Topological interactions account for border dynamics of murmurations and transit flocks

Abstract

Murmurations are one of nature’s most striking examples of collective behaviour. Despite extensive research the dynamics of individuals at the borders of these flocks remain poorly understood. These dynamics result in two unexplained phenomena: the tendency of birds to remain longer at the border than the way internal birds keep their position inside the flock; and the hardness (sharpness) of the borders. It has been suggested that the border dynamics can be attributed to selection pressures for advantageous behaviours such as prey evasion. Here with the aid of stochastic models we show that the observed border dynamics are an accidental but potentially advantageous by-product of topological interactions (when birds interact with a fixed number of neighbours) and that they do not arise with metric interactions (when birds coordinate with neighbours based on spatial distance). I find support for these predictions in an analysis of pre-existing telemetry data for flocks of jackdaws (Corvus monedula).

1. Introduction

Cavagna et al. [1] reported that starlings (Sturnus vulagis) in the interior of flocks (murmurations) change positions with their neighbours in time because of random fluctuations in their individual motions. That is, the mutual rearrangement of internal individuals can be explained in terms of a diffusion mechanism. A more complex dynamical process occurs at the border of the flocks, since birds on the borders swap position with their neighbours less often than internal birds do, resulting in hard (sharp) borders [1]. Cavagna et al. [1] suggested that these flocks may self-organize out of the selfish tendency of individuals not to stay at the border, a situation reminiscent of Hamilton’s [2] ‘selfish herd’ scenario (but see [1,3,4]). Cavagna et al. [1] noted rightly that ‘border dynamics [are] very fascinating and important’. Nonetheless, although border dynamics have been incorporated in models of flocking [5,6], no fundamental progress has been made in the understanding of border dynamics since the paper by Cavagna et al. [1].

Here, with the aid of stochastic models, we show that if, as in the case of starlings and jackdaws (Corvus monedula) [7–10], birds interact topologically (i.e. when birds interact with a fixed number of neighbours), then their mutual alignment will be strongest in the borders of the flock. That is, the flock’s ability to self-organize does not result out of the individual selfish tendency not to stay at the border, à la Hamilton’s [2] selfish herd scenario. It appears instead to be an accidental but potentially advantageous by-product of topological interactions, as increased alignment with their neighbours’ orientation could reduce the chance of predation through information sharing [11] or collective escape [12]. I find support for this prediction in an analysis of pre-existing telemetry data for flocks of jackdaws; a prediction which is shown to underpin both the observed tendency of birds to remain longer at the border than the way internal birds keep their position inside the flock and the hardness (sharpness) of the borders. I then show how the tendency for the density of birds at the borders to be greater than near the centre of the flock is a by-product of the effective cohesive forces that bind flocks together [13]. I also show how birds on the borders make the dominant contribution to the anisotropy of the nearest-neighbour distribution [7,14], i.e. make the dominant contribution to the higher probability of finding a neighbour next to, rather than in front of or behind, a focal bird.

2. Methodology

Here, the positions, $x_i$, and velocities (relative to the flock’s mean velocity), $u_i$, of birds in one-dimensional flocks are assumed to evolve jointly as a Markovian process described by the stochastic differential equations,

$${\displaystyle {\displaystyle d{u}_i=-\frac{u_i}{T}dt-k_1{x}_{i}dt-k_2\sum _j\left(x_i-x_j\right)dt+\sqrt{\frac{2{\sigma }^2}{T}}d{W}_i,}}$$ (2.1a)

$${\displaystyle d{x}_i=u_{i}dt,}$$ (2.1b)

where the subscripts denote individuals within the flock, T is a velocity correlation timescale, $\sigma$ is a velocity scale, $k_1$ and $k_2$ are effective spring constants, the summation is over the ith’s bird nearest neighbours, $d{W}_i\left(t\right)$ is an incremental Wiener process with correlation property $\overline{d{W}_i\left(t\right)d{W}_j\left(t+\tau \right)}=\delta \left(\tau \right){\delta }_{ij}dt$.

Models closely akin to equation (2.1) reproduce faithfully numerous properties of flocking [13,15]. The first term on the right-hand side of equation (2.1a) is a memory term that causes velocity fluctuations to relax back to their mean value. The second term on the right-hand side of equation (2.1a) is a mean acceleration towards the centre of the flock that keeps the flock intact. As observed in the case of jackdaws [13], the simulated birds behave on the average as if they are trapped in elastic potential wells. This is an emergent property of flocking [13]. The third term on the right-hand side of equation (2.1a) describes spring-like interactions between the ith bird and its neighbours, as evidenced in jackdaws wherein discrete pairs of individuals are tied together by spring-like effective forces [8]. As the strength of this coupling increases, so does the extent to which the motions of interacting individuals are coordinated. In the case of topological interactions the summation is over the nth nearest neighbours, whereas in the case of metric interactions the summation is over neighbours within the range of the interaction. For the case of topological interactions, simulated birds interact with their four nearest neighbours rather than with approximately seven of them as is the case for starlings and jackdaws [7,8]. This was done so as not to conflate the peculiarities of the border dynamics with the apparent ubiquity of the degree to which flocking birds interact topologically, i.e. it was done to show that the two characteristic properties of flocking are not related to one another. The fourth term on the right-hand side of equation (2.1a), the noise term, represents fluctuations in the resultant internal force that arise partly because of the limited number of individuals in the grouping and partly because of the non-uniformity in their spatial distribution. Examples of the simulated flight patterns are shown in the electronic supplementary material, data.

Model predictions are compared with pre-existing data for: the three-dimensional flight patterns of individual birds within flocks of starlings (Sturnus vulgaris) so-called ‘murmurations’ [7,10,16]; the transit flocks of jackdaws (C. monedula) (formed when jackdaws return to their roosting sites in the evening); and the collective anti-predator mobbing flocks of jackdaws [14]. These flight patterns were captured using stereometric photography and computer vision techniques.

3. Results

Using the methodology of [13], it can be shown that the positions and velocities of the birds are predicted to be statistically stationary, Gaussian with mean zero and homogeneous when birds are none interacting and when each bird interacts with all other birds in the flock. In both cases, the velocities of the birds are uncorrelated, i.e. $〈u_i{u}_j〉={\sigma }^2{\delta }_{ij}$. More generally, model predictions can only be determined with the aid of numerical simulations. Such simulations reveal that when each bird interacts only with its nearest neighbours, then the motions of the birds become correlated, and that this correlation is maximal in the borders of the flocks (figures 1 and 2). The correlations are largest when each bird interacts with one-half of the birds in the flock. This maximal correlation is unlikely to be attained in large flocks because it would necessitate that each bird keeps track of the movements of many other birds. For this reason, attention is hereafter focused on the case where each bird interacts with just a few of its nearest neighbours, as in the case of starlings and jackdaws [7–10]. By way of contrast, when the interactions are metric, i.e. when each bird interacts with all other birds within a prescribed interaction range, then the velocities of the birds are predicted to be uncorrelated throughout the flock (figure 1).

Figure 1.

Topological interactions are predicted to result in more strongly correlated movements in the borders of flocks. The average correlation between the velocity of the focal bird i and its nearest neighbour, bird $j\ne i\mathit{},$${\displaystyle ⟨u_i{u}_j⟩}$, is shown as a function of the position of bird i relative to the centre of the flock. ${\sigma }_x$ is the root-mean-square size of the flock, and ${\sigma }_u$ is the root-mean-square velocity of a flocking bird. Predictions are shown for flocks containing n = 10, 20 and 30 individuals. Each bird interacts with its four nearest neighbours. Predictions were obtained by numerically integrating the stochastic model, equation (2.1) with $k_2$ = 2 and with all other parameters set to unity. Shown for comparison (black dashed line) are predictions for a flock containing n = 10 individuals that have metric interactions with range ${\sigma }_x/4$.

Figure 2.

Increasing the number of topological interactions is predicted to result in more strongly correlated movements in the borders of flocks. The average correlation between the velocity of the focal bird i and its nearest neighbour, bird $j\ne i\mathit{},$${\displaystyle ⟨u_i{u}_j⟩}$, is shown as a function of the position of bird i relative to the centre of the flock. ${\sigma }_x$ is the root-mean-square size of the flock, and ${\sigma }_u$ is the root-mean-square velocity of a flocking bird. Predictions are shown for three flocks each containing n = 100 individuals with two, four or seven nearest-neighbour interactions. Predictions were obtained by numerically integrating the stochastic model, equation (2.1) with all parameters set to unity.

The predicted emergence of correlated velocities in the borders of flocks of birds that interact topologically finds support in an analysis of pre-existing telemetry data for flocks of jackdaws containing approximately 100 birds (figure 3). It is also consistent with the analysis of Cavagna et al. [17], who reported that the velocities of individual birds in starling flocks relative to the flock mean velocity reveal the presence of large domains wherein bird motions are nearly parallel. These domains seem to occur in and around the borders of the flock. Further indirect evidence for the model predictions comes from the fact that flocks of topologically interacting birds have hard borders [7] and from the analyses of et al. [1], reported on the border survival probability, $P\left(t\right)$, which is the probability that a bird initially at the border remains at the border for a time greater than t, i.e. it is the probability that the positional swap with a neighbour does not occur within a time t. The interior survival probability is defined in an exactly analogous way and was found to decay much faster than the border one. Moreover, Cavagna et al. [1], reported that the mutual rearrangements of internal individuals can be explained in terms of a diffusion mechanism. These observations are consistent with the predicted occurrence of correlated movements in the borders, and the uncorrelated (or weakly correlated) movements in the internal birds (figures 1 and 2). Indeed, model predictions for survival probabilities resemble observations (figure 4); albeit that the decay of the border survival probability is convex rather than concave, as observed [1]. A concave decay is predicted to occur if each bird interacts with its nearest neighbours but only if they are sufficiently close, as seems to be the case in transit flocks of jackdaws [9].

Figure 3.

Topological interactions result in more strongly correlated movements in the borders of flocks of jackdaws (C. monedula). The average correlations between the velocity each bird i and its nearest neighbour $j,$ ${\displaystyle C=⟨u_i{u}_j⟩/⟨u_i{u}_i⟩}$, are shown as a function of the position of bird i relative to the centre of the flocks. Results are shown for transit flocks wherein birds interact topologically and for mobbing flocks wherein birds have metric interactions [9]. Velocities are measured relative to the centre of mass motion of the flocks. Relative to their centre of mass, they are statistically stationary, as are murmurations and the simulated flocks. The dashed lines, added to guide the eye, are linear least squares regressions with slopes $0.01\pm 0.002$ and $-0.006\pm 0.008$. In contrast with the transit flock, for the mobbing flock the least square regression is not significantly different from a flat line. Data are taken from Ling et al. [9] and electronic supplementary materials, DataS2 and DataS3. The flocks contain approximately 100 birds.

Figure 4.

Border (■) and interior $\left(\mathbf{\Delta }\right)\mathit{}$survival probabilities. Predictions are shown for flocks containing 10 individuals. Each bird interacts with its four nearest neighbours. Predictions were obtained by numerically integrating the stochastic model, equation (2.1) with $k_2$ = 2 and with all other parameters set to unity.

As observed [7], the model predicts that the density of birds near the border of a flock is greater than in its interior (figure 5). Note, however, that in contrast with the former predictions, this prediction is not contingent on the mutual interactions being topological, as it is also found, albeit to a lesser degree, when individuals are non-interacting (figure 5).

Figure 5.

Predicted average nearest-neighbour distance as a function of the distance from the nearest border (solid line). As observed [7], this distance increases from the border to the centre. ${\sigma }_x$ is the root-mean-square size of the flock. Predictions are shown for flocks containing 10 individuals. Each bird interacts with its four nearest neighbours. Predictions were obtained by numerically integrating the stochastic model, equation (2.1) with $k_2$ = 2 and with all other parameters set to unity. Shown for comparison are predictions for non-interacting birds with $k_2$ = 0 (dashed line).

Ballerini et al. [7] were the first to report on the angular distribution of the nearest neighbours in bird flocks. To do this, for each bird (starling) in the flock, they first found the vector to its nearest neighbour and then measured the angle, $\theta$, between this vector and the average direction of motion of the flock. If the birds were distributed isotopically within the flock, then the distribution of ${\displaystyle \mathrm{c}\mathrm{o}\mathrm{s}\theta }$ would be constant and equal to one-half. For some flocks, the distribution is peaked at ${\displaystyle \mathrm{c}\mathrm{o}\mathrm{s}\theta =0}$, indicating that the nearest neighbours were more likely to be in the plane perpendicular to the average direction of motion; in other cases, the distribution was bimodal indicating a more structured distribution of the birds in space. In all cases, there was a lack of neighbours along the average direction of motion, i.e. P${\displaystyle \left(\mathrm{c}\mathrm{o}\mathrm{s}\theta \right)<1/2}$ for ${\displaystyle \mathrm{c}\mathrm{o}\mathrm{s}\theta \sim \pm 1}$. Ling et al. [14]. found directly analogous behaviours in transit flocks of jackdaws, reporting that there is a higher probability of finding a neighbour next to, rather than in front or behind a focal bird.

The results of numerical simulations reveal that such anisotropy arises spontaneously when birds interact topologically (figure 6). Model predictions are consistent with the observed form of the anisotropy when the mutual interactions are strongest in the average direction of travel (figure 6). In this case, birds in the borders of the flocks are predicted to make the largest contribution to the anisotropy. Despite appearances this modelling is not in conflict with the observations of [14] who reported that the long-range attraction in the direction perpendicular to the direction of travel is stronger than that along it. This is because birds within a flock behave on average as if they are trapped in an elastic potential well [13]. That is, each bird effectively behaves as if it is bound to the flock by a force that on average increases linearly as the distance from the flock centre increases. This results in apparent spring-like interactions between birds even when they are not interacting. For transit flocks of jackdaws, these are largest in the direction perpendicular to the direction of travel.

Figure 6.

Nearest-neighbour angular distribution. Predictions are shown for flocks containing 20 individuals. The three-dimensional movements of birds within three-dimensional flocks were simulated using three independent versions of the stochastic model (equation .21). Each bird interacts with its four nearest neighbours. Angular anisotropy is seen to be due almost entirely to birds on the borders of the flock. Predictions were obtained by numerically integrating three versions of the stochastic model, equation (2.1) with $k_2$ = 5 for the x direction and with $k_2$ = 2 for the y and z directions and with all other parameters set to unity. Angles of the vectors between each bird and its nearest neighbour were measured relative to (1,0,0) which effectively makes the x direction the direction of travel because then the angles are measured relative to the average flight direction as in [7]. ‘Border’ birds are the birds with maximal and minimal positions in the x, y and z directions.

The results of numerical simulations also show that bimodal nearest-neighbour angular distributions, as reported by Ballerini et al. [7], arise when in addition to the mutual long-range attraction between the birds there is also a sufficiently strong short-range repulsion.

4. Discussion

Cavagna et al. [1] suggested that the tendency of birds to remain longer at the border than the way in which internal birds keep their position inside the flock arises because ‘individuals on the border balance the tendency to exchange neighbours owing to motion, the availability of void space outside of the flock and the reluctance of internal neighbours to give up a more favourable position’. Here it was shown that the tendency of birds to remain longer at the border is a by-product of topological interactions and need not be attributed to any additional distinct behavioural responses resulting, for example, from selection pressures for advantageous behaviours such as predator evasion. Moreover, there is no requirement for birds to determine their depth within the flock from their visual field.

Ballerini et al. [7] reported that the simplest models of self-organized motion, which assume isotropic interactions between individuals, do not reproduce the observed angular anisotropy observed in flocks of starlings. This led them to suggest that the anisotropy of the nearest-neighbour distribution is an explicit consequence of the anisotropic character of the interaction itself, which should be incorporated into models if they are to reproduce the observed behaviour. Ballerini et al. [7] listed several biologically plausible candidate reasons for the lack of nearest neighbours in the front and back. It could, for example, be a consequence of starlings having lateral visual axes and a blind rear sector, or it could be a consequence of individuals reducing the chance of collisions by trying to keep more space between themselves and the individuals in front of them. Here, we showed that the angular anisotropy does, in fact, arise spontaneously when birds interact topologically. Nonetheless, the factors identified by Ballerini et al. [7] could be incorporated into the modelling and may modify the emergent anisotropic character that goes beyond the purely topological approach. We showed that the probability of finding a neighbour next to, rather than in front of or behind a focal bird is much higher in the borders of the flock than it is in the interior of the flock. This prediction awaits experimental verification but illustrates once again how advantageous behaviours can, in principle at least, arise accidentally and need not be attributed to selection. The emergent behaviour will be advantageous if, as expected, the birds have a blind visual angle, because then information transfer will be unidirectional if birds travel one behind the other, whereas when moving side by side the birds can see each other, which enables bidirectional information transfer. Information produced by environmental perturbations, such as the presence of predators, could then be transferred from the boundary to the bulk of the flock; a scenario investigated by Cavagna et al. [16].

The foregoing analyses call for a re-evaluation of the potential importance of biological drivers underlying the behaviour of birds in the borders of flocks. An important open question is how the border and interior dynamics emerge in flocks containing of the order of 1000 birds, as recorded, for instance, by Belarini et al. [7] and by Ling et al. [8]. Currently, such simulations are computationally prohibitive. Nonetheless, the key model prediction, namely, correlated motion of birds in the borders, was supported by analyses of jackdaw flocks containing approximately 100 birds.

The situation with flocking birds uncovered herein appears to be analogous to that of swarms of the non-biting midge Chironomus riparius and Anopheles gambiae mosquito; those advantageous emergent properties arise as accidental by-products of swarming behaviours rather than being the result of selection pressures for advantageous properties [18,19]. More pertinently, laboratory swarms of the non-biting midge C. riparius consist of a core ‘condensed’ phase surrounded by a dilute ‘vapour’ phase [20]. These two phases maintain distinct macroscopic properties even though individual insects pass freely between them. The emergence of such phases is predicted by models of non-interacting individuals [21]. As with flocking birds, novel border dynamics emerge from the same dynamics that govern the interior of the collective.

Finally, in the electronic supplementary material we show that topological interactions can, in fact, be attributed to (reinterpreted as) each bird being attracted towards the emptiest region of the space in their vicinity. The analysis is the first to account for why each starling within a murmuration interacts with just seven of its neighbours on average [7], as do jackdaws in transit flocks that have not formed lifelong pair bonds [8]. This warrants further investigation.

5. Summary

We showed that the distinctly different behaviours of interior birds and border birds arise spontaneously whenever birds interact topologically. In other words, the flock’s ability to self-organize does not result out of the individual selfish tendency not to stay at the border, à la Hamilton’s [2] selfish herd scenario. It appears instead to be an accidental but potentially advantageous by-product of topological interactions. The findings underscore the necessity of combining physics-based analysis and biological insight for understanding the dynamics of collective behaviour [22]. They are consistent with Sankey et al. [3] and Sankey [4], who argued against selfish herd dynamics in bird flocks.

Ethics

This work did not require ethical approval from a human subject or animal welfare committee.

Data accessibility

This study utilizes data from Ling et al. Behavioural plasticity and the transition to order in jackdaw flocks [9]. The base code is available as electronic supplementary material.

Supplementary material is available online [23].

Declaration of AI use

I have not used AI-assisted technologies in creating this article.

Conflict of interest declaration

I declare I have no competing interests.

Funding

The work at Rothamsted forms part of the Smart Crop Protection (SCP) strategic programme (BBS/OS/CP/000001) funded through the Biotechnology and Biological Sciences Research Council’s Industrial Strategy Challenge Fund.