Migratory birds in flight often group into V-like formations, that is, they tend to self-organize into a flight formation having very few individuals in lead positions while the others group behind them near the edges of imaginary trailing V’s whose vertices are the lead birds. While it seems to be a consensus that group flight provides greater protection against predators, the reasons for the particular grouping into laterally slanted, nearly straight lines have elicited two distinct lines of explanation, following the seminal works in  and , respectively.
The first one has evoked the aerodynamics of bird flight, whose essence is that each flying individual creates an upwash region behind it, just off the tips of its wings, such that another individual benefits greatly (in terms of requiring less exertion during flight) if it places one of its wings in that region. For relatively large birds, which are the ones that have been observed to group into V-like formations most often, this seems to be the main reason for formation flight [8, 6, 2, 19, 22]
. The second, competing line of reasoning has been that flying in a somewhat skewed position relative to the bird in front is crucial for an individual’s orientation, in addition to allowing unhindered visual communication and therefore helping avoid collisions. For relatively small birds, the aerodynamic benefits are less relevant and, moreover, the observed lateral and longitudinal separations between nearest birds seem to be correlated in a way that strongly supports this vision-related explanation.
In this study, we consider simple artificial birds and address the question of whether positioning rules exist that, during flight, can lead a flock to settle into some stable V-like formation. Such rules should be founded on a blend of the basic explanatory trends of placing more weight on aerodynamics- or vision-related justifications, thus consonant with the current belief that the traits that allow natural birds to take advantage of one or the other benefit evolved concomitantly . They should, in addition, be inherently distributed, so that only sensory information is to guide each bird’s course of action, and also motivated by the birds’ innate drive at flying as a flock. As we demonstrate in the remainder of this letter, rules with these characteristics do exist that are both robust (in the sense of ensuring that stable configurations can be expected to be reached) and flexible (allowing for a rich variety of V-like formations).
Our work is preceded by important related contributions, notably by the simple rules in , which in essence say that birds should avoid colliding with one another while attempting to maintain the same pace as the nearest birds as well as seeking to be positioned as near their center as possible. Such rules are not meant to give rise to V-like formations, only to the flight as a group itself. Adding to them the further rule that a bird should strive to keep some portion of the visual field unobstructed is reported to have caused some of the birds to coalesce into fragmented V-like formations , but the whole approach seems to have remained oblivious to the sensory input that birds derive from favorable aerodynamics. We also mention the fuzzy rules introduced in [10, 11], but they fail to induce V-like formations, except in the case of very special initial conditions. Less related to this study but very relevant nonetheless are mathematical analyses of flocking, both in the sense of  (cf., for example,  and , the latter partly based on ) and under the assumption of a V-formation (e.g., ).
2 Positioning rules
Our own set of rules is very succinct, comprising only three rules, each related to one of the three guiding principles discussed earlier (birds should flock, be afforded some unobstructed view in the direction of flight, and benefit from regions of upwash). They are the following.
- Rule 1 (coalescing rule):
Seek the proximity of the nearest bird.
- Rule 2 (gap-seeking rule):
If Rule 1 is no longer applicable, seek the nearest position that affords an unobstructed longitudinal view.
- Rule 3 (stationing rule):
Apply Rule 2 while the view that is sought is not obtained or the effort to keep up with the group decreases due to increased upwash.
These three rules are an informal expression of our understanding of how each of the guiding principles is to influence a bird’s actions. First of all, Rules 1 and 2 imply that we make a distinction between two modes of behavior, one succeeding, or sometimes alternating with, the other: birds are to seek first the closeness of the group, and only then being afforded some clear view. Secondly, Rules 2 and 3 imply that the sensory input that keeps the bird moving toward a relative position of clear view is not only visual but also originates in the ease with which it is keeping up with the group.
Rules 1–3 require further specification before we describe our computational experiments and their results. Note, first, that both Rule 1 and Rule 2 make implicit reference to the region of space where some entity is to be sought (the nearest bird in Rule 1, the nearest position with a clear view in Rule 2). We assume, in both cases, that this region is delimited, on the plane of movement and both to the right and left of the direction of movement, by an angle . We assume also that (this is in spite of the fact that many birds in nature have visual fields delimited by larger angles, since does not delimit a bird’s visual field but the region where it is to seek the input needed by Rules 1 and 2).
In a similar vein, Rules 1 and 3 allude, also implicitly, to the existence of a closed region surrounding each bird inside which proximity is attained (Rule 1) or upwash is found (Rule 3). Given the aforementioned constraints on , and disregarding as negligible the upwash that some authors claim may be generated by a flying bird even in the space just ahead of it [20, 1], we assume that this region is as follows, with reference to Figure 1 [in this figure, as in others to come, an artificial bird is depicted as a filled circle (the bird’s body) with two protruding straight-line segments (its wings)]. If is the bird seeking the proximity of another (say ) in Rule 1, then proximity is attained if the body of is in . As for finding upwash in Rule 3, all that is needed is that any portion of bird , body or a wing segment of any size, be in while no portion is in . In this case, the optimal relative placement of and (i.e., the relative placement for maximum upwash) occurs when the lateral separation between them is , where is a bird’s wingspan, assumed the same for all birds [8, 9, 17] (the birds’ longitudinal separation inside is thought to be irrelevant for maximum upwash [12, 8, 17]). So, for optimality, and must have laterally overlapping wings.
3 An algorithm and computational results
Our simulation algorithm operates on birds and runs from some initial positioning of the birds inside a unit square in two-dimensional space. Birds are in no way constrained to flying inside this unit square, even though by Rule 1 it seems reasonable to expect that a fixed-size square exists which, at all times, encompasses all birds. They may then be thought of as inhabiting unbounded two-dimensional space.
We assume that the lateral and longitudinal directions are parallel to the original unit square’s sides. In order to move according to Rules 1–3, a bird’s position relative to the others’ must undergo lateral as well as longitudinal displacements. Except for the occurrence of these displacements, we assume that birds have no lateral speed and that they all fly at the same longitudinal speed, giving rise to a common velocity vector that we henceforth refer to as the baseline velocity. Of course, each bird’s actual velocity vector at a given instant is the sum of the baseline velocity and the vector comprising the bird’s lateral and longitudinal rates of displacement at that instant.
For the sake of increased computational efficiency during simulation, it then suffices to subtract the baseline velocity off all birds’ velocity vectors and handle displacement rates only. To an observer looking down on the birds orthogonally to the plane of movement and moving synchronously with them (i.e., at the baseline velocity), a sequence of snapshots of the group will seem like resulting from the birds’ movements relative to one another only, even though they are indeed traveling through space and capable of changing their velocity vectors, both in magnitude and in direction, through time (forthcoming Figures 3 and 4 show some of these snapshots).
Each simulation runs through time steps of fixed duration, so the handling of displacement rates may be further simplified so that only the actual displacements per time step need in fact be handled. We implement displacements by simply adding (or subtracting) a fixed amount (laterally) or (longitudinally) to a bird’s position per time step.
At each time step, a bird follows Rule 1, if applicable, by altering its position via or . If Rule 1 is not applicable, then applying Rule 2 (and controlling its application via Rule 3) requires a precise criterion for seeking “the nearest position that affords an unobstructed longitudinal view.” Let us call a gap, in the current time step, any lateral span that contains the lateral coordinate of no portion of any bird. A gap is maximal if it is contained in no other gap; maximal gaps, therefore, may be regarded as inducing longitudinal stripes of empty space, each stripe delimited by some bird’s wing tip on at least one side. Our algorithm implements the criterion required by Rule 2 as follows. Let be the bird in question and consider all maximal gaps that are at least wide (this includes the two outermost gaps, which are infinitely wide). The position that seeks is given by the nearest wing tip delimiting such a gap, provided the width- extension of that wing into the gap can be seen by without obstruction. Aiming at that position, is applied as needed, and eventually Rule 1 may once again become applicable. As for what remains unspecified of Rule 3, our algorithm implements the lateral separation by directly (i.e., Rule 2 is applied if upwash has not been found or is not optimal).
Collisions are avoided at all time steps whenever attempting a displacement. Two birds are allowed to get as close to each other as having no lateral separation between them or a longitudinal separation of . If the desired displacement infringes this, then it is not effected and the bird simply chooses randomly between a longitudinal displacement of or , which is only applied if possible. In our algorithm, this possibility of displacing a bird longitudinally by represents the only source of longitudinal deceleration.
We have conducted extensive simulations for and (the occurrence of V-like formations in large flocks is inherently difficult and is rarely observed in nature ,111Except for relatively small birds , in which case V-like formations, as we have noted, seem not to originate primarily from aerodynamics-related gains. In fact, large formations of relatively small birds are characterized by lateral separations that are inherently incompatible with , the optimum predicted by aerodynamics-based analyses. thence our moderate choices for the value of ). All simulations used and the values listed in Table 1 for the distance-related parameters. For a quantitative evaluation, we concentrate on the five indicators listed next, for which results were obtained as averages over independent simulations, each one starting at a random placement of the birds inside the unit square.
- Time for stabilization:
Number of time steps until all birds stop moving relative to one another (taken as for the simulations in which this does not happen).
- Number of lead birds:
How many birds intersect no other’s upwash regions.
- Number of unconnected groups:
A set of birds is counted as an unconnected group if no bird outside it intersects any of its own birds’ upwash regions, and conversely [e.g., the situation in Figure 4(c) contains two groups].
- Number of straight-line segments:
A straight-line segment joins a trailing bird—one whose upwash regions no other bird intersects—to either a lead bird or a “bifurcation” bird—one whose upwash regions are intersected by two other birds (e.g., the situation in Figure 2 contains two straight-line segments).
- Mean distance to nearest straight-line segments:
Each of the birds contributes one distance to the mean, unless it is a lead or “bifurcation” bird, in which case it contributes the average distance to the two nearest segments.
Except for the time for stabilization, all indicators refer to the end of the simulation.
Two snapshot sequences, in the sense discussed earlier, are shown in Figures 3 and 4. The former sequence depicts, for six time steps during the same simulation, the evolution of the birds’ positions from their initial randomly determined situation through the eventual V-like formation. The latter sequence is a sampler of the formations achieved after time steps. The averages for the five indicators are shown in Figure 5.
|Lateral size of upwash region (cf. Figure 1)|
|Longitudinal size of upwash region (cf. Figure 1)|
|Lateral displacement per time step|
|Longitudinal displacement per time step|
|Margin for longitudinal collision|
4 Discussion and further work
The plots in Figure 5 reveal that, for , the average formation into which the birds settle seems to become stable relatively quickly, and also to be evocative of the canonical V formation: one lead bird (hence one single group), and two straight-line segments. Decreasing to delays stabilization significantly while allowing for an increased number of lead birds and of unconnected groups, and consequently of straight-line segments as well.222We remark that our results for , when compared to those for , are qualitatively representative of what we have observed down through roughly , which singles out as a sort of special case. Behind all these effects are the facts that, with the smaller angle, a bird tends to have a more limited set of choices for the nearest bird needed in Rule 1, and also fewer maximal gaps for use in the implementation of Rule 2. Upon reaching stability, however, the number of birds clustered around each straight-line segment tends to be smaller, and thus so does the mean distance between the birds and the nearest segments.
Figure 5(f) offers the possibility of a closer glimpse into ten (out of the ) simulations for and . The figure indicates that, even though at this reduced angle stability is on average harder to achieve, there are simulations in which the mean distance to the nearest straight-line segments drops nearly vertiginously. This suggests, as in fact we have confirmed by examining all the data more closely, that the average time for stabilization is being influenced by some simulations that did not fully stabilize and contribute each time steps to the average.
While we believe our results to support the view that Rules 1–3 are successful in helping explain the emergence of V-like formations, we also recognize that the possibilities for further investigation are very numerous. These include a more detailed study of how different parameter values in our simulation algorithm affect the results,333For example, allowing to deviate from its purported optimal value near , or yet varying the value of , is expected to influence the various angles in the resulting V-like formations. and also the possibility of studying other algorithmic approaches to realizing the same set of rules. Some of these other approaches might, for example, introduce a new parameter to quantify the “unobstructed longitudinal view” mentioned in Rule 2; such a parameter would, essentially, substitute bounded regions of space for the maximal gaps of our current algorithm, which are longitudinally unbounded. They might also adopt a perturbation model to continually effect small fluctuations in the birds’ positions so that stability would never really be achieved; interesting questions in this case would be whether V-like formations would still occur, how persistent they would be, and how they would change with time.
To finalize, we remark that Rules 1–3 themselves may be put through ever more stringent tests. One clear possibility is to consider them in three-dimensional space and to investigate whether roughly planar V-like formations still emerge. We find it at this point unclear whether the three rules will hold unchanged, since a three-dimensional proximity model will be needed for use with Rule 1, and likewise three-dimensional models of vision obstruction and of flight aerodynamics for use with the other two rules.
We acknowledge partial support from CNPq, CAPES, and a FAPERJ BBP grant.
-  M. Anderson and J. Wallander. Kin selection and reciprocity in flight formation? Behavioral Ecology, 15:158–162, 2004.
-  J. P. Badgerow. An analysis of function in the formation flight of Canada geese. The Auk, 105:749–755, 1988.
-  C. J. Cutts and J. R. Speakman. Energy savings in formation flight of pink-footed geese. Journal of Experimental Biology, 189:251–261, 1994.
-  G. W. Flake. The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation. The MIT Press, Cambridge, MA, 1998.
-  L. L. Gould and F. Heppner. The vee formation of Canada geese. The Auk, 91:494–506, 1974.
-  F. R. Hainsworth. Precision and dynamics of positioning by Canada geese flying in formation. Journal of Experimental Biology, 128:445–462, 1987.
-  F. Heppner. Avian flight formations. Bird-Banding, 45:160–169, 1974.
-  D. Hummel. Aerodynamic aspects of formation flight in birds. Journal of Theoretical Biology, 104:321–347, 1983.
-  D. Hummel. Formation flight as an energy-saving mechanism. Israel Journal of Zoology, 41:261–278, 1995.
-  I. Lebar Bajec, N. Zimic, and M. Mraz. Fuzzifying the thoughts of animats. In Proceedings of the 10th International Fuzzy Systems Association World Congress, volume 2715 of Lecture Notes in Computer Science, pages 195–202, 2003.
-  I. Lebar Bajec, N. Zimic, and M. Mraz. Simulating flocks on the wing: the fuzzy approach. Journal of Theoretical Biology, 233:199–220, 2005.
-  P. B. S. Lissaman and C. A. Shollenberger. Formation flight of birds. Science, 168:1003–1005, 1970.
-  R. Olfati-Saber. Flocking for multi-agent dynamic systems: algorithms and theory. IEEE Transactions on Automatic Control, 51:401–420, 2006.
-  R. Olfati-Saber and R. M. Murray. Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49:1520–1533, 2004.
-  J. M. V. Rayner. Fat and formation in flight. Nature, 413:685–686, 2001.
-  C. W. Reynolds. Flocks, herds, and schools: a distributed behavioral model. Computer Graphics, 21:25–34, 1987.
-  P. Seiler, A. Pant, and K. Hedrick. Analysis of bird formations. In Proceedings of the 41st IEEE Conference on Decision and Control, volume 1, pages 118–123, 2002.
-  P. Seiler, A. Pant, and K. Hedrick. A systems interpretation for observations of bird V-formations. Journal of Theoretical Biology, 221:279–287, 2003.
-  J. R. Speakman and D. Banks. The function of flight formations in greylag geese Anser anser; energy saving or orientation? Ibis, 140:280–287, 1998.
-  T. Sugimoto. A theoretical analysis of formation flight as a nonlinear self-organizing phenomenon. IMA Journal of Applied Mathematics, 68:441–470, 2003.
-  J. Toner and Y. Tu. Flocks, herds, and schools: a quantitative theory of flocking. Physical Review E, 58:4828–4858, 1998.
-  H. Weimerskirch, J. Martin, Y. Clerquin, P. Alexandre, and S. Jiraskova. Energy saving in flight formation. Nature, 413:697–698, 2001.