis of high spatial-temporal resolution GPS data of three pigeon flocks, we extract the hidden interaction principle by using a newly emerging machine learning method, namely the sparse Bayesian learning. It is observed that the interaction probability has an inflection point at pairwise distance of 3–4 m closer than the average maximum interindividual distance, after which it decays strictly with rising pairwise metric distances. Significantly, the density of spatial neighbor distribution is strongly anisotropic, with an evident lack of interactions along individual velocity. Thus, it is found that in small-sized bird flocks, individuals reciprocally cooperate with a variational number of neighbors in metric space and tend to interact with closer time-varying neighbors, rather than interacting with a fixed number of topological ones. Finally, extensive numerical investigation is conducted to verify both the revealed interaction and decision-making principle during circular flights of pigeon flocks.

Figure 1

Two-dimensional (2D) projection of the circular movement in flock $A$. To avoid confusion, only trajectories of three pigeons in the flock are exhibited as an illustration.

Figure 2

Fitting errors of individual positions (IP) and filtered circular centers (FCC) corresponding to distinct recovered durations. Due to the fact that a flock of pigeons interact dynamically with switched neighbors, the fixed dictionary matrix cannot capture the time-varying interindividual interactions for a longer period. Thus, errors grow larger with increasing recovered durations.

Figure 3

Degree distribution and connectivity probability of interaction networks based on the SBL method for individual positions (IP), circular centers (CC), and filtered circular centers (Filter). (a) In and out degree distributions of the pigeon flocks. For each flock, individuals typically interact with a limited number of neighbors and never interact with all the other members. (b) Connectivity probability of the interaction networks. Durations indicate the number of consecutive instants of interaction networks merging into a union. Connectivity probability is calculated as the proportion of connected situations. Clearly, connectivity probability of both directed and undirected cases grow fast along increasing duration.

Figure 4

(a) Probability distribution of pairwise metric distances of interacting neighbors (solid lines) and all the other individuals (dash lines) corresponding to the three pigeon flocks. The interaction probability distribution evolution curves have an inflection point, after which they decay strictly along increasing pairwise metric distances, whereas for the case of remainders including both interacting and noninteracting individuals, the curves have a shifting in the positive $x$ axis, which means that interacting neighbors detected based on SBL are closer to the focal one. (b) Position distribution of interaction in flocks $A,B$, and $C$. The red arrow indicates the velocity direction, and colors demonstrate the normalized interaction frequencies. It is observed that individuals tend to interact with neighbors located right or left, but not along their velocity directions. (c) Position distribution of remainders in flocks $A,B$, and $C$. Analogously, we show the 360-degree position distribution of the remainders. It is observed that, on average, all the others locate on sides during circular motions.

Figure 5

Probability distributions of the system status, i.e., the directional index of the entire flock. Therein, the sign “+” and “$-$” represent the individual counterclockwise and clockwise rotations, respectively. With the reduction and increase of the index values, the entire flock gradually changes rotational direction from counterclockwise to clockwise and the reverse, respectively. Significantly, it is observed that the spontaneous changes of rotational directions follow a Gaussian distribution.

Figure 6

Comparison between experimental data and numerical study results. 100-s long (1000 time steps) segment of flight trajectories from flock $A$. (a)–(c) The distances between the average circular centers and loft, average curvature radius, and distances between the average positions and loft, respectively. If the pigeon flock flies relatively straight for three consecutive instants, the average curvature radius will grow larger in synchronized manner with peaks in (a) and (b). The evolution tendency in numerical study is well consistent with the empirical data.