From Feathers to Formations: A Mathematical Analysis of Starling Murmurations

Keywords: Networks, Scale-Free Topologies, Murmurations, Geometry of Avian Physiology, Egg Geometry

Abstract

This paper explores the mathematical properties and relationships at play within bird murmurations. Studying the characteristic formation patterns of murmurations reveals practical methods of predator evasion. The rapid amplification of propagating dynamic signals within the murmurations ensures a quick rate of information transmission and therefore greater odds of survival. Birds’ general awareness of the group’s dynamics and formation patterns within the murmuration informs subsequent movements to improve overall coordination. Furthermore, modelling murmurations as a network of interconnected information-processing nodes reveals a linear behavioral correlation model between birds within the murmuration. Noise within the system, individual behavioral fluctuations, is hypothesized to be an essential component of biological networks. Finally, the geometrical properties of avian physiology reflect the complex mechanisms governing biological development and growth.

Introduction

Birds are renowned for their remarkable diversity of form and behavior. This diversity is remarkably displayed by the formation of murmurations: large assemblies of coordinated birds driven by predator evasion (Fig. 1). Relatedly, their features on an individual level have been refined through evolution to serve essential ecological functions: from the intricate patterns in feather geometry that enhance flight to the structural complexities of nests that provide rigid shelter for offspring. This essay explores the fascinating dimensions of avian biology, examining how features such as egg and beak geometry as well as flock dynamics contribute to the adaptability and efficiency of bird species. Using mathematical models, we analyze the wave-like patterns of flocks and their coordinated responses to threats. Scale-free correlations observed in flock movements further reveal the emergent order in bird movements, suggesting that simple individual actions can lead to complex, ordered group dynamics. The common denominator of all adaptations within bird species is the enhancement of survival and the minimizing of costly metabolic processes. This paper explores how collective behavior is merely another instance of birds’ continuous efforts at behavioral and physiological optimization.

Fig. 1. European starling murmuration (Solkær, 2022).

Scale-Free Correlations

Collective Animal Behaviour

The sharing of information within a network containing a set number of nodes can be accomplished in a variety of ways. A network node consists of a branch in a network which processes data and relays it to neighbouring nodes. The rules which govern the transfer of information in a network are referred to as a topology. Topology hence relates the diverse ways in which nodes are connected. 

Several topologies can govern the transfer of information in a network (Cavagna et al., 2010). A hierarchical architecture is one in which adjacent nodes are not mutually interconnected but are instead all connected to a single node which governs the behaviour of the group. Contrastingly, distributed and scale-free topologies refer to networks wherein adjacent nodes interact to varying degrees: nodes are solely connected to their direct neighbours in a distributed network, whereas there is a greater variety of connections in the network in a scale-free model. It follows, especially in the study of collective animal behaviour, that groups of coordinated animals are analogous in function to networks. Studying the topologies which govern such animal networks is fundamental to understanding the physiological mechanisms underlying collective animal behaviours and gaining insight on the evolutionary advantage of such behaviours. 

Starling murmurations are formed because of positive chemical feedback loops and are maintained by each individual starling closely monitoring the spatial and temporal distributions (i.e. position and velocity) of its six or seven closest neighbours (Vallee, 2021). The latter defines the interactions between adjacent nodes in the so-called murmuration network: how does the whole network behave? The collective animal behaviour manifested in starlings stems from a self-organizing process as opposed to a centralized one (Cavagna et al., 2010). Local interactions between starlings, namely between neighbours within the murmuration, are reflected throughout the entire network. Thus, the topology of the starling murmuration network is scale-free: there exists a correlation between individuals’ behaviour within the group regardless of the distance separating them.

Additionally, the correlation length is the spatial span of the correlated nodes in the network (i.e. nodes communicating either directly or indirectly). This correlation length can be much greater than the range of interaction between adjacent nodes and is influenced by the level of noise within the system. In any case, noise within the network corresponds to external disruptions or internal failures which corrupt the signal being relayed. In a scale-free network, the correlation length corresponds to defined domains of nodes which observe positive (reinforcing) correlations. As all nodes are interconnected in a scale-free network, domains extending past the correlation length of a particular domain are anticorrelated, carrying out the ‘inverse’ or ‘opposite’ behaviour. 

Mathematical Analysis and Study of Scale-Free Correlations

In their study, Cavagna and colleagues monitored starling (Sturnus sturnidae) murmurations during a period of two years over a roosting site in Rome, Italy (Cavagna et al., 2010). Through numerical analysis, they extracted the spatial distribution as well as the velocities of starlings within each murmuration to compare their scale-free network model to actual data. Murmuring flocks of as little as 122 individuals and as large as 4,268 individuals were observed.

A flock’s cohesion, namely its degree of order, is represented by the polarizability, which is given by Equation 1. 

N denotes the number of starlings and v_i denotes the velocity of each starling within the murmuration. Equation 1 consists of the sum of all velocity unit vectors divided by the number of starlings within the murmuration and its values range between 0 and 1. Equation 1 will yield zero if the unit vectors all point in different directions and will yield values equal or near 1 if there is a strong correlation between the velocities of individuals within the flock. Similarly, the difference between the starling’s velocity within the murmuration and the net velocity of the starling murmuration (starling’s velocity in the murmuration reference frame) is given by Equation 2. 

Where u_i denotes the relative velocity vector. By construction, ∑u_i is zero. Indeed, the net velocity of the starlings in the murmuration’s reference frame must be zero because the net velocity of all starlings in the regular reference frame corresponds to the murmuration’s velocity. From their collected data, Cavagna et al. found that the average polarizability of the twenty-four observed flocks was 0.96 ± 0.03. Furthermore, they were able to observe the presence of strongly correlated domains through their data within a network of strongly correlated individuals (Fig. 2). This reflects the influence of local interactions on the group’s culminating behaviour.

Fig. 2. Data relating to the motion of a murmuration of starlings. A) Scaled velocity vector map of the starlings within the murmuration. B) Scaled relative velocity vector map of the starling with respect to the frame of reference of the murmuration, clearly displaying two strongly correlated domains in the far-upper extremities. C) Probability distribution of the norm of the velocity vectors (A) and their fluctuations (B). Norm of the velocity fluctuations is on average one degree of magnitude smaller than that of the velocities (Cavagna et al., 2010).

Another parameter used to characterize the degree of cohesion within the flock is the correlation function. This function evaluates the average scalar product between velocity fluctuation vectors of starlings within a confined radius. This is accomplished with the Dirac δ-function, which goes to infinity at zero and has a value of zero everywhere else. The Dirac δ-function ‘selects’ a pair of birds at distance r to compute the scalar product of their velocity fluctuations. The correlation function is expressed as follows:

Where c₀ normalizes the function to a value of 1 at r=0, and u_i * u_j is the scalar product in three dimensions. The correlation function will evaluate the span of a correlation domain. It follows from the properties of the scalar product that the correlation function will yield larger values (either negative or positive) when the degree of correlation between velocity fluctuations is very high. Likewise, it will return a value of zero when there is no net correlation within a group. Figure 3A reveals the correlation between starlings' behaviour within a murmuration. At small distances, and up to the correlation length ζ, starlings will mutually reinforce one another’s behaviours: they carry out similar means of correcting their velocities to correspond with the group. Relatedly, starlings at a greater distance than the correlation length correct their behaviours oppositely. The presence of correlation and anticorrelation domains is a trivial consequence of setting the sum of all velocity fluctuations to zero. However, the presence of a single correlation domain and a single anticorrelation domain is not trivial. This reflects an inherent mechanism of coordination within murmurations to maintain cohesion. Indeed, there are multiple ways by which a map of vectors can be arranged to sum to zero. Realistically, however, the presence of two domains merely reflects how murmurations are constantly rotating. Both domains correcting velocity in opposite directions is akin to a force couple: two opposite forces create innate torque which will generate constant rotation in space.

Fig. 3. On the right, the value of the correlation function with respect to the distance separating starling pairs. ζ is the correlation length. On the left, the value of the correlation length as a function on the size of the flock. The correlation length is linearly correlated to the flock size L (Cavagna et al., 2010).

Also obtained from experimental data, Figure 3B plots the correlation length, namely C(r = ζ) = 0, as a function of flock size. Directly, one can observe that its magnitude is not fixed to some set number of birds or a particular distance: rather, it is linearly scaled with the size of the flock. Linear scaling of the span of correlation between nodes in a network directly corresponds to a scale-free topology. Relatedly, the actual degree of correlation between starlings can be interpreted conceptually through the following equation:

Equation 4 expresses the degree of correlation between starlings as a scaled power-law. That is, the correlation is proportional at some given power to the actual distance r as well as the relative distance with respect to the size of the flock. Here, f(x) is a scaling function which consists of the dimensionless relation between the actual distance and the flock size. Hence, as presented in Figure 3, the correlation function, with respect to the ratio of distance and flock size, is also a linear relation. Its derivative is given by Equation 5 and is also plotted in Figure 4. Furthermore, the value of the correlation function at large distances within the flock, the limit of the distance as it approaches infinity, is given by Equation 6.

As inferred from Equation 5, the value of γ is the determining factor as regard the degree at which starlings separated by some distance will be correlated. The greater its value, the more negative the slope of the correlation function will be at a smaller distance between starlings within the flock, indicating a rapid and complete loss of correlation. Contrastingly, at small values of γ, the derivative of the correlation function approaches a constant value of negative one, and its value asymptotically approaches one at very large distances. Cavagna and colleagues concluded, from Figure 4 and with Equation 5, that the average value of γ amongst all observed starling murmurations was γ = 0.19 ± 0.08, which effectively corresponds to no decay of correlation, regardless of the murmuration’s span. It is thus a matter of fact that two birds within a 100 m murmuration and separated by 10 m are as strongly correlated as two birds separated by 100 m in a murmuration spanning 1 km. 

Fig. 4. The correlation function with respect to the dimensionless quantity r / ζ. Its value decreases at an approximately constant rate, as presented in the sub-plot of slope with respect to correlation length (Cavagna et al., 2010).

The mechanisms enabling scale-free correlations in animal systems are well-documented. Computer algorithms, based on a simple set of rules, can mimic the murmuring behaviours of birds through velocity vectors within a vector field; individual vector nodes are instructed to adjust their alignment and speed based on that of neighbouring vector nodes which are initially triggered stochastically. Rather, the perplexing nature of bird murmurations relates to the inevitable presence of noise within any system spanning large distances. Noise in bird murmurations is brought forth by slight alignment errors on an individual basis, which corrupts the communication signal. Naturally, it might be hypothesized that minimizing noise within the network would induce higher order and correlation within the system. However, a certain level of noise within a biological-based network such as this one is critical to amplify signals throughout the system. Decreasing the noise beyond a certain threshold within the murmuration would render individuals more insensitive to adjustments in their neighbours’ velocities, a phenomenon referred to as behavioural inertia (Cavagna et al., 2010). Hence, achieving a non-decaying correlation occurs by implementing an optimal noise level within the system to both maximize order and correlation. Bird flocks evidently achieve such a critical noise level as informed by the near-zero value of γ, the power law exponent. However, the evolutionary and physiological means underlying this feat are yet to be elucidated.

Collective Movement

Flock Morphology

Starling murmurations tend to maintain constant proportions regardless of flock density, size or composition during regular flocking events. Flocks are thinner on the vertical axis, and wider on the horizontal plane (Ballerini et al., 2008). To measure the aspect ratios of a flock, consider I₁ as the thickness of the flock in the vertical axis. I₂ and I₃ are located on two mutually orthogonal axes on the horizontal plane, with I₂ corresponding to the smaller magnitude, and I₃, the longer magnitude (i.e. I₁ <  I₂ < I₃). As determined through multiple studies (Ballerini et al., 2008), the aspect ratios I₂ / I₁ and I₃ / I₁ yield constant values despite minor fluctuations across multiple flocks. Relatedly, it was found that the thickness I₁ is linearly correlated to the cubic root of the flock’s volume (Fig. 5). Equation 7 provides a general expression for volume V of a murmuring starling flock.

Substituting with the provided ratios, we obtain

Considering I₂ / I₁ and I₃ / I₁ as constants yields

With a new constant β.

Fig. 5. Thickness I₁ plotted against V^1/3. The thickness is proportional to the cubic root of flock volume. Aspect ratios I₂ / I₁ and I₃ / I₁ are plotted against  V^1/3, fluctuating around a constant value. These indicate that flock proportions remain constant regardless of volume change (Ballerini et al., 2008).

The unit vector I₁ described as “yaw” was found to be nearly parallel to the G unit vector (gravity), as the cosine of the angle between yaw and G bordered the value 1, indicating that the flock’s horizontal plane is parallel to the ground. Thus, normalizing the flocks’ center of mass velocities and calculating their dot product with gravity yield values nearing zero. This evidently suggests that flock velocity during flight is perpendicular to gravity, leading to level flight on the vertical plane (Fig. 6). There is no correlation between the velocity vector and the unit vectors I₂ and I₃, suggesting that the elongation of a flock is not correlated with its velocity. There are several hypotheses regarding the thin morphology of murmurations on the horizontal plane. First, gravity renders vertical elongations energetically unfavorable. Instead, lateral stretching of the flock is favored. Second, a vertically elongated flock would experience more drag than a laterally stretched flock, reducing the efficiency of collective flight (Ballerini et al., 2008). No satisfactory hypothesis could be formulated to explain why the lateral elongation of the I₃ axis is uncorrelated with the direction of velocity. Regardless, maintaining constant flock proportions despite variation in densities and volumes may help preserve consistent flock properties and cohesion over time.

As depicted in Figure 6, flock morphology remains constant during turning sequences.

Fig. 6. A flock produces a turn. A) The flock’s trajectory on the horizontal plane. B) The flock’s trajectory on the vertical plane, which demonstrates how the flock remains parallel to the ground during turning events. C) The angle between velocity V and I₂ (red) as well as I₃ (blue), which varies with time. This depicts a change in the direction of the velocity: from almost parallel to the I₂ axis at the beginning of the turn, to being parallel to the I₃ axis at the turn’s end. D) Depicts the rotation of the flock in 2-space: the axis in red is associated with I₂, and the blue axis denotes I₃. Velocity is indicated by the black arrow and rotates from initial alignment with the red axis to ultimate alignment with the blue axis. Thus, the birds at the front of the flock end up on the right side of the flock after the flock enacts a left turn. The absolute orientation persists throughout a turn. E) The peak in centripetal acceleration (depicting a gaussian curve) during a turning event. The angle between I₁ and G increases during the turn, and the angle between I₁ and V decreases, indicating that I₁ tilts toward the direction of the turn (Ballerini et al., 2008).

Since the flock’s horizontal plane becomes tilted during a turn, the flock experiences drag, like an aircraft “banking” its turns, contributing to its lift. However, flocks are aggregates of individuals and hence do not experience global lifts. It can thus be hypothesized that flocks execute tilted turns to facilitate neighbor detection, as such maneuvers align individuals along a common visual plane.

Topographical Models

Starling murmurations can form intricate and distinct patterns in the sky, ever-changing and evolving into new formations of various densities. Starlings are well-known for their diverse collective escape maneuvers in the face of danger. The level of perceived threat by its members and the flock’s prior configuration are factors which influence a murmuration’s topographical model of escape (Storms et al., 2019). A low-level threat corresponds to the absence of a predator in the flock’s vicinity. A medium threat corresponds to the predicament in which a predator is in proximity or following the flock. Finally, a high-level threat describes the situation in which a raptor attacks. Notable escape maneuvers have been identified through starling observation. Wave events are the most eminent and correspond to the propagation of pulses of dark bands throughout the flock.

Wave Events

Wave events facilitate rapid information transfer within the murmuration. The propagation of a repetitive and continually amplifying movement throughout a group allows for information to be transferred at a faster rate than that resulting from the monitoring of the more disorganized dynamics of individual members. This makes wave events effective at decreasing the probability of success of predator attacks. Indeed, a wave’s speed of propagation was found to be on average 13.4 m/s, which is faster than the flock’s speed (10.6 m/s) and comparable to the predator’s speed (11 to 15 m/s), thus making this maneuver a viable and effective technique for predator evasion (Hemelrijk et al., 2015). Wave bands are propagated away from the predator, from one end of the flock to the other, as information is derived from the source. Dynamic waves can serve to confuse predators and have been found to occur before or after attacks of medium speed, as well as upon the detection of medium threats. The exact mechanism which underlies the waves of agitation of starlings is still a mystery. Relatedly, models of wave propagation, such as that developed by Hemelrijk et al., must further investigate whether these waves of agitation consist of density waves, formed by the modulation of flock density through birds’ acceleration, or orientation waves, which are formed by the shift in orientation of the birds’ bodies when rotating to the side. Starlings’ dark and uniform plumage prevented the identification of the birds’ orientation in the sky due to lack of differences in color between their dorsal and ventral areas, contrary to certain other species. However, the visible change in a bird’s wing surface area from an observer’s point of view could be sufficient means to indicate whether it has rotated to its side (Fig. 7).

Fig. 7. Projected area of simulated bird model when viewed from the side. The maximum projected area corresponds to a 90° rotation, and the minimum projected area corresponds to level flight (Hemelrijk et al., 2015).

By observing comparable avian species, the hypothesized rolling motion of starlings during wave events was categorized as either a “zigzag” wherein the bird rolls from one side to the other or a “zig” where the bird rolls solely to one side (Fig. 8).

Fig. 8. Bird trajectory when performing a “zigzag” (A), and a “zig” consisting of rolling to the left and back (B) (Hemelrijk et al., 2015).

Interestingly, Hemelrijk and colleagues’ simulation of a falcon attack from the murmuration’s left-side yielded a variety of results:

Fig. 9. Flock response to raptor attack from the left, depicted by images displayed in chronological order. The empirical data is represented by a), all images below result from the model. No wave is formed in b) where the zig is not repeated, nor c) where the maneuver consists of accelerating forward. A double band is propagated throughout the flock when the zigzag is performed in d), as opposed to a single band when the zig is performed in e), which corresponds to empirical data. No wave is formed if the birds are modelled as spheres while performing the zig f) (Hemelrijk et al., 2015).

The appropriate wave only results from the performing of “zigs”, demonstrating that wave events should be generated by orientation waves and not density waves, as the simulation fails to produce waves when the birds solely accelerate forward or when the surface area of the birds are modelled to be constant. It was also demonstrated that wave speed is positively correlated with the number of monitored neighbors an individual startling is responsible for since a greater ability to perceive repeating patterns allows for faster information transfer. As the number progresses from 2 to 7 neighbors, the wave speed increased to around 15 m/s, like empirical data which agrees that starlings monitor their six or seven closest neighbors to ensure the flock’s cohesion and synchronization. Wave speed is also positively correlated with distance between neighbors. Since reaction time remains constant, a larger distance between individuals allows for transfer of information across larger distances within the same timeframe. Conversely, wave speed is not correlated with flock size. Relatedly, there are two possible explanations as to why forward accelerations did not produce wave events. First, due to the avoidance of collisions. Indeed, starlings may not desire to approach one another past a certain distance, rendering the formation of a density wave impossible. In fact, the very basis by which murmurations form is the maintaining of a constant distance and relative velocity with one’s neighbors. Density waves would thus violate one of the fundamental rules ensuring flock cohesion. Second, the act of rolling on one’s side can be performed faster than the act of accelerating forward, potentially too slow for the formation of waves (Hemelrijk et al., 2015). Changing velocity requires merely the adjustment of the wing’s distribution in space, whereas changing speed requires the expense of additional energy, which birds cannot afford as a regular occurrence.

Blackening

Other notable formations include blackening, where part of the flock or its totality darkens due to compacting. Blackening occurs before or after predator attacks or during situations of medium threat. A flock’s tendency to blacken is independent of predator speed, angle of attack, and repetitiveness of attack, and is the most observed maneuver (Storms et al., 2019).

Flash Expansion

Flash expansions are starlings’ escape maneuver of choice in situations of high threat where the raptor attacks at high speeds, from above rather than the side or below (Fig. 10). It consists of the sudden outwards motion of the birds within the murmuration, leading to the expansion of the flock. This maneuver can be performed four to ten times faster than other tactics, naturally conveying its critical importance in situations of immediate danger.

Risks associated with flash expansion consist of a reduction of flock cohesion, and the splitting of the original flock into smaller sub-flocks, which are more susceptible to predator targeting. Hence, to avoid the greater risk of individual predation, starlings tend to resist splitting after flash expansions: 78.3% of observed flash expansions did not result in splits (Storms et al., 2019).

Fig. 10. Attacks followed by flash expansion with respect to the hunting approach (Storms et al., 2019).

Additional Maneuvers and Formations

Murmurations will regularly perform four other maneuvers in addition to those described above (Fig. 11). First, vacuole corresponds to the formation of a hole in the flock. This formation is the least observed, making up only 0.6% of all escape methods. Secondly, splits correspond to the splitting of a flock into sub-flocks. The tendency to split is assumed to be correlated with the size of the flock, with larger flocks exhibiting a greater eagerness to split than smaller flocks, as smaller flocks risk aggressive predation. Thirdly, the merging of two sub-flocks, designated by the formation “merge”, allows for a decrease in targeting risk, aiding in survival success. Finally, a thin string of individuals that connect two parts of a flock is called a “cordon”. This pattern is seldom used, making up only 4.4% of the observed patterns of escape. Flock dilution represents the increase of distance between neighbors, resulting in density dilution. In situations of low threat, murmurations adopt a somewhat diluted configuration. The rate of occurrence of various maneuvers is directly correlated to its associated risk and survival benefit (Fig. 12).

Fig. 11. Various methods of collective escape employed by starlings. A are wave events, b and e, are flash expansions, c and f are vacuoles, d and g display cordons (Storms et al., 2019).

Fig. 12. The frequency of escape patterns among starlings. Blackening is the most common, followed by wave events, splits, merge, flash expansion, cordon, and vacuole (Storms et al., 2019).

The use of certain escape maneuvers was found to be correlated, and often succeeded by others, with higher frequencies of attack corresponding linearly to higher frequencies of escape among starlings (Fig, 13). Falcons prefer consecutive attacks of five second intervals, rendering the combination of escape strategies and a smooth transition even more important. Flash expansions are strongly correlated to attacks, with 25% of attacks leading to their use. Only 21.7% of flash expansions lead to splits, and 83.3% of flash expansions stem from attacks. Splits are strongly correlated with subsequent merging. 54% of flock dilutions lead to blackening, and 28.3% of blackening lead to wave events (Storms et al., 2019).

Fig. 13. Sequence of escape maneuvers. Arrows represent 5 second intervals. A thicker arrow indicates a stronger correlation between two events than chance. Numbers preceding an arrow represent the ratio of that event that resulted in that secondary event, and numbers succeeding an arrow represent the portion of the secondary event that stems from the primary event (Storms et al., 2019).

Physiological Geometry

Feathers

Studying the geometry of a bird’s feathers is critical in the analysis of flight optimization. Thus, as explored in Murmurations Physics, the structure of a feather consists of a central shaft called the rachis, which serves as a “backbone” (Fig. 14). Extending laterally from the rachis are barbs. The barbs then branch into smaller barbules, which interlock via microscopic hooks called barbicels. In conjunction, the arrangement creates a smooth continuous surface, known as the vane, that simultaneously maintains flexibility and rigidity. The feather tapers at the tip, with a broader base. The specific geometry not only streamlines airflow but allows it to be lightweight yet sturdy (Hall 2014).

The aerodynamic properties of feathers are governed by the principles of fluid dynamics, which describe how air interacts with the bird’s body during flight. Lift, a critical flight force, can be expressed in terms of air density, velocity, surface area, and the lift coefficient C_L.

Feathers are shaped and angled to maximize C_L ­ to generate significant lift even at low speeds. Herein lies the relevance of multiple different feather geometries, which are tailored to serve distinct purposes based on morphological attributes.

The symmetry of contour feathers, those that cover the bird’s body are critical for balanced flight. They are primarily responsible for streamlining the bird’s overall body and reducing drag (Hall, 2014).

Fig. 14. Parts of the feather (The Editors of Encyclopaedia Britannica, 2024).

It is important to note that flight feathers have an asymmetrical structure. In these feathers, for instance the primaries or the secondaries, the vane on the leading edge (side facing direction of flight) is narrower while the trailing edge is broader. This asymmetric structure is crucial for generating efficient lift and thrust. The broader trailing edge provides a larger surface area to capture air and generate lift.

Beaks

The structure and shape of bird beaks, particularly the upper surface and tomium (cutting edge), are essential for understanding their functional morphology and evolution. (Mosleh et al., 2023)

Fig. 15. (A) The left is a bird skull modeled by micro-computed tomography scans. The right is the upper surface of the beak. The yellow line represents the midsagittal curve of the beak. The red line is the tomium of the beak. D_B is the beak depth, W_B is the width of the beak, L_B is the characteristic length of the beak. k_x is the curvature of the beak along the x-axis and k_y is the transverse curvature of the beak. (B) The tomium is projected onto the xz-plane where it fits a parabolic shape. (C) The centerline of the beak is characterized by length L_C and curvature k_C. At the cross-section along the centerline, D_C is the depth and W_C is the width of the beak. (D) The morphospace of beaks is defined using two dimensionless variables: k_x^~ and S^~. (E) The tomium shapes are compared using two dimensionless variables: a_T^~ and k_T^~. (F) The aspect ratio W_C / 2L_C and dimensionless variable k_C^~ characterize the beak morphospace. (Mosleh et al., 2023)

The skull and beak structure were modeled using micro-computed tomography scans (Fig. 15). Panel A shows the upper surface of the beak with the red-highlighted section representing the tomium. The tomium is the cutting edge of the beak which separates the upper and lower beak surfaces. The upper surface of the beak is oriented so that the origin is positioned at the beak tip shown on the right side of Panel A. The yellow line is the midsagittal curve which is captured by a paraboloid equation:

Where L_B is the beak’s characteristic length, D_B is the beak depth, and k_x is the curvature along the x-axis. This equation models how the shape of the beak changes along its length from the base to the tip. To standardize comparisons between beaks of different sizes and species, we define three dimensionless variables:

By examining samples in the morphospace in Panel D, it is observed that the region is primarily occupied by honeycreepers, of genus Cyanerpe, suggesting a growth pattern specific for them. (Mosleh et al., 2023)

The mathematical model for the shape of the upper surface is given by:

Where U is the upper surface, x is the position along the length of the beak (from base to tip), and y is the transverse position across the beak (from side to side). The term (a_Ux - k_xx²) represents the curvature along the length of the beak, (k_tip - Sx)y² represents the transverse curvature across the width. Here, k_tip represents the curvature at the tip and Sx describes how quickly the curvature decreases along the length moving away from the tip. Therefore, (k_tip - Sx) = k_y(x), where k_y(x) is the transverse curvature near the tip. This model captures how beaks are generally more rounded at the tip with high transverse curvature and flatter at the base. (Mosleh et al., 2023)

The shape of the tomium is projected onto the x-z plane and fits a parabolic form:

The 3D tomium curve can be derived using the Z_U(x,y) and Z_T(x,y) equations. The depth of the beaker at a given point x is defined as the difference in height between the upper surface and the tomium curve at y = 0:

The width of the beak at a given point x can be determined by finding where the upper surface and tomium curve intersect at y = W(x) / 2, wherein W(x) is represented by the following equation:

To standardize the shape of the tomium across different species, dimensionless parameters are computed:

When looking at the normalized tomium parameters in Panel E, most points fall on the line k_T^~ = 0.5 + a_T^~, suggesting that the depth of the tomium below the tip is about half the overall beak depth for most species. However, species like Maui parrotbill and Eurasian bullfinch deviate from this and have their unique beak shapes. (Mosleh et al., 2023)

In Panel C, the beak centerline, k_C , is used to analyze the role of centerline curvature in beak shape variation. Additionally, the length, width, and depth of the centerline are denoted as L_C, W_C, and D_C. To standardize curvature across different beak shapes, we define a dimensionless variable k_C^~ = L_Ck_C and an aspect ratio W_C / L_C. Further analysis reveals a strong correlation between the beak’s aspect ratio S^~ and the bird’s diet, with a correlation coefficient of 0.73. The aspect ratio relates to the beak’s width and length. Species that feed on nectar that have narrower beaks have small aspect ratios while species that eat fruit and seeds that have wider beaks have large aspect ratios. This suggests that wider beakers have greater bite forces. Additionally, research proves that the sharpening rate of the beak, , also shows a correlation with diet, with a correlation coefficient of 0.46. This shows that fruit and seed-eating birds typically have higher sharpening rates, allowing the beaks to withstand high forces. The ecology of bird species has shaped their beaks to meet specific dietary needs. Thus, different diets require different beak shapes and structural adaptations (Mosleh et al., 2023).

Nest

Indeed, studies suggest the existence of a correlation between bird beak morphology and nest material choice. The study involved the use of random forest models which are suited for complex and non-linear data to explore different response variables. Three types of response variables related to nest-building were tested: a binary indicator of whether the species used a specific material, the primary material category used by each species, and a reduced data for species that only use one material type. The model was built with 5000 decision trees and used predictor variables like beak morphology and body size. The primary nest material prediction had an accuracy up to 59.8% for species that only use one material type. Adding ecological predictors such as diet and material availability further improved accuracy of material use within a range of 63.8% - 96.8%. (Sheard et al., 2023)

To address potential biases and phylogenetic relationships, the phylogenetic generalized linear models (GLMs) were used to make clear phylogenetic dependencies within binary response data. Unlike random forest models, GLMs assume a linear relationship between predictor and response variables. GLMs allowed researchers to interpret how beak traits correlate with nest material use after accounting for evolutionary history. The results were that traits like body size and high flight ability had moderate correlations with nest material choice. However, phylogenetic simulations indicated 66.4-96.9% of the observed material use could contribute to shared evolutionary history. GLMs combining with the random forest models provides a comprehensive view of factors affecting nest material choice. It highlights how both morphology and evolutionary history jointly shape these ecological behaviors (Sheard et al., 2023).

Egg Geometry

Starlings typically have eggs in a clutch size of 4 to 5. They average 6.4 g and are about 30 x 21 mm. These specific dimensions contribute to the incredible balance between the function and strength of a bird’s egg (Fig. 16). Eggs are optimized as architectural domes to distribute forces evenly across the shell, protecting them from external forces. The shape also maximizes internal volume for nutrient storage (Sharp, 2023).

Fig. 16. Depicted are starling eggs, smooth with pale blue coloring and occasionally brown speckles (Sharp, 2023).

Birds’ eggs are categorized into four basic geometries: circular, elliptical, oval, and pyriform (pear shape) (Fig. 17). While the former three shapes are mathematically well-defined and are represented by quadric equations, no such mathematical equation initially existed for the latter, namely the pyriform. This restricted researchers’ ability to derive a universal egg equation (Narushin et al., 2021).

Fig. 17. Basic egg shapes. (A) circular, (B) elliptical, (C) oval, (D) pyriform (Narushin et al., 2021).

However, recent efforts to properly characterize an egg’s geometry have yielded Equation 19. Hence, y defines the level set curve depicting the shape of the egg in 2 dimensions.

L is the egg length, B is the egg’s maximum breadth, and w is the parameter that indicates the distance between two vertical axes corresponding to the maximum breadth and length of the egg (Fig. 18). Additionally, x is an expression in terms of w, B and L (see Appendix). It is important to note that when L = B, the level set curve assumes the shape of a circle. Furthermore, when w = 0, y takes the form of an ellipse. The equation was derived with the intent of encompassing all possible egg shapes, thus solving the egg equation problem.

Fig. 18. Parameters of the universal egg equation. D_L/4 is the egg diameter at point L/4 from the pointed end (Narushin et al., 2021).

The egg equation’s derivation, and its ensuing mathematical representation of the pyriform egg geometry, can lead to a better understanding of the specific mechanical strengths and advantages of different types of eggs. For instance, pyriform eggs allow for maximum incubation surface against the mother’s brood patch which provides a selective advantage on account of a more energy efficient thermoregulation process. Additionally, compared to symmetrical egg shapes, pyriform eggs are less likely to roll away, which consists in a crucial adaptation for birds like starlings, which occasionally nest closer to the ground.

Conclusion

Birds have optimized their murmuring behaviors by regulating critical noise levels within the murmuration. Sufficient noise, namely a baseline number of individual aberrations within the flock, is essential to amplify external signals. These amplified signals generate large dynamic patterns within the murmuration which serve to resist predation. As a plain manifestation of bird murmurations being more than the sum of their parts, the formation of recurring patterns within the flock during flight reflects a certain collective conscience of the group. Birds must move themselves, but they can only do so successfully when coordinating with the entirety of their flock. This fact is further emphasized upon considering that murmurations exhibit a scale-free behavioral correlation. Thus, collective awareness results in the adoption of various efficient murmuration patterns for the survival of the flock. Directly, the minimization of the murmuration’s height in the vertical axis and the maximization of its width in the horizontal axis results in less drag being exerted on the birds and therefore a more efficient means of locomotion.

Mathematical models and geometric analyses deepen our understanding of avian biology, particularly in the context of starling murmurations. Indeed, the mechanisms underlying information transmission within murmurations are highly optimized and in many ways reflect the protocols which humans have implemented in computer systems to facilitate computationally intensive tasks. Networks composed of interconnected nodes receive and process information to conduct binary operations in deep learning: in the case of murmuring starlings, these binary operations consist of staying put or modulating their velocities with respect to those of their neighbors. While manifestations of biomimetics in technology are prevalent, they in many ways fail to compare to those of biological systems which have been fine-tuned and improved over the course of millions of years. Studying the mutual interactions between organisms, whether it be the emerging properties of a collective or a mutualism, can only serve to deepen our own prowess in engineering and technology.

Appendix

The following expression in terms of w, L and B is a key relation in the universal egg equation. The values of w, L and B are defined in the Egg Geometry section.