Whose energy cost would birds like to save? a revisit of the migratory formation flight

Line formation of migrating birds is well-accepted to be caused by birds exploiting wake benefits to save energy expenditure. A flying bird generates wingtip trailing vortices that stir the surrounding air upward and downward, and the following bird can get a free supportive lift when positioned at the upward airflow region. However, little to no attention has been paid to clarifying birds’ interests in energy saving, namely, do birds intend to reduce their individual energy consumption or the total energy of the flock? Here, by explicitly considering birds’ interests, we employ a modified fixed-wing wake model that includes the wake dissipation to numerically reexamine the energy saving mechanism in line formation. Surprisingly, our computations show that line formation cannot be explained simply by energy optimization. This remains true whether birds are selfish or cooperative. However, line formations may be explained by strategies optimizing energy cost and either avoiding collision or maintaining vision comfort. We also find that the total wake benefit of the formation attained by selfish birds does not differ much from that got by cooperative birds, the maximum that birds can attain. This implies that selfish birds are still able to fly in formation with very high efficiency of energy saving. In addition, we explore the hypothesis that birds are empathetic and would like to optimize their own energy cost and the neighbors’. Our analysis shows that if birds are more empathetic, the resulting line formation shape deviates more from a straight line, and the flock enjoys higher total wake benefit. Author summary Migratory birds can achieve remarkable performance and efficiency in energy exploitation during annual round-trip migration flight. Theoretical and experimental results have shown that this might be achieved because birds fly together in formation with specific shapes, e.g. the noticeable V formation, to utilize the aerodynamic benefits generated by their flock mates. However, it is still unclear whether energy-guided behavior indeed can lead to these formations. We show that the special formation adopted by migratory birds cannot be explained purely by the energy exploitation mechanism, and that birds’ vision performance and collision avoidance very likely also play important roles in the formation emergence. Our results imply that birds fly together in formation because of energy saving, but the specific shape of the formation depends on non-aerodynamic reasons. The research provides further understandings of the emergence of migratory formation and the energy saving mechanism of animal groups. It may also indicate that wing flapping, currently not considered, has an important effect on the way birds exploit aerodynamic benefits from others during the formation flight.

harm the whole group, with the prisoner dilemma as a typical example. Hence, to 63 maximize the whole group survival, migratory birds may have evolved after several 64 million years to behave somehow collaboratively in order to save the energy expenditure 65 of the entire flock. Some works have unconsciously adopted the selfish bird 66 assumption [16,28,[34][35][36]45]. However, the longitudinal and/or lateral distances to the 67 front neighbor of the bird in the formation are often decided by assuming implicitly that 68 each bird only exploits the upwash from the front neighbor. This is unrealistic since 69 every bird is immersed in the wake field induced by the vortices of all other birds. 70 Though the wake from distant birds might be neglected, it is argued that a bird can still 71 get wake benefit from the back neighbor [22,30,31]. As for cooperative birds, even 72 though many papers took the energy saving of the group as the flight performance 73 index, very few discussed the optimal formation shape. This might be because the wake 74 model used in these works ignores the slow decay of the vortex strength [19,[42][43][44], and 75 predicts thus that the total energy saving is unchanged when birds slide in the flight 76 path, so that a wide range of formations are equivalent in terms of energy savings. 77 However, the wake dissipation may overturn this conclusion. Although [45] has noticed 78 the wake dissipation, the Gaussian decay model used there for the vortex dissipation 79 makes the vortex strength decays too fast with respect to experimental results [49], 80 which may have an important impact on their results. Moreover, [45] omitted the bound 81 vortex [22,31,32] that may also be important in the near field of the bird, and assumed 82 the same optimal relative position of each bird to its front bird. Since bird interests in 83 energy savings straightforwardly influence their behavior, the neglect of bird interests in 84 the existing literature renders the conclusion that birds adopt the line formation 85 because of energy saving questionable. 86 In view of these, the actual reasons behind the emergence of line formations remain 87 unexplained, and several important questions arise naturally: does the line formation 88 emerge due to migrating birds saving energy expenditure following a certain behavior 89 pattern? If not, should other factors, like visual contact and/or collision avoidance also 90 play roles in determining the formation configuration? Could birds only behave selfishly 91 in formation flight, or is cooperation necessary for birds in forming the line formation. 92 To answer these questions, we examine how the combination of aerodynamic wake 93 benefit, visual contact or collision avoidance with bird interests in energy saving, 94 determines the emergence of the line formation. This is accomplished by testing if there 95 exist stable bird position configurations that correspond to line formations and 96 maximize birds wake benefits under different behavior assumptions and/or 97 non-aerodynamic factors. Our wake model is based on the most basic and widely used 98 horseshoe model [22,28,[31][32][33][34][35]45] and also characterizes the slower vortex dissipation in 99 the close formation, closer to the real wake data [44,49]. Moreover, our wake-induced 100 model is accurate in the sense that each bird is affected by the wake induced by all 101 other birds. In this situation, it is unclear whether the line formation assumed to have 102 the same relative position of neighboring birds [16,22,28,31,36] could appear. only reduces slightly (by around 5%) compared with that of cooperative birds. Since the 115 aerodynamic wake benefit is quite complex, we speculate that birds mostly behave 116 selfishly in forming the line formation, in consideration of the trade-off between the 117 complexity of information processing and energy efficiency. Finally, taking into account 118 both vision performance and collision avoidance, we focus on an intermediate situation 119 between selfish birds and cooperative birds, where birds show empathy by also taking 120 into account the wake benefit of their neighbors. We show that if birds are more 121 empathetic, the formation shape is close to that of cooperative birds with more 122 variability, though the flock achieves higher energy efficiency. 123

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Model 125 We consider a flock of n + 1 birds in a steady-state horizontal formation flight on the 126 same plane: the flight direction, height and velocity of the formation remain constant. 127 We focus on the left echelon formation, where each bird is positioned behind and to the 128 left of its front neighbor, since echelon is the most common shape [14,21]. We assume 129 that birds have the same size and weight and nearly fly in an echelon shape. The birds 130 are ordered from front to back, with the leading bird labeled 0, and the ith closest bird 131 to the leader labeled i. Two birds are neighbors if the difference of their order is one. 132 Since only relative positions matter in the formation shape, we set the origin of our 133 frame of reference at the center of the leader. The x axis of the frame is chosen towards 134 the flying direction, and the y axis points in the right-hand direction, perpendicular to 135 the x axis. The position of bird i in the frame is denoted as Since we assume that the flock is close to an echelon shape, x i ≤ x i−1 and 137 y i ≤ y i−1 for each i = 1, 2, ..., n.

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As aforementioned, migration birds flying in a line formation may lower their energy 139 consumption by exploiting the wake phenomenon: the flight of a front bird stirs the 140 surrounding air upward and downward. The birds that position themselves properly 141 relative to their predecessor can gain beneficial free lift from the upward airflow, leading 142 to drag reduction. We treat each bird as a fixed wing and characterize the wake benefit 143 by the supportive airflow velocity. The latter is extended by a modified horse-shoe 144 vortex model, including the phenomenon of slow wake circulation strength decay [44]. 145 In detail, the wake of a constant-velocity wing consists of a finite bound vortex around 146 the wing and two semi-infinite vortices that start at the wingtips and extend 147 downstream. The vertical airflow induced by the vortices points downward right behind 148 the bird and upward in the region outboard from the wingtips. Note that there also 149 exists upward airflow at the front of the wing. For a bird i, we take the averaged 150 vertical airflow velocity over its wing induced by the wake of another bird j as the wake 151 benefit f (p ij ), with p ij = p i − p j the relative position of bird i to j [22,31,45] (See 152 Method for details). Figure 1 shows the function f (·). The wake benefit f i (p) of a bird i 153 from all other birds j in the flock, with p the vector of all birds' positions, is assumed to 154 be the sum of f (p ij ). Note that we neglect the sidewash and the rolling moment on the 155 bird induced by the non-even distribution of the vertical air velocity from the wake of 156 another bird [19], which is realistic according to [22].  [9] and see Method for notations). The wake benefit function f (p) peaks slightly before the first bird. After the first bird, it peaks around two lines y = y * = −1.340 and y = −y * = 1.340. More specifically, in the fourth quadrant ((−∞, 0) × (−∞, 0)), around the line y = y * and along the streamwise direction (negative x direction), f (p) increases quickly to maximum, then decays very slowly. A slight deviation of the y coordinate from y * leads to a large decrease of wake benefit. When |y| ≥ |y * |, the larger the lateral distance of birds is, the flatter f (p) is along the x direction, and the smaller the x that makes f (p) peak is. Also if |y| ≥ |2y * |, the wake benefit of the second bird varies little when its lateral distance to the first bird changes.
The maximum of f (p) in the fourth quadrant is (x * , y * ) = (−2.601, −1.340). A: The wake benefit f (x, y). B: Contour of f (p) and gradient field in the vicinity of the leader. The arrow represents the local direction at which the wake benefit function increases. C: Slice of wake benefit f (x, y)-Front view. D: Slice of wake benefit f (x, y)-Left view for negative y.
the total wake benefit J(p), the sum of the wake benefit f i (p) of all birds (including the 164 leader), are maximized. Birds could also show empathy to the birds close to them; this 165 constitutes an intermediate between purely selfish and fully cooperative. In this The barrier region and collision alert region of bird j to another bird i. The orange ellipse represents the barrier of bird j. If the center of any other bird i locates within this region, then i collide with j. Hence this should be avoided in any case. The yellow region represents the collision alert region. If the center of bird i locates within this region, it is aware that a collision with bird j can happen if not moving properly. situation, each follower may optimize not just its wake benefit, but also the neighbors'. 167 In the later, we will first elaborate on selfish and cooperative birds, then discuss the 168 empathy situation shortly. 169 We also examine the effect of collision avoidance in the generation of a formation.

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This is very relevant in our context as energy-guided behavior may in some cases lead to 171 very small bird separations that actually constitute collisions. We model each bird as an 172 ellipse with the semi-major axis b l and the semi-minor axis b s , which align with the y 173 and x axis, respectively. See Fig 2A. The value of b l and b s depend on the wingspan 174 and bill-to-tail distance of the bird. As shown in Figure 2B, to avoid the collision with 175 bird j, the center of bird i must be outside the orange ellipse, whose boundary sets a 176 barrier of bird j to other birds. Others should realize the possible future collision, hence 177 a yellow region, slightly enlarged from the orange ellipse, is used to represent the 178 collision alert region of bird j. We characterize the barrier and the collision alert region 179 by a potential function B c (p ij ) (see Method for details), which satisfies several  We also test whether visual communication influences the formation emergence. As 184 argued in [37], the bird may need to align its front neighbor around one of the vision 185 axes (see Figure 3A) to avoid eye discomfort. This requirement seems too restrictive, 186 since the cone cells, which determines the vision performance, does not fade abruptly 187 around the fovea area [39,40]. Hence we assume conceptually that each bird eye has a 188 vision comfort zone (see Fig. 3A), which is a sector with spanning angle 2θ > 0, 189 centered at the eye and evenly divided by the vision axis. If the front neighbor of a bird 190 locates within the vision comfort zone, the bird feels no eye discomfort, therefore 191 behaves normally. However, if the front birds locates outside of this sector, then the 192 bird feels the vision distortion and tries to eliminate it by repositioning. To characterize 193 the vision comfort, as shown in Figure 3B, we assume that the bird eyes locate at the 194 front end of the semi-minor axis of the ellipse that represents the bird body, neglecting 195 the bill-to-eye and the eye-to-eye distance of a bird, since they are often small compared 196 to wingspan and bill-to-tail distance (For Canada goose, the ratios of the eye-to-eye and 1 9.6 , respectively) and only have effects on final results if birds are too close where 199 the formation is inconsistent with observations. We assume that the bird takes the 200 center of another bird as the position of that bird. The vision comfort is quantified by a 201 function B v (ϕ ij ), where ϕ ij is the vision angle of bird j in bird i's eyes. The function 202 B v (ϕ ij ) is constructed such that if |ϕ ij | ≤ θ, B c (ϕ ij ) = 0, implying no vision discomfort, 203 while if |ϕ ij | > θ and gets larger, the function B v (ϕ ij ) becomes more negative (see 204 Method for details). Although we allow B v (ϕ ij ) to approach −∞, this scenario cannot 205 happen in the current setting as explained in Method. We associate to each follower a 206 vision comfort function. During the long-term formation flight, the follower adjusts its 207 position to maximize the vision comfort function. The vision angle can be calculated 208 from the relative position of a bird to its front neighbor, hence B v is also the function of 209 the relative position of two neighboring birds. In the paper, the parameters for vision 210 angle are selected artificially or partly from Canadian goose [37]. Note that other 211 meticulous models for the vision comfort could also be possible, but the main effects 212 remain the same. when an object locates out of the vision comfort zone (green zone), the bird feels the vision discomfort. B: Two birds and the right vision comfort zone. If bird j locates outside green zone, bird i will try to reposition to make bird j close to this zone.
We examine whether bird formations emerge from birds optimizing their wake 214 benefits by checking if there exists some stationary point for the game or the 215 optimization problem of maximizing wake benefits. When birds are selfish, the egoistic 216 equilibrium designates the Nash equilibrium of the game of birds maximizing f i (p).

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When birds collaborate to maximize J(p), the solution of the optimization problem is 218 called the cooperative equilibrium. An equilibrium can be unstable or marginally stable; 219 birds can thus hardly maintain the corresponding formation shape due to the pervasive 220 disturbances, e.g., wind or perception noise. Hence, we are focusing specifically on the 221 stable equilibria that correspond to line formations, and not interested in others. We 222 ignore how birds adjust their positions to reach the equilibria, but we can at least 223 compute the equilibria (when they exist) numerically. In this paper, we use a 224 gradient-based method to detect the existence of such equilibria. Starting from multiple 225 initial points, we iteratively update the estimate vector p k+1 ∈ R 2n of the equilibria 226 along the gradient of the local wake benefit and the total wake benefit, respectively, 227 based on the current estimate p k . Let us recall that gradient is popular in optimization 228 March 17, 2023 7/26 problems [41] and also the computation of equilibria of games [1,4]. It represents what 229 birds would do naturally if they could feel the gradient of the individual or the total 230 wake benefit, which probably happens in the selfish situation since the upwash gradient 231 is proportional to the rolling and pitching moments felt by the bird. Even so, we note 232 that birds do not need to follow the gradient of wake benefits in actual formation flight. 233 Additionally, there may exist multiple equlibria and some are missed in our 234 computation. However, since multiple different initial bird positions are used in our 235 computations, the stability margin of the potential equilibria should be small and such 236 formation would then be easily devastated by noise and disturbances. We also examine 237 whether the stable equilibria exist if birds maximize wake benefit with or without 238 including collision avoidance or/and vision comfort. This is done by adding the gradient 239 of the barrier function B c (p) or/and the vision comfort function B v (p) to the 240 gradient-based search. Throughout the paper, the parameters related to the wake model 241 are taken from Canada geese [9]. However, this does not imply that our conclusion 242 cannot hold for other birds. In fact, as shown in Method, the scaled optimal positions of 243 birds (the ratio of bird positions over the wingspan) obtained from the pure wake 244 benefit maximization do not change with the variation of bird parameters. 245 We focus on whether there exists a stable line formation that is energy optimal.

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There are multiple mechanisms by which birds could achieve such a formation apart 247 from gradient guided action. For instance, optimal behavior could be learned over time, 248 evolution, social pressure etc. [5][6][7], but we do not consider these here. Also the 249 assumption of fixed leader position seems to violate the selfishness of birds, since as 250 shown later the leader can get more wake benefit if it is closer to the followers. However, 251 assuming that the leader also optimizes its wake benefit by adjusting position would 252 reduce the whole flock speed gradually to zero, an unrealistic situation. On the other 253 hand, it has been recognized that during migration flight, birds often take turns in 254 leading formation [46]. Hence the current leader can also enjoy the wake benefit from 255 other birds in the future. Furthermore, the situation that parenting birds always take 256 the leading position has been observed [8]. In the paper, we do not address the question 257 that how selfish birds switch positions to maximize their long-term wake benefits but 258 rather than focus on the optimal formation at a longer time scale over which no birds 259 switch position. Finally, we stress that no specific kinematics of birds is considered.

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Hence the bird positions at the obtained equilibria can be interpreted as the limit for 261 more realistic bird kinematics models.

Insufficiency of energy optimization in formation emergence
263 Surprisingly, we find that energy optimization alone is insufficient to explain the 264 emergence of realistic line formation, for both selfish and cooperative birds. The 265 simulation shows that no stable egoistic equilibrium for selfish birds exists and the 266 cooperative equilibrium found for collaborative birds is inconsistent with empirical 267 observations. In this subsection, we only present the situation of one leader and two 268 followers, since the results for more followers lead to identical conclusions.

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For the case of selfish birds, the non-cooperative game has no equilibrium within the 270 valid zone of the wake model. This is reflected as the non-convergence of the and 1, respectively (see Fig. 1D). Hence a quasi-equilibrium considered in the following: 280 each of these two birds locates at the best relative longitudinal position (x * ) to the front 281 neighbor that maximizes the wake benefit from the front neighbor, same as 282 in [16,24,28,[34][35][36]45]. However, this quasi-equilibrium has a trend to drift downstream 283 due to the wake benefit of bird 1 from bird 2 (f (p 12 )). Although this benefit f (p 12 ) is 284 much smaller than the wake benefit of bird 1 from the front neighbor (f (p 10 )), the 285 former increases more than the latter decreases when bird 1 moves backward. This 286 implies that near equilibrium, bird 1 would like to move backward (downstream) as that 287 would increase its wake benefit f 1 (p) = f (p 10 ) + f (p 12 ). Meanwhile since bird 2 has no 288 back neighbor, it would like to maintain the relative longitudinal position to bird 1 that 289 maximizes its wake benefit f 2 (p). In cycle, bird 1's backward motion would "push" bird 290 2 back, which in return attracts bird 1 to move backward again. Hence the almost 291 equilibrium cannot be maintained. This explanation can be extended to the case with 292 more than 3 birds, in particular to the last two followers, since from Fig. 1D if the 293 distance of each bird to its front neighbor is around |y * |, it gets the wake benefit mostly 294 from neighbors.  As for the cooperative case, there does exist a cooperative equilibrium, where the y 296 coordinate of bird 1 and 2 is almost y * and 2y * , respectively, and the x coordinate of 297 both birds equal to zero. The corresponding configuration are birds equally distributed 298 on a laterally extended line, where birds longitudinal positions are the same and the 299 lateral distance of each two neighboring birds equals |y * |. This can be regarded as a 300 large wingspan bird. This confirms the observations that: 1) When x = 0, the wake 301 benefit f (x, y) attains the maximum at y * on the negative half y axis, as shown in Fig 302  1C; 2) When the lateral positions of birds 1 and 2 are fixed to y 1 = y * and y 2 = 2y * , 303 the total wake benefit J reaches the maximum at x 1 = x 2 = 0, as shown in Fig. 5. The 304 second observation can be explained if we consider the Munk's stagger theorem [32], 305 which states that "a collection of lifting surfaces (birds) may be translated in the 306 streamwise direction (x direction) without affecting the total induced drag of the system 307 (flock) as long as the circulation Γ of every wing (or lift) is unchanged." The induced 308 drag of each bird can be decomposed into the self-induced drag and the mutually 309 induced drag [31,32], where only the latter relates to the net vertical air velocities induced by other birds wakes. The mutually induced drag of each bird is proportional 311 to its wake benefit [31]. Hence one can also conclude that as long as the vortex 312 circulation Γ of every wing is unchanged, the total wake benefit of the flock does not 313 change if birds slide along the streamwise direction. This conclusion holds for constant 314 circulation. However, to count the slow dissipation of wake [44,45]   If birds cooperate to optimize the total wake benefit and try to avoid colliding with 333 each other, then there exists one cooperative equilibrium, whose corresponding 334 formation configuration for a flock with 1 leader and 10 followers is demonstrated in Fig 335  6A. Hence being aware of collision risk contributes to the echelon formation emergence. 336 This conclusion can be understood for general collision avoidance models. As shown in 337 the previous subsection where collision avoidance is excluded, to maximize the group 338 wake benefit, the lateral distance of neighboring birds should be around |y * | and birds 339 should be cohesive longitudinally. The forbidden region around each bird induced by 340 collision avoidance enforces a non-zero longitudinal separation between neighboring 341 birds when their lateral distance is around |y * |. Hence each follower cannot be too close 342 to its front neighbor longitudinally, although that would increase the group's wake 343 benefit. Indeed, the computation shows that the conclusion is robust to the choice of is less than that of other followers to their front neighbors. This is still reasonable since 365 in the real birds formation, the longitudinal distance of neighboring birds has a large 366 variation [35]. From this, we conclude that the echelon formation can emerge if each 367 bird optimizes its own wake benefit and maintains the vision comfort together.

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Moreover, a closer look at S1 Table shows The table shows whether the equilibria exist under the factor of collision avoidance or/and vision comfort. The mark ✓ (×) represents that the equilibrium exists (not exists). "CA" and "VC" stand for collision avoidance and vision comfort, respectively.
When collision avoidance and vision comfort are both combined with energy benefit 380 optimization, the echelon formation emerges for both cooperative and selfish birds: both 381 egoistic and cooperative equilibria exist. The egoistic equilibria for the four cases of vision comfort is not included. Table 1 summarizes the factors that affect the 385 appearance of equilibria. At all the equilibria, the lateral distance of neighboring birds 386 is almost the same (closely around 0.893 wingspan, same as in [22,28,31]), but the 387 longitudinal distance of neighboring birds varies much. Even so, the total wake benefit 388 J (excluding the value of the barrier potential and the vision comfort) of the egoistic 389 equilibrium for all four cases of (λ, θ) does not differ too much from that of the 390 cooperative equilibria, see Fig. 6B). A wider vision comfort zone and a more 391 front-biased binocular vision cause the total wake benefit of the flock to decrease more. 392 The small variation of the total wake benefit J even when the longitudinal position of 393 birds varies in a large range, is due to the slow decay of the wake benefit function f (p) 394 on the line y = y * along the streamwise direction.

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As mentioned in Model and also well recognized in [30,31]. Fig 6B shows that the 396 leader bird gains wake benefit from other birds, even though it is much smaller than the 397 wake benefit of followers. The wake benefit of every bird, except the last one, at the 398 egoistic equilibrium is no greater than that at the cooperative equilibrium. This is 399 because, in the case of selfish birds, no follower cares about others birds. The decrease 400 of the wake benefit of the last bird for the cooperative equilibrium is more tricky, and 401 we try to explain it hereafter. Note that the last bird can receive the wake benefit only 402 from front birds. Based on Fig. 1C and 1D and birds relative positions at the equilibria 403 in Fig. 1A, the benefit of the last bird induced from the closed front bird (the second to 404 last bird) for the cooperative equilibrium is less than that for the Nash equilibrium, bird i that is far away from its front neighbor in the lateral direction, e.g.,

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|y 0 i − y 0 i−1 | ≥ 2|y * |, then the leader and the first i − 1 followers would form a sub-group, 423 while the last n − i + 1 birds would form another subgroup. Within each sub-group, the 424 relative lateral position of each bird to its front neighbor is y * . The flock split is caused 425 by the vision discomfort mitigation and the slowing change and small value of the wake 426 benefit f (p) when |y| ≥ 2|y * |: due to the latter reason, birds i, i + 1, ..., n nearly receive 427 no wake benefit from birds 0, 1, ..., i − 1 and their incentive to get closer to i − 1 428 laterally is very weak. Hence bird i, i + 1, ..., n can be considered as a free moving group, 429 with i the leader, and will drift back along the stream direction, since from Fig 1D the 430 leader gains more wake benefit if it is closer to the back neighbor. However, to keep the 431 front neighbor i − 1 close to bird i's vision comfort zone, bird i cannot move back too 432 much, and would stop somewhere, making the back birds also stop. The above analysis 433 shows that line formations can easily break apart if any two neighboring birds are not 434 close laterally, which can be caused by turbulence or some wind shear. Hence it would 435 be hard for large scale flocks to keep the formation intact. This also poses a problem for 436 birds to initiate a line formation and implies that birds have a certain knowledge about 437 the optimal position relative to the front birds, possibly after long time learning and 438 evolution.

439
The effect of empathy on energy saving 440 We now discuss an intermediate case where birds are empathetic and concerned about 441 the energy expenditure of neighbors. Empathy has been observed in some birds, e.g, the 442 social interaction of bystanding greylag geese involved with paired partners within the 443 same family [47]. Furthermore, [48] showed that paired birds tying together and 444 interacting with fewer close birds may save more energy for the pairs. One way to 445 interpret this is that birds show empathy to other close-related birds and care about 446 their performance during flight. Though this is only found in bird clustering, we may 447 expect this also happens for formations of migratory birds of the same family. Note that 448 not all migratory formations are composed of members of the same family, hence, birds 449 empathy is unlikely the key reason behind line formation. This also explains why we 450 consider empathy separately from the selfish and cooperative birds assumptions. We 451 model the empathy by assuming that each follower maximizes its wake benefit plus the 452 weighted wake benefit of the front and back neighbors, with the weight h ∈ [0, 1].

453
Increasing h implies that birds care more about their neighbors. By restricting h ≤ 1, 454 we assume that each follower cares about neighbors no more than itself. Followers 455 maximizing the mixed benefits form another non-cooperative game, whose Nash 456 equilibrium is named as the empathetic equilibrium (EmE2) in this paper. We also 457 consider the case where each bird only shows empathy to its front neighbor, and thereby 458 maximizes its wake benefit and the weighted wake benefit of its front neighbor.

459
Similarly, this leads to another game whose empathetic equilibrium (EmE1) may exist. 460 We note that [47,48] point out the possibility of empathy with more birds in the 461 formation, which would lead to more complex models. Nevertheless our model already 462 accounts for empathy towards agents with which the main aerodynamic interactions 463 take place, and should therefore account for the main effects. 464 We compute the empathetic equilibria also by the gradient method, in which the 465 gradient of the barrier function B c and the vision comfort function B v are combined 466 with the gradient of the mixed wake benefit. Note that our model of empathy assumes 467 that all the birds have the same degree of empathy and birds can access their neighbors' 468 wake benefits and their gradients with sufficient accuracy. One should bear in mind that 469 birds can never "feel" or measure the wake benefit of neighbors. They may get these 470 information only by communication, e. g., honk, through which the exchanged 471 information are always coarse and noisy. However, we stress again that our focus is the 472 equilibria, not the emergence of birds empathy and the way that birds seek the benefit. In the following we only focus on the situation where vision discomfort 476 mitigation and collision avoidance also influence birds behavior. 477 We observed that when h is close to 1, multiple equilibria appear for both empathy 478 models, see EmE2 with h = 1 in Fig 7A and EmE1 with h = 0.8, 0.9, 1 in Fig 7C. Also 479 there is a trend that as h increases, EmE2 and EmE1 shift from the egoistic equilibrium 480 towards the cooperative equilibrium obtained in the previous subsection, and the 481 longitudinal distance of some birds to their front neighbors decreases, even though this 482 is not obvious for EmE2, see Fig 8 and S1. In contrast, the lateral distance of 483 neighboring birds is almost the same as that of the egoistic equilibrium or the 484 cooperative equilibrium. For the situation where birds are empathetic to both front and 485 back neighbors, the further down the bird is in the formation (except the last bird), the 486 higher its wake benefit will be, see efficient, since the difference between the total benefit at the empathetic equilibria and 492 that at the cooperative equilibrium is smaller, see Fig. 8. Interestingly, this figure also 493 shows that the energy efficiency losses for all EmE1s are smaller than that for all 494 EmE2s when h ≥ 0.6. This means that the flock with followers being only empathetic 495 to their front neighbors get more wake benefit or saves more energy than the flock with 496 followers being empathetic to both the front and back neighbors. All in all, Fig. 8

497
shows that if empathy does exist in close-related birds, then these birds flying together 498 × 100%, wherep * is the cooperative equilibrium obtained for the case with (λ, θ) = (15.7 • , 45 • ) in the previous subsection and p is the birds positions corresponding to a formation shape. The dashed line represents the energy loss of the group at the egoistic equilibrium. could save more total energy than if they were purely selfish.

500
Even though energy saving has been known to play an important role in the formation 501 flight of large migration birds, the question of how birds exploit this benefit is still open. 502 Using a modified fixed-wing model for bird flying, we revisit the the emergence of the 503 common migration formation, by testing whether any stationary echelon formation 504 shape can be reconstructed numerically when birds are selfish or cooperative somehow 505 in maximizing the wake benefits. We demonstrated that the hypothesis of birds purely 506 optimizing their aerodynamic wake benefits is not sufficient to produce a formation that 507 is similar to the daily-observed ones, for both cooperative and selfish birds. On the 508 other hand, collision avoidance and vision discomfort mitigation can assist the flock in 509 creating the line formation. Collision avoidance does help cooperative birds, which 510 optimize the total wake benefit of the entire flock, but not selfish ones, which maximize 511 their own wake benefits, toward forming an echelon formation. As a contrast, mitigating 512 vision discomfort or enhancing vision could play a part in creating echelon formations 513 for selfish birds, but not cooperative birds. Moreover, if reducing collision risk and 514 maintaining vision comfort are assumed to simultaneously play roles, then echelon 515 formations appear for both cooperative and selfish birds, as well as for birds that are 516 empathetic and maximize their own and the neighbors' wake benefits. Hence, we 517 conclude that the motivation of birds during long-term flights is to optimize the wake 518 benefit (or reversely reduce energy consumption). However, the well-known formation 519 shape depends on non-aerodynamic factors: the collision avoidance and/or the vision 520 constraint. Across and between the constructed formations for all the cases of self, 521 cooperative and empathetic birds, the lateral distance between neighboring birds is 522 almost the same, but the longitudinal distance between neighboring birds differs much. 523 In addition, we found that the total wake benefit of the flock for the formation 524 obtained by considering non-aerodynamic factors is most maximized for cooperative 525 birds, followed by empathetic birds, and then selfish birds. However, the differences 526 among the three cases are quite small, even though the longitudinal position of birds 527 across the three situations differ much. Hence, compared to the cooperative situation, 528 the energy efficiency for migration flocks following partially cooperative, or even purely 529 selfish strategies is very high. This is due to the fact that in the three situations, each 530 follower gets the most wake benefit from the front neighbor, which does not change 531 much when the follower moves along the streamwise direction. It happens because the 532 wake benefit generated by the upwash behind a flying bird peaks along two lines that 533 extend in the streamwise direction, and varies vastly around these two lines in the 534 lateral direction but decays very slowly along the streamwise direction. This also causes 535 the absence of line formation emergence for selfish birds that purely optimize wake 536 benefits. Due to the feature of the wake benefit, if the bird moves backward along the 537 streamwise direction, the wake benefit increment due to the back neighbor is larger than 538 the loss of wake benefit due to the front neighbor. This, along with the fact that the 539 last bird in the flock achieves the maximal wake benefit when it keeps an almost 540 constant longitudinal distance to the front neighbor, leads to a cyclic dilemma for the 541 last two followers: the second to last follower's backward motion increases it own benefit 542 but reduces that of the last follower, while the backward motion of the last follower 543 increases its own benefit but reduces the second last follower's. information for optimization. However, as mentioned before, the communication among 560 birds is probably very coarse due to wind noise and birds may not have sufficient ability 561 to process information. Considering these, it is undoubtedly a difficult task for birds to 562 cooperatively find the positions that most reduce the energy cost of the group the most. 563 For empathetic birds, each follower also needs to obtain the wake benefit of its 564 neighbors', hence the problem of coarse communication again, which could prevent the 565 current simulation result to hold. In contrast, selfish birds only need to perceive the 566 gradient of their own wake benefits, hence we conjecture that the assumption of selfish 567 birds is closer to the reality when the flock size is large. Another argument for this 568 conclusion is that although birds behave selfishly, the energy efficiency of the flock does 569 not deteriorate much. Hence, birds may lack the motivation to switch or learn to 570 behave collaboratively. The reconstruction of the echelon formation for selfish birds 571 relies on the assumptions that vision comfort zones exist for migrating birds and birds 572 eyes are relative immobilized during the long-time migration flight. We have not made 573 experiments to justify these assumptions and quantify the vision comfort model.

574
However, we anticipate they are qualitatively correct based on the experimental 575 research in [37,39,40].

576
The failure in reproducing the echelon formation without considering the 577 non-aerodynamic factors may also be caused by the simplicity of the current models. 578 We have indeed regarded each bird as a fixed-wing, and only take the averaged vertical 579 aerial velocity generated by other birds as the wake benefit. A faithful model would 580 account for the bird's kinematics, flapping gait and the resulting aerodynamics [51].

581
Additionally, the wake of a flapping bird is unsteady by nature and it will more likely plane along the x direction and assume that the air density is ρ. If approximating birds 588 by fixed-wings, the most widely used model to represent the wake of the leading bird is 589 the horseshoe model, in which the wake consists of a finite bound vortex and two 590 trailing vortices, The forming of these vortices is briefly explained as follows and can be 591 found in [32,50].

592
To support the bird weight, a difference of the air pressure between the lower and This circulatory flow is the finite bounded vortex, with length a = π 4 b and circulation Γ. 604 However, as mentioned in Introduction, the trailing vortex strength decays slowly as it extends downstream [44,45]. To account this decay, we used a modified horseshoe model, given hereafter. Suppose the leading bird is at the origin [0 0] ⊤ , the vertical airflow velocity v(p) generated by the vortices at p = [x y] ⊤ can be given as where v b and v t are the vertical velocities induced by the bound vortex and the trailing 605 vortices, respectively, r c (x) = (r c (0)) 2 + D f U |x|, r c (0) is the vortex core radius at 606 x = 0 and taken as 0.02b in this paper, and D f is the diffusion term. The introducing of 607 D f allows the vortex core to expand as it gets away from the wing along the streamwise 608 direction. This modification enable us to incorporate the decay phenomenon of the 609 trailing vortex circulation. We select D f = 5.25 × 10 −5 U b such that r c (x) increases 610 from 0.02b to approximate 0.05b when |x| varies from 0 to 40b, which is fairly realistic 611 according to the empirical data [49].

612
Then consider a following bird with the same size and weight as the leader. Let the center of the follower be p = [x y] ⊤ . We neglect the momentum induced by the vertical airspeed, and characterize the wake benefit of the following bird generated by the leading bird as follows, In the simulation of this paper, all the birds are assumed to have the same size and For bird i = 0, 1, ..., n with position p i ∈ R 2 , its net wake benefit is assumed to be where p ij = p i − p j is the relative position of bird i to bird j, and f (p ij ) is the wake 617 benefit of bird i induced by the wake generated by bird j.

618
When birds are selfish and only maximize their own wake benefits, we denote the egoistic (Nash) equilibrium by is the position of bird i at the equilibrium. According to the definition of Nash equilibrium, if p * exists, it should satisfy where p * ij = p * i − p * j and f ′ (·) denotes the first derivative of f (·).

619
The cooperative equilibrium is the point at which the following total wake benefit 620 function is maximized, .., n, the cooperative equilibrium. Since J is differentiable,p * should satisfy the following condition Invariance of scaled equilibria under parameters variation 623 One can show that the equilibria defined in the previous subsection is scale-invariant, in 624 the sense that the scaled versions of the equilibrium remain unchanged when one 625 modifies parameters such as the air density ρ, bird weight W , wingspan b and velocity 626 U . Such modifications have thus no effect on the general shape of the formation.

627
To see this, we first note that multiplying the wake benefit function f (p) in (2) by the non-argument constant 1 Γ does not change the equilibria. Now let x ′ = x b and y ′ = y b and p ′ = p b be the scaled position variables, then Recalling a = πb 4 , r c (0) = c 1 b, with c 1 = 0.02, D f = c 2 U b, with c 2 = 5.025 × 10 −5 , and noticing r c (x) = r 2 the two terms in the last integration of the above equation can be written as From these, we know that f (p) Γ does not contain the parameters mentioned at the 628 beginning of the subsection. This shows that if birds purely maximize wake benefits, the 629 existence of equilibria for the benefit maximization game and total benefit optimization 630 does not depend on these parameters. Moreover, the equilibria (if they exist) normalized 631 by the wingspan do not change as these parameters vary. Finally, we note that if 632 collision avoidance or vision comfort are also considered, the equilibria would depend on 633 the wingspan and bill-to-tail distance of birds, which may vary for different bird species. 634 Barrier function 635 As mentioned in Model, we model each bird as an ellipse, with the semi-major axis b l and the semi-minor axis b s , where b l and b s depend on the size of birds. It is easy to know that the space occupied by any bird with center p c = [x c y c ] ⊤ can be given by the following ellipse, Now consider a bird j and any other bird i, with the relative position to bird j being p ij = [x ij y ij ] ⊤ . To avoid the collision with bird j, as shown in Figure 2B, the center of bird i should satisfy the condition below, The yellow ellipse in Fig 2B that defines the collision alert region of bird i with respect 636 to bird j. 637 We use two potential functions modeling the barrier. Let p ij = [x ij y ij ] ⊤ . Barrier where k c ≥ 0, D = x 2 ij /(2b s ) 2 + y 2 ij /(2b l ) 2 ,b l > b l determines the collision alert region. 640 gradient method [41] by incorporating the gradient of the barrier function B c or/and the vision comfort function B v . In detail, for a flock with one fixed leader and n followers, let p 0 ∈ R 2n be the initial vector of followers' positions. The iteration for searching the egoistic equilibrium can be given as p k+1 i =p k i + cF ′ i (p k ), i = 1, 2, ...., n (10) where F i (p k ) is given as in (3), u i(i−1),v (p k i(i−1) ) and u ij,c (p k ij ) are given as follows, where s i(i−1) = tan ϕ i(i−1) , with ϕ i(i−1) denoting the vision angle of bird i − 1 in the 661 eye of bird i. The variable µ v , µ c ∈ {0, 1} indicate that whether the collision avoidance 662 or/and maintaining vision comfort are taken into account in the egoistic equilibrium 663 search.

664
Similarly, the algorithm for search the cooperative equilibrium with or without including the factor of collision avoidance or/and vision comfort can be given as whereF i (p k ) is given as in (4).

665
If max i=1,2,...n |F ′ (p k )| ≤ 0.0001 (max i=1,2,...n |F ′ (p k )| ≤ 0.0001) are satisfied within 666 sufficiently number of iterations, or p k stays around a point for a large number of 667 iterations, the algorithm (10) ( (12)) is considered as converging, otherwise not. We  The table shows the maximum of the absolute value of the gradients of the wake 686 benefit and the vision comfort function of each bird at the egoistic equilibrium for the 687 four pairs of angle parameters. We can see that the gradient of the vision comfort is 688 small and the same magnitude as the gradient of wake benefit. This shows that a little 689 effort of birds vision discomfort mitigation is capable to help the emergence of the 690 echelon formation.