Ethics Statement

The office of the rector of the centre for international relations of Tribhuvan University (Kathmandu, Nepal) certified the research expedition entitled “Study on the behaviour of the Giant honeybees: Observations and Recording of behaviours at the nesting site” in Chitwan district of Nepal.

Experimental Site and Recording

The shimmering behaviour of Giant honeybees (Apis dorsata) was studied under field conditions in Chitwan, Nepal, during two expeditions in February 2009 and November 2010. To obtain a detailed view of the movements of individual bees within the entirety of the bee curtain [7], [14] the motions of hundreds of surface bees were measured simultaneously in the three directions of space. This was achieved by an adaptation of the stereoscopic imaging principle [27] with its fundamental algorithms [16]: segmentation [28], matching [29], [30] and reconstruction [31], [32] by tracking and triangulation (Fig. 3A,B). Two video cameras (type DALSA Falcon 4M60) delivered black-and-white images with a resolution of 2352×1728 pixels (px). From the given working distance of 2 m, and with a calibrated focal length of 53 mm, about two thirds (700 mm in diameter) of the nest were recorded, whereby one px represented ∼ 0.30 mm in metric real-world coordinates. Therefore, the characteristic abdomen width of 6 mm of a bee was imaged by roughly 20 px. The cameras were able to capture 60 frames per sec (fps) and so resolve the abdomen-flipping phase of an individual bee of 200 ms within 12 frames (for further details see [16]).

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Figure 3. Schematic of the evaluation process to analyse properties of the propagation process in shimmering.

The light-grey flow charts (A–G) address the single-agent analysis from image acquisition, stereoscopic imaging of individuals, wave incident detection, to the synchronization, categorization, and pooling of wave incidents. The black arrows and dashed lines on the right side symbolize that the stereoscopic analysis produced further data for hundreds of agent bees simultaneously. (A) The experimental nest was captured by two frame-locked video cameras positioned at an angle of 30° two meters in front of the nest. A single recording session lasted 15 s and included up to two shimmering waves which spread across the nest surface. Shimmering was elicited by a dummy wasp. (B) In the offline data assessment phase the acquired images were processed as follows: Segmentation distinguished single agent bees in the densely packed clusters of bees on the surface of the nest in each of the paired images, stereo matching identified corresponding agent bees in both paired images. These two processes enabled stereo tracking of the agents in subsequent frames throughout whole film scenes and the triangulation of their thoracic positions regarding the three dimensions of space (x,y,z) [14]. (C) The stereoscopic analysis delivered identities of agent bees in subsequent image sequences. The arrival of the wave front at an individual agent was recognized by a movement detection algorithm (see Fig.1B, white square). The grey arrow on the right with the stop bar symbolizes that a minority of identified agents did not participate in shimmering and was not evaluated further. (D) The detection of active participation in the wave allowed synchronization of wave incidents of individual agent bees at different positions of the nest. (E,F) Four categorizations of participating agents were discerned to define single focus bees: the wave direction dir_WAV (F₁); the wave strength level (c_WS) as a measure of the response strength, which refers to the maximal _relXYmov values of a wave incident (F₂); the trigger direction dir_TRIG gives the angle of the triggering neighbour in the near neighbourhood (F₃); the oval area with its displacement from the centre shows the direction of the maximum activity of neighbours in the far neighbourhood (F₄); in (F₅), the paradigm of direct-sector and opposite-sector analysis was introduced. (G) Pooling of synchronized and categorized 3D data (Δx, Δy, Δz) and wave strength values _relXYmov. (H-J) Using the parameters defined in F_1–5, three criteria of bucket bridging were assessed (dark-grey flow charts H,I: linearity, continuity, graduality) and the respective hypotheses (directed-trigger-, active-neighbours- and gradual-transfer-) were tested (J_1–3). The panels in D, F and I illustrate the results of the processes given in the grey or black boxes (see referenced figures and movies for details).

https://doi.org/10.1371/journal.pone.0036736.g003

Identification of Surface Bees

In a Giant honeybee nest, the colony members are arranged at both sides of the central comb in several layers. A nest has several functional regions (Fig. 1) such as the mouth zone, the attachment zone, the rim zone and the quiescent zone [14], [20], [33], [34]. Shimmering behaviour is displayed primarily by the surface bees in the quiescent zone [7], [35]. More than 600 individually identified surface bees per frame were tracked (Fig. 3A,B; the identification of the positional coordinates defines them as “agent bees”), summing up to 14549 wave incidents which were documented in successive film sessions (defined as a sequence of images tracked continuously in the course of a single experiment). The identification of individual agent bees throughout a single session is challenging because Giant honeybees in a colony are extremely similar in morphology, are densely clustered, and show rapid movements in 3D during their abdominal flipping [16]. In addition, an individual bee that has sensed an incoming wave front due to the 3D movements of her neighbours is free to decide whether or not to join the wave, and if she joins, whether to raise her abdomen strongly or weakly. Weak participation of individuals in shimmering is difficult to detect by automated analysis. Furthermore, it is critical to distinguish active “movements”, i.e. abdominal flipping, from passive “motions” caused by the surrounding bee curtain.

Definition of the Wave Directions

The architecture of the bee curtain of Giant honeybee nests is determined by three characteristics: first, by the orientation of surface bees which hang, particularly in the quiescent areas, with their heads upwards and their abdomens downwards (Fig. 1B; [7], [14]); second, by the particular structure of the nest, which suspends freely from overhead structures to which it is attached; and third, by the polarity between the nest centre of the mouth zone and the quiescent periphery. Four key directions of wave spreading were defined (dir_WAV = 1-4; Table 1), namely two horizontal directions (from Right to Left [_fromR_toL], from Left to Right [_fromL_toR]) and two vertical directions (from Bottom to Top [_fromB_toT], from Top to Bottom [_fromT_toB]), numbered clockwise, starting with _fromR_toL. For that, shimmering waves in these key directions were identified visually from recorded movies.

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Table 1. Definitions of categories of directions (dir) and sectors around a focus bee.

https://doi.org/10.1371/journal.pone.0036736.t001

Assessment of the Motion Strength at the Individual Level

For each identified agent bee (Fig. 3A,B) a set of data was available frame by frame [16]. We introduced the parameter _relXYmov (Fig. 3C,D) to categorize the motion strength of individual agents throughout the experiment [16]. This measure allowed the assessment of the precise time of the arrival of a wave at an agent’s position. A reliable trigger criterion was found by detecting the luminance changes in two sequential frames (f_i-1, f_i) in a pixel-wise subtraction creating a difference image [16]. A region of interest (ROI; Fig. 1B, Fig.4A) of 60×60 px around an agent bee was defined, with the centre point positioned in the middle of the thorax corresponding to 18×18 mm in real-world coordinates. The size of the ROI was chosen in conformance with the mean side-to-side distances between surface bees [16] covering 80% of the agent bee, including most of the abdomen, and also considering the stretched-wing condition where the two wings on each side are presented separately without overlapping, which is characteristic of surface bees. A larger sensor field would have interfered too much with the motion activities of the neighbours. We recorded 8-bit px values regarding the differences in luminance (Δ lum) between two successive frames assessed by pixel-operated subtraction [16] with the references as black [Δ lum = 0] and white [Δ lum = 255].

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Figure 4. The time courses of wave strength of focus bees and their triggering neighbours.

(A) Details about the assessment of the image-based _relXYmov values (see Methods); yellow square gives the 60×60 px region of interest (ROI) around the selected focus bee; the red circle defines the near neighbourhood (r < 40 mm); the white lines in the background show the eight sectors of neighbourhood (dir_Nh = 1–8). (B) Time courses of the _relXYmov values of focus bees at different wave strength levels (dir_WAV = _fromB_toT; c_ws = 1-6; n = 2908 wave incidents). The contours of the green areas show arithmetic means, vertical grey bars show mean errors. Black lines are means of _relXYmov values of the triggering neighbours (radius_Nh<40 m) regarding the paired, focus bee-related wave incidents; mean errors are not displayed here. Red vertical bars signify the time points t₀ defined by the rapid onset of shimmering activity (quantified by the ordinate _relXYmov values) of the focus bees. (C) How long does it take that information of shimmering is transferred from the neighbours in the far neighbourhood (radius_Nh<100 mm) to the focus bees? Estimation exemplified for dir_WAV = _fromR_toL; abscissa, wave strength level; ordinate, time interval in ms in which information has been transferred, with means (red dots) ± SE (black vertical bars), n = 14 threshold levels (_relXYmov = 1.0–2.3 in steps of 0.1) for the neighbourhood activity; ordinate values are calculated by weighted interpolation and cross correlation of the time courses between focus bees (fb) and their trigger neighbours (nb) of the same wave strength level (N_fb = 2824; N_nb = 29237).

https://doi.org/10.1371/journal.pone.0036736.g004

These Δ lum values represented motion activities of a selected agent bee regarding the whole body within 16.67 ms. These motions were quantified by the parameter _relXYmov and include positional changes in horizontal (x-) and vertical (y-) directions, but also movements of head, abdomen, and extremities, such as legs, antennae or wings. This value is therefore affected by various behavioural contexts: first, by passive deflections of the whole body in x-, y- and z-directions which originate from the immediate neighbourhood [16]; second, by active movements of the body parts such as observed in flickering [36] or shimmering [14], and third, by locomotor activities of the whole body such as moving around on the nest surface or penetrating into deeper layers of the nest. For quiescent conditions the Δ lum values typically were on average 3 per px, which summed up to _relXYmov = 1.08 * 10⁴ for the whole 3600 px of the ROI. Massive shimmering activity reached on average tenfold _relXYmov values. Therefore, the _relXYmov value was an appropriate parameter to quantitatively describe the level of the individual participation of an agent bee in shimmering waves, i.e. the wave strength (ws) of a given wave incident [16].

Determination of the Time Zero of a Singular Wave Incident

When a wave front reaches an agent bee, weak passive deflections are detectable, but when an agent bee starts raising her abdomen the ws-value sharply rise and peak within 40 ms (Fig. 3C). This effect was utilized, first, to determine the starting time (t₀) of an individual shimmering incident. The time point zero t₀ was defined one frame before the time t₁ at which the sharp rise of _relXYmov value was detected (t₀ = t [f₁–1]). Second, this rapid rise of the _relXYmov value was also utilized to synchronize the multitude of shimmering incidents which took place, simultaneously or successively, at the various agents of the whole nest. For that, we defined an additional threshold level of Δ ws_th = 10 to identify an agent bee which participated in the shimmering wave. An agent must have shown Δ ws-values which exceeded this threshold for at least five successive frames (Δ _relXYmov (t₁,…t₅) > Δ ws_th), to exclude noise-triggered processes. Both rules were important because the parameter _relXYmov was also used to categorize eight levels of participation of agent bees in shimmering (c_ws = 1–8). The assignment of c_ws = 1 served as the minimum level of shimmering activity of an agent bee (Fig. 3C,D) and unambiguously excludes sub-threshold shimmering activities.

The Concept of the Focus Bee

The participation in shimmering varied strongly from agent to agent. This individual bias is caused by a series of agent-specific factors such as the location in the nest, the individual’s age, the direction of wave spreading at the agent’s location, the level of the passive participation (i.e. motion without abdominal flipping) or of the active participation (i.e. motion caused by abdominal flipping at variable strength), and lastly the participation of the neighbours of the selected agent in the wave. Therefore, we introduced the concept of the focus bee (fb) to enable the pooling and statistical evaluation of the data in particular. For every focus bee, we defined two areas of neighbourhood (Fig. 1A,B), the near neighbourhood (radius_Nh<40 mm) and the far neighbourhood (radius_Nh<100 mm), and selected those neighbour bees (nb) which participated in shimmering before the wave front had arrived at the focus bee. As a consequence, an agent bee was treated in the evaluation either as focus bee or as neighbour bee of a previously defined focus bee.

Triggering the Participation in Shimmering

For a given focus bee, the _relXYmov value (Fig. 3C,D) was utilized to define the arrival time (t₀) of the shimmering front and also the level of participation in shimmering (c_ws-value). For the nest mates around a given focus bee, the rapid rise of the _relXYmov value at t₁ = t[f₁] was used to assign the potential trigger neighbour (tn) for a given shimmering incident by considering three criteria: (a) The trigger neighbour had already participated in the same wave, but only for a maximum of 5 frames (Δt≤88 ms) before the focus bee itself had started her shimmering activity. (b) The trigger neighbour was closest of all other candidates to the focus bee and was positioned in her near neighbourhood which made up not more than 5–7 agents. This rule excluded those bees from analysis which generated daughter waves [22]. Such surface bees were flipping their abdomens singularly or in a small group and were farther away from other active bees than defined by the near neighbourhood. (c) Furthermore, the angular position of the trigger neighbour (α_tn) defined the trigger direction (dir_TRIG) of the focus bee. For that, eight sectors of neighbourhood (dir_Nh = 1–8) circularly around the focus bee were defined in steps of angular ranges Δα_Nh = 45° (Table 1). Consequently, the trigger direction of the focus bee was defined by the angular sector (dir_TRIG = 1–8) in which the trigger neighbour was positioned (Table 1).

Relative Time Scales

The time point t₀ (Fig. 3D) of a focus bee defined the frame before the focus bee displayed the supra-threshold movement of abdomen flipping. This reference established a time scale for both partners in the trigger process, the triggered focus bee and her triggering neighbour. The time courses of the movements of both agents (fb,tn) were registered and synchronized to the time zero of either the focus bee herself (t₀[fb]) or in the case of the triggering neighbours (tn), to the time zero of the focus bee to which they had been referred to. This synchronization conjoined the movements of both agents by the positional (x,y,z) parameters and by the wave strength (_relXYmov). Such wave processes of identified focus bees and of their neighbours collected at different locations and times were consecutively synchronized and sorted according to behavioural categories such as wave strength c_ws, main wave direction dir_WAV and trigger direction dir_TRIG (Table 1).

Stimulation by Dummy Wasps

Waves were elicited using a cable car device [16] with a dummy wasp positioned at the outer sun-oriented nest side below the mouth zone (shown on the bottom right side of the experimental nest in movies S1, S2, and S3). Using this computer-controlled device [16], [35], the dummy wasp could be moved at variable velocities (0.1 - 0.5 m/s) and directions (Fig.3A). The dummy consisted of Styrofoam (L×W×H: 40×15×15 mm) with white, yellow and black painted stripes. The dummy was fastened to the cable car by a thin thread, so that is was freely swinging with the length axis horizontally. In the experiments described here, the dummy wasp was moved at an angle of 90° to the nest surface. For more intense stimulation we used the manual technique of moving a rod with a swinging dummy of the same geometrical pattern as under cable-car conditions.

Categorization of Active Participation in Shimmering

Individual agent bees were identified on the surface of a Giant honeybee nest by stereoscopic imaging (Fig. 3A,B; [11]) for 66 manually selected shimmering waves (Fig. 3C). Twenty-two independent data sets of four wave directions (dir_WAV = 1–4) were analysed. Only a subset of 53.06±3.05% of identified agent bees were found to participate actively in shimmering, defined by flipping their abdomens at an above-threshold strength level (c_WS = 1-6). Bees that did not participate above this threshold were excluded from further analyses. Focus bees (Fig. 3E) received the mechanical cues from their nest mates in the near neighbourhood. The category of focus bees constituted 75.72±2.90% (n = 22 data sets of 10715 identified bees) of those bees which actively contributed to shimmering. The complementary part of 24.28% active bees flipped their abdomens without having received a mechanical cue from their immediate neighbours. These bees were likely to be members of an alternative spreading mechanism [22] and were excluded from further evaluation (Fig. 3E–H).

The Time Courses of Wave Incidents of Focus Bees and Their Triggering Neighbours

By definition, the focus bees started their participation in shimmering at t₀, after they had received triggering cues from their active neighbours. Prior to t_0, the shimmering activities rose slowly within 200 ms (Fig. 4B) which was caused in the arrival phase of the shimmering front by the movements of the neighbours [16]). This passive motion of the focus bees can be considered as a potential decision-enabling cue which may trigger, or at least influence, the start and strength of the subsequent participation. In particular at higher wave strength levels, the subsequent course of _relXYmov values displayed damped oscillations, which were due to resonance conditions of the bee curtain [16].

Fig. 4B also shows in black lines the curves of the neighbours in the far neighbourhood of the focus bees assembled in their trigger direction. For that, only those neighbours were selected that had flipped their abdomens at the same wave strength as the focus bee. This enabled the assessment of the time for the information to spread from the neighbours to the focus bees which lie at 47.29±1.12 ms (Fig. 4C; mean ± SE; n = 6 wave strength levels; averaged for threshold levels of 1.0≥_relXYmov≥2.3, Fig. 3C). For small and higher wave strength values (c_WS = 1; c_WS>4) the transfer times are significantly shorter. Taken in account the average distance of 62 mm between the focus bee and neighbours in the far neighbourhood, bucket bridging achieved a speed of 1.311±0.030 m/s for this step of wave propagation.

Testing Linearity of Wave Propagation

In shimmering, the directed-trigger hypothesis.

(1a)can be tested at each focus bee by the coincidence of trigger direction and wave direction (Fig. 3H₁). We proved this aspect of linearity (Fig. 3I₁) using four independent data sets (dir_WAV = 1–4; Table 1) of selected sample waves. In detail, we checked whether and how the angular positions of the triggering neighbour coincided either with the direction from where the wave came (conforming to equation 1a, ) or with the opposite direction, into which the wave spread.

(1b)Strict linearity in wave propagation would support that the participation of the focus bee is triggered from the side where the wave came from, but not that the trigger process is launched from the opposite direction.

In a first step of analysis, the focus bees were screened for their participation in shimmering. In all four data sets (Fig. 5A) the normalized counts _reln_fb were Gauss-distributed around mid-level strengths (c_ws = 2–4); the reference (_reln_fb = 1.00).

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Figure 5. Correspondence of trigger direction and wave direction in individual focus bees addressing linearity of wave spreading in shimmering.

The ordinates show the rate of wave incidents assessed at individual focus bees (_reln_fb) and normalized for the maximum number of incidents per wave direction (_reln_fb = 1 for MAX n_fb [dir_WAV]). (A) Distribution of wave incidents regarding the four wave directions (dir_WAV = 1–4). The focus bees were distinguished according to their individual wave strength levels (c_ws = 1–6) in different colours; for evaluation, only those focus bees were selected that had been triggered by an immediate neighbour in the near neighbourhood (radius_Nh<40 mm); red, from Right to Left (_fromR_toL), n = 2825 wave incidents; green, from Bottom to Top (_fromB_toT), n = 2908; orange, from Left to Right (_fromL_toR), n = 2433; blue, from Top to Bottom (_fromT_toB), n = 1256. (B-C) distributions of wave events regarding the trigger directions of individual focus bees (dir_TRIG = 1–8; for definition, see Table 1). Angular sector lines in (B) and abscissa in (C) show the trigger angle α_TRIG. (C) The curves show the regression polynomials (for specification see Table 2); the grey vertical bars refer to the main wave directions coded by the median as α_WAV (dark grey) and the tolerances of ± 45° (bright grey).

https://doi.org/10.1371/journal.pone.0036736.g005

(2a)was the total number of wave incidents at a given wave direction. We then pooled the data of all wave strength levels (c_ws = 1–6), and sorted them according to their trigger direction (dir_TRIG = 1-8; Table 1). The data sets showed (Fig. 5B) oval-shaped angular distributions with clearly positioned minima and maxima (Fig. 5C). Furthermore, the maxima referred to those trigger angles (α_TRIG) that conformed to the direction from where the wave came (equation 1a), while the minima were displayed at those trigger angles that conformed to the angle where the wave spread to (Equation 1b). Both results were replicated with four independent data sets, and reliably support the directed-trigger hypothesis which addresses the linearity principle in spreading (for test statistics see below).

However, the results of Fig. 5 also demonstrate that the propagation process is affected by a strikingly high level of fuzziness, unpredictability and non-linearity. In detail, the normalized counts _reln_fb correlated along the trigger angle with the polynomial regression curves revealing maxima of 0.152±0.004 and a distance between minima and maxima of 0.055±0.004 (n = 4 data sets of dir_WAV; further specifications of the polynomials of Fig. 5C, see Table S1). An average base quantity of focus bees (0.097±0.002; n = 4 | dir_WAV = 1–4) was observed in each of the eight angular sectors of trigger direction. This means that for a socket of nearly 80% of focus bees, of a total of 100% per behavioural condition (n = 24 | c_ws = 1–6; dir_WAV = 1–4), the trigger angles did not correlate with the main wave direction.

Here, the question arises, to which extent the polynomial regression curves (Fig. 5) may then signal linearity in the propagation of shimmering waves. We confirmed this by a simple test considering the counts of those angular sectors (n_s) which coincide in wave and trigger direction in the four data sets (dir_WAV = 1–4). Linearity would here be obvious if the _reln_fb-values above median level (see ordinate values in Fig. 5) referred to the sectors of wave direction (s_WAV ≡ s_TRIG), proving coincidence of both directional paradigms (dir_WAV ≡ dir_TRIG) in the shimmering process. To test this, we expanded the critical directional window by α_WAVnew = α_WAVorig ± 45° (which increased the numbers of critical sectors in the comparison by the factor of 3). This is legitimate because the manual selection of wave direction had a similar level of tolerance (see Methods). Our results proved linearity because both quantities of n_s .(2b)coincided here at 100% (Fig.5C). In other words, the majority of focus bees were triggered from the direction where the wave came from(2c)and the minority of the focus bees were triggered from the direction where the wave was spreading to

(2d)In a next step, the level of linearity in wave propagation was estimated in two ways (A,B). For estimate_A, we integrated the percentages of participations above the fuzziness level (which was defined by the minima of the polynomials; Fig. 5C). Estimate_A referred to 20.99±2.38% (n = 4 data sets of dir_WAV) of the total of identified wave incidents and signifies the overall variability in directedness in spreading. For estimate_B, we summed the percentages of participations only above the median level of _reln_fb = 0.1253±0.0015 (n = 4 data sets of dir_WAV; equation 2e; Fig. 5C).(2e)

This estimate_B was 6.15±1.05% (n = 4) of identified wave incidents and included only the excess quantities of wave incidents, which were primarily responsible for the directedness of shimmering waves.

Summarizing, the data strongly support the directed-trigger hypothesis. However, the rates of focus bees conforming to equation 1a and considered by estimate_B were only 6.15% of all identified wave incidents. Therefore, the principle of directedness in spreading was extremely outnumbered by the complementary part of 93.85% of focus bees which were triggered from random directions, independently of the main propagation direction of the waves.

Testing Continuity of Wave Propagation

We used the sample size of the far neighbourhood (radius_Nh<100 mm) of focus bees to characterise continuity in propagation by the circular distribution of nest mates which had been active in shimmering before the focus bee herself started to participate (Fig. 3I₂) addressing the transmittance of information over a distance that was greater than that between immediate neighbours. The active-neighbours hypothesis is proved in Fig. 6 which displays the respective results of normalized data of four independent tests (dir_WAV = 1-4). The relative numbers of active neighbours (_reln_Nh) in the far neighbourhood.

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Figure 6. Proof of linearity and continuity in wave spreading of shimmering as documented by neighbour-bee related analysis.

(A) Arithmetic means (curves) and SE (vertical bars) of the relative numbers of shimmering neighbours (_reln_Nh) in the far neighbourhood (radius_Nh<100 mm) around the focus bees regarding to the wave strength levels c_ws = 1-5; abscissa: Δα_{TRIG | Nh}, the deviation from the trigger angle in degrees at which the focus bees were activated to participate in shimmering, therefore Δα_{TRIG | Nh} = 0 refers to the trigger angle (trigger direction). Four data sets representing different spreading directions of shimmering waves (for colour codes, see Fig. 5): _fromR_toL, n_nb = 138190 neighbours; _fromB_toT, 105356; _fromL_toR, 159417; _fromT_toB, 93199; (B) arithmetic means (curves) of the relative numbers of shimmering neighbours in the far neighbourhood (_reln_Nh), considering four classes of wave strength levels were considered: darkest grey lines, c_ws = 4-5; brightest grey lines, c_ws = 1–2. The curves show the respective regression functions of the means (see Table S2); SEs are not shown. (C) Angular diagrams of relative numbers of shimmering neighbours (_reln_Nh ) of only those focus bees selected for mid-level wave strength activation (c_ws = 3): _fromR_toL, n_nb = 4652 neighbours; _fromB_toT, 4458; _fromL_toR, 7182; _fromT_toB, 2295; background sectors give the trigger angles α_TRIG of the focus bees (C₁) focus bees which were triggered in the direction where the wave had come from (equation 1a); (C₂) focus bees which were triggered in the direction opposite to that under C₁ (equation 1b).

https://doi.org/10.1371/journal.pone.0036736.g006

(3a)for Nh<100 mm with max _reln_Nh(c _BEHAV) = 1.0

(3b) (3c)

For a better survey on the intrinsic data structure, we offer three subsets of diagrams: Fig. 6A refers to the data pool for the wave strength levels c_ws = 1-5, Fig. 6B distinguishes between different groups of wave strength levels (for details, see Table S2), and the circular diagrams in Fig. 6C exemplify only data for the specific wave strength level c_ws = 3.

In total, 48 sectors of 20 behavioural conditions (c_ws = 1-5, dir_WAV = 1–4) were evaluated. In all the four data sets the numbers of active neighbours were found to peak in the trigger directions (Δα_{TRIG | Nh} = 0°). This means that the focus bees were triggered exactly (P<0.001; t-test) from those directions where most of the neighbours had been active immediately before. The maxima of the graphs (Fig. 6A) increased above a level of _reln_Nh = 0.80, which corresponded to the number of n_Nh = 16.63±0.42 (mean ± SE) active neighbours per focus bee. For the angular sectors adjacent to the peaks, the graphs dropped down to _reln_Nh = 0.60, which corresponded to 11.81±0.087 neighbours per focus bee and 71.97% of the maximal numbers of active neighbours. The complimentary value of 28.03%.(3d)gives a usable estimate of the excess rate of neighbours found in the trigger directions of the focus bees. With the reference of the trigger angle of active neighbours (equation 3b) this result delivers an estimate of continuity in wave propagation which was fourfold higher than the linearity aspect detected in the context of trigger direction and near neighbourhood (Fig. 5C).

The curves of the normalized _reln_Nh values were robust regarding wave strength (Fig. 6B). Interestingly, the preferences for the wave directions were remarkably dominant (Fig. 6A-C) which is exemplified in Fig. 6C for focus bees acting at mid-level wave strengths (c_ws = 3). Here, the directional dependencies of the numbers of actively shimmering neighbours are displayed under two different conditions for the four regimes of wave directions (dir_WAV = 1-4): In the left column, focus bees were selected that corresponded in trigger direction and wave direction (according to equation 1a). The angular graphs are oval-shaped with significant (P<0.001; chi square test) preference for the wave direction. The values differed between max and min values by Δ_reln_Nh = 33%. In the right columns, the samples refer to focus bees that were triggered from the direction opposite to the wave direction (according to equation 1b). These wave incidents showed a much weaker but still significant (P<0.001; chi square test; Δ_reln_Nh = 22.65%) preference for the wave direction.

Testing Graduality in Propagation

There are several ways how surface bees may participate in a shimmering wave. It may happen (a) as an all-or-none decision, to participate, for example, at full wave strength, if the surrounding activity had exceeded a certain threshold level. The circumstances of participation may be even more complex, if it were true that (b) a single focus bee has the freedom to decide whether or not to participate, and if she does, at which strength level. Finally, (c) it can also be a matter of graduality whereby focus bees respond with variable wave strengths dependent on the activity level of the surrounding neighbours.

If a focus bee receives the signal to start abdominal flipping as a gradual message from her surrounding neighbours, she has, in theory, two options: first, she could match the strength of her abdominal flipping according to the numbers of neighbouring bees that flipped their abdomens immediately before. This aspect has already been demonstrated in two independent test sets: we showed (a) that the rate of shimmering incidents was higher if the focus bee was triggered in wave direction (Fig. 5B,C), and (b), that the focus bee must have received stronger mechanical cues from her neighbours in her trigger direction, because at this angle more neighbours had been active (Fig. 6). Both findings, although highly significant, referred only to a minority of focus bees, at a proportion of less than 10% in Fig. 5B,C or up to a quarter in Fig. 6A,B of the full number of identified wave incidents.

As a second option, a small number of very active neighbours may also elicit a strength level that could override smaller activities of a greater number of neighbours. To address this aspect, we correlated the wave strengths of focus bees with those of their neighbours in the far neighbourhood. To simplify this survey, we concentrated on the neighbours in only two defined directional sectors of neighbourhood: (a) on the sectors of the trigger direction (ds: direct-sector analysis), and (b) on the sector opposite to the trigger direction (os: opposite-sector analysis). We then assessed the correlations (r_ds, r_os) that referred to the comparison of the wave strengths of focus bees with those of their neighbours in the selected sectors. The correlations either delivered proportional effects, if the wave strengths of focus bees correlated positively with those of their selected neighbours (with k>0).(4a)or they delivered antagonistic effects, if the wave strengths of focus bees correlated negatively (with k<0 in equation 4a).

In detail, we pooled the data of identified focus bees and their neighbours under 184 test conditions (n_fb = 10911; n_nb = 496162; c_BEHAV = 184; dir_TRIG = 1–8; dir_WAV = 1–4; c_ws = 1–6) and received positive correlations under both (ds, os) conditions (r_ds = 0.0594±0.0174; r_os = 0.0381±0.0175; means ± SE; n_r = c_BEHAV), but the regression coefficient was only slightly higher for the direct-sector analysis (P = 0.1944, t-test). In a next step, we assessed the regression data separately for each wave strength level (Fig. 7A,B; c_ws = 1–6) and achieved significantly positive values under weak and higher strength levels (e.g. for c_ws = 5: r_ds = 0.1600±0.0426) which also delivered significant differences between ds- and os-analysis (P = 0.0017; t-test; c_BEHAV = 24: dir_TRIG = 1-8, dir_WAV = 1–4; n_fb = 1 203; n_nb = 25084), in particular for the strong participation in shimmering (c_ws = 5). These results document that focus bees behave in a gradual and thus, directed manner if they responded at high wave strengths to the mechanical cues of those active neighbours from the trigger directions in the far neighbourhood. Similar responses are documented if the mechanical cues of the triggering neighbours were considered in the near neighbourhood (Fig. 7B). Here the correlation between the _relXYmov values of the focus bees and their triggering neighbours were also higher at higher wave strengths, and there was also a sink of regression values at mid-strength-levels. However, at threshold conditions (c_ws = 1) the correlations were extremely low.

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Figure 7. Proof of graduality in the wave strength between focus bees and their neighbours.

Abscissa, wave strength levels of the focus bee (c_ws [fb] = 1–6). (A) How are the focus bees influenced from their neighbours in the far neighbourhood? Ordinate, the regression coefficients resulting from the comparison of wave strength values (_relXYmov) of focus bees (fb) and those of their nest mates (nb) in the far neighbourhood (r<100 mm); the data were pooled from all sets of wave directions (dir_WAV = 1–4); white columns refer to direct-sector analysis, black columns refer to opposite-sector analysis (for further definition, see text and equations 4a–d); the white curve refers to the distribution of direct-sector data (polynomial: a₂ = 0.0324, a₁ = -0.1763, a₀ = 0.2379, R² = 0.975; n = 5 | c_ws [fb] = 1–5). (B) How are focus bees influenced from their immediate triggering neighbours? Ordinate, regression coefficients resulting from the comparison of _relXYmov values of focus bees (fb) and those of their triggering neighbours (tn) in the near neighbourhood (r<40 mm), considering all wave incidences separately per wave direction (dir_WAV = 1 to 4); polynomial: a₄ = −0.0037, a₃ = 0.0668, a₂ = −0.3926, a₁ = 0.896, a₀ = 0,5691, R² = 0.9994; n = 6 | c_ws [fb] = 1–6); means (full circles) and SE (bars). (C) Numbers of wave incidences selected for B.

https://doi.org/10.1371/journal.pone.0036736.g007

Summarizing, the overall graduality in wave propagation displayed positive correlations between shimmering activities of focus bees and those of their neighbours. However, the overall achieved value with r_ds<0.06 can be used to estimate the occurrence of graduality in agents participating in shimmering. The coefficient of determination R² = r_ds² is here a measure of the goodness of the fit. Therefore, only 0.36% of agents participating in shimmering are controlled by the principle of graduality. If the wave strength levels were considered separately, the results proved graduality (Fig. 7A; equations 4a) in particular for focus bees acting more forcefully (c_ws = 5). Here, the regression coefficients achieved values of r_ds = 0.16 (R² = 0.0256) representing 2.56% of the selected agents. These results also differed significantly between ds- and os-analyses. Such polar conditions preferentially occur at the very frontline of shimmering waves [14], [22]. Conversely, the bees at mid-level wave strength (c_ws = 2–3) which constituted up to 85% of all agents (Fig. 5A, Fig. 7C), were not identified as candidates responsible for gradual propagation.

Properties of Wave Propagation in Shimmering

Shimmering behaviour is one element of the defence behaviour in Giant honeybees and has repelling properties upon potential predators [3]-[7]. When observed by naked eye, shimmering waves originate from particular nest areas around the mouth zone [10], [14] and propagate to the periphery in a Mexican-wavelike [21] way, spreading seemingly linearly and continuously across the nest, mostly with a clearly visible front line (movie S1). In this study, we investigated three paradigms of this wave propagation, which can be assigned as linearity, continuity and graduality (summarized in Fig. 3H-J), which affect direction, velocity and gain of the bucket-bridging processes [14] in shimmering.

We applied a set of recording techniques and analytical approaches that enabled us to investigate the propagation features of shimmering waves with four single-agent-based concepts (Fig. 3): (1) the (re-) identification of single agent bees at the nest surface and tracing them throughout the recorded sessions (Fig. 3B, [16]); (2) the detection of the participation of agent bees in shimmering, defining the starting time and the relative time scale of individual abdominal flipping episodes (termed as wave incidents), enabling synchronisation of wave incidents in subsequent pooling (Fig. 3C,D); (3) the measurement of the movement strength to quantify the participation in a wave (Fig. 3F₂); and (4) the implementation of the concept of the focus bee as the addressee of the trigger process (see Methods), who receives directional information from the triggering neighbours in the near neighbourhood or from other agents in the far neighbourhood (Fig. 3F₃). All directional properties assessed for a focus bee lastly refer to the partitioning of her neighbourhood into eight angular sectors (Fig. 3F_3,4).

Self Organization in Shimmering

Evidence is emerging that shimmering possesses a series of properties which categorize this behaviour as a self-organized process [11]. Two points to consider are, first, the adaptive value of the signal for external addressees. It is known [7] that shimmering can change in appearance and goal setting if a single parameter changes. For example, we found that the distance between a wasp and a Giant honeybee nest alone causes a bifurcating [11] situation for the attacking wasp to assess the shimmering pattern either as defensive or attractive. The second point to consider is the underlying mechanism of pattern formation. In this context, this paper gives a series of hallmark data to quantify the spatial and temporal propagation mechanisms.

Self-organised collectives such as densely clustered surface bees in Giant honeybees [11], [37] guide their actions by simple behavioural rules. One of the simplest rules that play a role in wave propagation generally [11] is that agents acquire information by monitoring their nearest neighbours. Here, this principle has been addressed for three paradigms of wave propagation in shimmering, linearity, continuity and graduality. The question was if and in which way these paradigms concur with the concept of self-organization [11]. For wider comparison, we consider three collective animal behaviours, fish schooling [38], bird flocking [11], [39], and fire-fly flashing [11], [40]–[42]. All these behaviours are supposedly self organized [11] and utilize neighbourhood parameters, but differ essentially in their specific design and goals.