CT54.A6.CombinedSwarms — the science of this version

This document describes the simulation as it is in CT54.A6, and only that. It does not cover earlier versions. How to run the program is in the Audit (CT54.A6.CombinedSwarms_AUDIT.md / .html); the build history is in the context file (CT54.A6.CombinedSwarms.md).

0. In one page

The program simulates a honeybee reproductive swarm in the minutes after it leaves the hive. The bees mill in a loose cloud, find the queen by her scent and gather around her. Nothing is scripted: each bee follows a few local rules, and the timing, size and shape of the result come out of them.

There are five options:

Options 1–4 model behaviour that researchers have measured and described. Option 5 is an experiment. It is based on what the author once saw a swarm do. No published study describes it, and §6 sets out plainly how it differs from what is known.

Time is compressed about 8×. A real swarm takes 10–20 minutes or more to condense into a cluster (Seeley 2010). In the simulation the cluster forms over about 2–3 minutes of simulated time. The scent rates and smell thresholds set the pace (§4.5), not a timetable.

1. What a reproductive swarm is

• Colony division. A honeybee colony reproduces by dividing. In late spring a crowded colony rears new queens. The old queen and roughly half to two-thirds of the workers then leave together, about ten thousand bees, ~1.5 kg. The rest stay behind with a new queen (Seeley 2010; Clemson HGIC).
• Preparation. Before leaving, the workers fill their crops with honey, fuel for the move and for building comb later. The queen is slimmed down beforehand so that she can fly: workers feed her less, and she lays fewer eggs (Wikipedia, Swarming (honey bee)).
• The exodus. At the signal (worker piping and buzz-runs), the bees pour out and take to the air. They fly around one another in a cloud about 10–20 m (30–60 ft) across (Seeley 2010).
• The bivouac. The cloud does not go far. Within about 30 m of the hive the bees settle into a temporary cluster, usually hanging under a branch. They stay there for hours, occasionally a day or more, while scouts look for a permanent home (§8).

The simulation covers the milling cloud and the forming of the cluster (options 1–4), or the forming of the flying ball (option 5). Choosing a new home and flying off are not simulated (§8).

2. The scene

• Setting. An old, partly collapsed wooden arbor (a pergola) stands at the edge of a summit high above the Adriatic, about an hour before sunrise. The swarm gathers under its sagging front beam, about 9 ft up.
• What the bees can hang from (options 1–4, the "Hangs from" setting).
• A thin vine stem (the default), 1.8 in across, slung in a loop under the beam. Swarms most often hang from branches (Seeley 2010).
• The beam's flat underside, a rough timber face 0.55 ft wide. Old wood gives the bees' claws plenty of grip, and the beam shelters the cluster from above.
• The sky is the sky seen from 41.7° N (the latitude of Monte Gargano) on an autumn morning before dawn. The waning crescent moon hangs low in the east, with Venus (the morning star) and Jupiter, the last points of light left as the sky brightens. The dawn brightens with simulated time, so a movie of the same run always looks the same. The planets' positions are plausible for such a morning, not an ephemeris for a particular date.
• Artistic licence. Swarms issue in the warm middle of the day, usually between about 10 am and 2 pm (Minnesota Hobby Beekeepers Association), and bees can barely fly below about 10–13 °C. A summit before dawn is not when or where a swarm would fly. The setting is a choice for the picture, not part of the science.
• The measuring pole (option 5). A pole 1 in across and 10 ft high, with a white band every foot (red at 5 and 10 ft), stands near the queen's second hovering spot. It is there to judge the ball's height and size.

3. The bees

3.1 Who is in the swarm

Drone numbers in swarms vary widely. The 3 % and 15–17 mm are reasonable illustrative values, not measurements.

• Default numbers: 7,500 bees for options 1–4 and 5,000 for option 5 (the Bees selector offers 500–20,000). A real prime swarm is about 10,000.
• In the physics each bee is a sphere with the same volume as its body (an ellipsoid of length L and width W): diameter (L·W²)^⅓, about 6 mm for a worker. Ocko & Mahadevan (2014) model a bee the same way (≈ 0.65 cm).

3.2 How they are drawn

• Body. Abdomen, thorax and head, in honeybee proportions of the bee's own length.
• Eyes. Five, as a real honeybee has: two compound eyes on the sides of the head and three ocelli in a triangle on the crown.
• Wings. In flight a honeybee's fore and hind wings are hooked together (by the hamuli) and work as one surface, so each side is drawn as one clear wing. At about 230 beats a second no frame rate can follow the stroke, so each frame shows a different phase, as in real footage. Hanging bees fold their wings; fanning bees hold them half spread.
• Colours are stylised. Narrow black and broad gold metal bands, red eyes, crystal wings. Real honeybees are banded amber-brown and black.
• Visible size. In the wide shot a distant bee is drawn at least about 3 pixels long (at 1080 lines; the HUD shows the factor; E turns it off). The close-up window shows bees at their true size.

4. The model shared by all five options

4.1 Scent in the air

• The air around the site is a 3-D grid of 1-ft cells (64 × 28 × 64 ft). Every 1/30 s the scent in each cell is updated by finite differences:
• diffusion with an effective, turbulent diffusivity of 2 ft²/s (outdoor air mixes far faster than molecular diffusion);
• decay at 0.08 per second (chemical breakdown and dilution by the moving air);
• open edges, with the ground reflecting.
• The queen's pheromone is the glue of a swarm. Queen mandibular pheromone (9-ODA, 9-HDA and others) keeps the bees together; a swarm that loses its queen does not settle for long (Winston et al. 1989, in NCBI; Seeley 2010). In the model the queen releases it all the time: weaker in flight, at full strength once she has stopped.
• Nasonov scent is the workers' "lemony" aggregation signal (geraniol, citral and others; Schmidt 1994, 1999, in NCBI). Workers that reach the queen raise the abdomen, expose the gland and fan. Fanning bees blow the scent away from the queen, making a relay that draws the rest of the cloud in (Nguyen et al. 2021).
• Options 1–4: 35 % of the bees on the surface of the cluster fan, and their scent is placed 1 ft out from the cluster. More settled bees means more fanners and a plume that reaches farther, so more bees are recruited. This positive feedback is why clustering speeds up and then finishes.
• Option 5: 20 % of the bees in the ball scent as they fly (§6.3).
• S shows a slice of the scent field.
• Units are arbitrary. Only the ratio of the emission rates to the bees' smell thresholds matters.

4.2 Milling: the cloud

• Before a bee smells enough scent it mills: erratic flight with random, persistent turning and changes of speed. These are Ornstein–Uhlenbeck processes, with a mean speed of 13 ft/s and a top speed of 24.6 ft/s. That top speed is the cruising speed of an unloaded worker, ≈ 7.5 m/s (Wenner 1963).
• Neighbours.
• Keep clear: turn away from any flying bee closer than 3 in.
• Stay with what you can see: turn toward where more bees are, within a few feet (4-ft cells).
• Stay with the swarm: beyond 70 % of the cloud's radius, measured from the flying bees' centre, turn back. The radius (20 ft across-ways, 7 ft up and down) shrinks with the cube root of the number still flying, so the cloud keeps its density as it empties.
• The cloud is about 35–40 ft across, inside Seeley's 30–60 ft.

4.3 Smelling and following

• Every bee has its own threshold. Most are sensitive, and there is a long tail of reluctant bees. When the scent where a bee is passes its threshold, the bee becomes attracted: it turns up the scent gradient (chemotaxis) and flies slower and steadier as it closes in. This threshold-triggered gradient following is the rule in the Nguyen et al. (2021) model.
• Seeing. Close in, an attracted bee flies at what it can see: the cluster (within 1.5 ft of its edge, options 1–4) or the queen (within 6 ft, option 5).

4.4 The queen

The queen mills with the cloud, calmer than the workers. Then she leaves it: * options 1–4: at 12 s she makes for the support and alights under the middle of the beam. Real queens land "apparently at random" (American Bee Journal); the stem stands in for wherever she lands. * option 5: at 20 s she flies to a point in the open air and hovers there (§6.3).

4.5 What sets the pace

The scent emission rates, the decay and the bees' thresholds were tuned so that the whole event takes 2–3 minutes of simulated time instead of the real 10–20 minutes. ?threshold= and ?fan= change the pace.

5. Options 1–4: the hanging cluster (documented)

5.1 What real clusters are like

• Size. Swarm clusters are typically 10–30 cm across and hold on the order of 10,000 bees (Thermoregulatory morphodynamics…, J. Exp. Biol. 2022). Experimental clusters of 3,725–9,700 bees were weighed by Shishkov et al. (2022).
• Packing. The bees' own volume fills only part of the cluster. The packing fraction (bee volume ÷ cluster volume) is about 0.2–0.8, higher when cold (Ocko & Mahadevan 2014). Clusters roughly double their volume between 15 °C and 30 °C (Heinrich 1981, as reported in J. Exp. Biol. 2022).
• Structure. The cluster has a dense outer mantle, several bees thick, over a looser core with passages. The bees hang from one another in chains (festoons). The chains are inside the mass, not dangling outside it. In rain the surface bees lie like shingles (Peleg et al. 2018).
• Temperature. The core is kept at about 34–36 °C. In cool air the mantle is held at about 17 °C by shivering bees and the cluster contracts: less surface, higher density (Heinrich 1981).
• Forces. Shishkov et al. (2022) measured the forces inside real swarms with X-ray CT:
• one bee's grip can hold 35 ± 14 bees;
• the bees near the attachment carry on average only 3.8 bee-weights, about 11 % of what they could;
• the load on a layer grows as (mass below)^1.5;
• density is highest near the attachment and falls toward the tip.
• Shape depends on conditions (Peleg et al. 2018; Heinrich 1981; J. Exp. Biol. 2022):
• mild and calm: "an inverted pendant cone", elongated — the classic beard;
• cool: the cluster contracts, "reduces its surface area and increases its density" — compact;
• hot: it nearly doubles in volume, widens where it is attached and flattens — broad;
• shaken from side to side (wind): it spreads "into a flatter conical form"; bees sense the strain and spread their attachment — swaying;
• support: the shape "is also influenced by the surface to which the cluster is clinging".

5.2 How the model forms the cluster

1. Landing where it touches. An attracted bee lands at the point where it touches the queen or a bee already settled (two equal-volume spheres touching at the packing spacing). Nothing reserves a place for it. It lands there only if it can take a hold above it; otherwise it flies on. So bees build up on the underside and the cluster grows downward.
2. Grips. Every settled bee has up to three grips: on the support (one bee per foothold) or on a settled bee above it. A bee never hangs from a bee below it, and at most four bees can hang from one bee. A grip is a link of about one body length (a bee hangs by its legs). Footholds on the support are taken only while the cluster is founding, or next to support-holders that carry more than a quarter of their grip strength. Grip strengths are drawn from Shishkov et al.'s 35 ± 14 bee-weights.
3. Load. Each tick, every bee passes its own weight plus everything hanging from it up to its holds, shared equally. A bee whose load exceeds its grip strength re-grips to a reachable hold above. K shades the bees by load ÷ grip strength, G draws the grips, and N shows the chains inside the cluster.
4. The shell and the outline. The settled bees fill an envelope, a shape of revolution hanging from the support around the queen. Its volume comes from the bees: their own volume divided by the packing for the air temperature (0.6 at 15 °C falling linearly to 0.3 at 30 °C) in the core and 0.62 in the shell. Inside it the bees pack under a light gravity. The outer ~3 bee-widths are the shell: packed tighter than the core, and drawn facing up. Bees outside the envelope are drawn back each step, so the surface is smooth.
5. Its proportions come from the weather setting:

On the flat beam the cluster spreads along the wood: it is no longer than 0.9 × its width, and its top is wider.

Honesty note. These proportions are chosen to match the observations in §5.1. They are not derived from the bees' mechanics. The bees fill the outline, pack, grip and form their chains on their own; the volume is computed; the outline's shape is given. 6. Wind (option 4). Gusts change smoothly every 1.7 s. They push the hanging bees across the beam, more the lower they hang, and the envelope leans with them, so the cluster sways. They also drift the milling cloud.

5.3 Results (seed 3906, 7,500 bees)

These sit within the 10–30 cm (4–12 in) reported for real clusters. With 7,500 bees they are toward the small end, as expected for a swarm smaller than the typical 10,000.

6. Option 5: the hovering ball — an experiment based on the author's observations

Status: an experiment, not a documented behaviour. Option 5 is built from the author's own observation of a swarm. No published study that could be found describes a dense ball of honeybees flying around a queen who hovers in one place, attached to nothing. Everything below about how option 5 behaves is a model of what the bees would have to do to produce what the author saw. It is not a description of known swarm behaviour.

6.1 What the author saw

Years ago the author watched a swarm. There was first the large cloud, about 10 ft across. Then there was a dense, roughly spherical ball of bees 12–18 in across that seemed not to be attached to anything, with the bees still flying. Later there was a similar ball in a different place. The author did not see the ball form, nor see it move from one place to the other.

6.2 What the literature says

• Swarms in flight are sparse. Beekman, Fathke & Seeley (2006) photographed swarms in flight:
• large swarms: clouds 8–12 m long, with 30–80 bees per m³ (mean 50);
• small swarms: 17–31 bees per m³;
• the swarm as a whole moved at 1.7–3.2 m/s; single bees flew at up to 9–10 m/s.

The bees in a flying swarm are tens of centimetres apart. * Clustering means landing. In studies of swarm aggregation, workers that find the queen stand on a surface, face her, expose the Nasonov gland and fan scented air away from her. The cluster forms on something solid around her (Nguyen et al. 2021; SICB abstract on flow-mediated pheromone signalling). * The queen's scent works in flight. It holds the swarm together and draws it to settle (Seeley 2010). * Hovering. Honeybees can hover. A queen holding one spot in still air for minutes has not, as far as could be found, been described. * Queen balling is something else. It is documented, but it is workers clumping tightly around a queen on the comb or the ground, not in the air.

What this means for the experiment. * Density. 5,000 bees in a 15-in ball is about 190,000 bees per m³, roughly 2,000 to 10,000 times denser than any measured swarm in flight. The bees are about 17 mm apart, centre to centre: about one wingspan. * Energy. A worker in flight burns sugar fast. Hovering for many minutes would drain the honey the swarm carries for its move (Seeley 2010). * A possible explanation. A ball that looks free-floating could be a cluster on something small or hidden (a thin twig, a wire, a leaf), with the outer bees still flying and crawling over it so that it looks unattached. The two balls in different places could be an ordinary short relocation: a swarm that has settled can lift off and re-cluster a short distance away, and beekeepers often see this. * Why the experiment is still worth running. The observation stands as an observation. Option 5 tests what it would take for bees to stay close around a queen in open air, and what the ball would do if she moved.

6.3 How the experiment is modelled

1. The queen. She mills with the cloud for 20 s, then flies at 6 ft/s to her station, 7 ft up and 4 ft in front of the arbor, in still air. She holds it, wandering slowly about it (rms about 1 in), and releases her scent at the full rate.
2. The scent. The same air grid as §4.1. Around a point source its steady plume falls off roughly as e^(−r / 5 ft) / r. In-flight scenting: 20 % of the bees in the ball expose the Nasonov gland and add a little scent where they fly. There is no directed fanning stream, because fanning is something bees do standing (§6.2).
3. Joining. A milling bee whose threshold is passed follows the scent uphill (§4.3). Within 6 ft it sees the queen and flies straight at her. Within 1.6 ft it changes to the slow, agile flight of the ball. A bee that loses the scent (below half its threshold) drops back to milling.
4. In the ball each bee has a velocity and feels: * a pull toward the queen: 24 ft/s² inside 0.35 ft, plus 80 ft/s² for every foot beyond; * crowding: a push down the gradient of the density of ball bees, stiffening once the bees are closer than a comfortable spacing of 1.35 body lengths; * its own erratic turning; * its own slow cruising speed: 1.6 ft/s (about 0.5 m/s) on average, 0.7–4 ft/s. Bees can fly slowly and hover.
5. Excursions. Now and then (0.4 % of the bees each second) a bee darts out for 1–3 s and comes back. Bees flung beyond 5 ft follow the scent in again.
6. The ball's size is not set anywhere. It comes out of two balances: the bees' speed against the pull (a bee flying at speed v against a pull g stays within about v²/g), and personal space against crowding, which makes a bigger swarm a bigger ball.

6.4 Results (seed 3906)

• With 5,000 bees: 18 % are in the ball at 60 s, 61 % at 90 s and 96.5 % at 120 s. After that about 98 % fly around the queen in the ball; about 50 are always on their way in or out, and about 30 keep milling.
• The ball comes out at the author's 12–18 in because the pull, the slow speed, the spacing and the crowding strength were tuned to it. The in-flight scent rate was tuned so that the ball gathers in a few minutes.

6.5 Moving the queen

On Move queen (G) she flies slowly, 0.8 ft/s, 10 ft to the measuring pole; pressed again, she flies back. The Follow setting chooses how the ball's bees keep with her:

• By sight (the default): every ball bee is pulled toward her. The ball moves with her almost as one body, lagging less than an inch. This follows from the rule: a pull strong enough to hold 5,000 bees in a 15-in ball is far stronger than a slow move can stretch.
• By scent: a bee knows only the scent where it is. The queen's pheromone has to diffuse to the ball, so she flies out of the ball. The ball keeps its size and shape but trails about 2.5 ft behind her, then closes on her 2–6 s after she stops. This needs the queen's pheromone kept apart from the workers' Nasonov scent (they are different chemicals, and bees tell them apart; Seeley 2010; Nguyen et al. 2021). With one shared smell, the ball, which makes far more scent than she does, keeps to itself and lets her go.

Which is closer to real swarms? Over metres, flying swarms are steered by sight: a few fast streakers fly through the cloud toward the goal and the rest follow (Beekman et al. 2006; Schultz, Passino & Seeley 2008). Over centimetres, inside a dense mass, scent and touch matter more. A real ball would most likely follow by a mix of both: it would lag more than the sight rule and less than the scent rule, and some bees would lose contact.

6.6 What is tuned or not real in the experiment

• Tuned to the author's observation: the pull, the slow speed, the spacing, the crowding strength.
• Not known or not real: the density (§6.2); a queen hovering for minutes; no collisions between bees, and no aerodynamic wash between bees this close (at 17 mm they would fly in each other's downwash); the energy cost.

7. The sound

• Where the buzz comes from. A flying bee's hum is its wingbeat, a fundamental with many harmonics. Published wingbeat frequencies run 190–250 Hz (American Bee Journal, "Sounds of the Hive"). In swarms, buzz-running bees make 180–250 Hz, and bees pressing on one another add a broader noise up to ~500 Hz with a component near 5 kHz (Z. vergl. Physiol.).
• Many bees. Bees are not in step, so their sounds add in power: N bees are √N times as loud as one, not N times. Loudness falls as 1/distance.
• A passing bee changes pitch (Doppler): up to about 2 % at the cruising speed of 24.6 ft/s against a speed of sound of ~1,125 ft/s.

How the program makes it (synthesised, no recordings):

In a movie the same sound is re-created sample by sample for every movie frame, so it is locked to the picture.

Not real (yet): the timbre is synthetic; no wind, ground reflection or air absorption; no worker piping before lift-off.

8. What comes next in nature (not simulated)

The cluster is only a bivouac. Usually within a few hours, occasionally after two or three days: 1. Scouts — a few hundred of the oldest foragers, 3–5 % of the swarm — search for a cavity of about 40 litres with a small entrance (Seeley 2010). 2. A scout that finds one comes back and dances on the surface of the cluster. Better sites get longer dances. 3. The choice is made at the site: when about 10–20 or more scouts are there at once, they sense a quorum (Seeley & Visscher 2003). 4. Those scouts pipe on the cluster. The mantle bees warm their flight muscles to about 35 °C, which takes about an hour (Seeley & Tautz 2001). 5. Lift-off: within a minute or so the cluster dissolves into the air, from the surface inward. 6. The flight: the cloud is steered to its new home by streakers flying fast through its upper part (Schultz, Passino & Seeley 2008).

9. Time, determinism and the movie

• Simulation step. Fixed at 1/120 s. The simulation uses a seeded integer random generator and only + − × ÷ and √ (cube roots by Newton's method, no trigonometry), in a fixed order, and its checkpoints include the scent field. The same seed gives a bit-identical swarm on every machine and in every browser.
• Speed. On screen, ×1 is real simulated time. The simulation and the camera follow the clock, whatever the frame rate. A computer drawing fewer than 4 frames a second falls behind; the HUD shows how fast the screen is running.
• The movie. Every movie frame is exactly 1/30 s of the simulation at the chosen speed, and the sound is made for that same 1/30 s. The movie therefore plays at the speed the screen shows when it is not recording, whatever the computer's speed. It is drawn at the screen's own resolution by default.

10. Summary: tuned, measured, given

11. Sources

• Beekman, M., Fathke, R. L. & Seeley, T. D. (2006). How does an informed minority of scouts guide a honeybee swarm as it flies to its new home? Animal Behaviour 71, 161–171. — swarm clouds in flight hold 17–80 bees/m³. PDF
• Clemson Home & Garden Information Center — Frequently Asked Questions About Honey Bee Swarms. Fact sheet
• Heinrich, B. (1981). The mechanisms and energetics of honeybee swarm temperature regulation. J. Exp. Biol. 91, 25–55. Article
• Thermoregulatory morphodynamics of honeybee swarm clusters (2022). J. Exp. Biol. 225, jeb242234. — clusters 10–30 cm, ~10,000 bees; ~2× volume change with temperature. PDF
• Minnesota Hobby Beekeepers Association — Honey Bee Swarms. — swarms usually fly between 10 am and 2 pm. Page
• Mechanisms of Swarming in the Honey Bee Superorganism, American Bee Journal. — the queen lands "apparently at random". Article
• Sounds of the Hive — Part 1, American Bee Journal. — wingbeat 190–250 Hz. Article
• Neurobiology of Chemical Communication, ch. "Chemical Communication in the Honey Bee Society" (NCBI Bookshelf). — queen pheromone and swarm cohesion (Winston et al. 1989); Nasonov components; synthetic Nasonov attracts swarms (Schmidt 1994, 1999). NCBI NBK200983
• Nguyen, D. M. T. … Peleg, O. (2021). Flow-mediated olfactory communication in honeybee swarms. PNAS 118, e2011916118. — scenting bees fan scent away from the queen; threshold-triggered gradient following. PNAS
• Ocko, S. A. & Mahadevan, L. (2014). Collective thermoregulation in bee clusters. J. R. Soc. Interface 11, 20131033. — bee ≈ 0.65 cm sphere; packing 0.2–0.8. PMC3869176
• Peleg, O., Peters, J. M., Salcedo, M. K. & Mahadevan, L. (2018). Collective mechanical adaptation of honeybee swarms. Nature Physics 14, 1193–1198. — the pendant cone at rest; flatter when shaken; shape depends on the surface. Peleg lab
• Schultz, K. M., Passino, K. M. & Seeley, T. D. (2008). The mechanism of flight guidance in honeybee swarms: subtle guides or streaker bees? J. Exp. Biol. 211, 3287–3295. Article
• Seeley, T. D. (2010). Honeybee Democracy. Princeton University Press. — ~10,000 workers, 1.5 kg; cloud 10–20 m; cluster ~30 m from the hive; 10–20 min to condense. JSTOR
• Seeley, T. D. & Tautz, J. (2001). Worker piping in honey bee swarms and its role in preparing for liftoff. J. Comp. Physiol. A 187, 667–676. doi:10.1007/s00359-001-0243-0
• Seeley, T. D. & Visscher, P. K. (2003). Choosing a home: how the scouts in a honey bee swarm perceive the completion of their group decision. Behav. Ecol. Sociobiol. 54, 511–520. PDF
• Shishkov, O., Chen, C., Madonna, C. A., Jayaram, K. & Peleg, O. (2022). Strength-mass scaling law governs mass distribution inside honey bee swarms. Scientific Reports. — grip 35 ± 14 bee-weights; 3.8 at the attachment; load ∝ mass^1.5. Article
• SICB abstract: Honeybee swarms use a flow-mediated pheromone signaling scheme to coordinate aggregation. sicb.org
• The sounds produced by swarming honey bees, Zeitschrift für vergleichende Physiologie (Springer). Springer
• Swarming (honey bee) — Wikipedia. Article
• Wenner, A. M. (1963). The flight speed of honeybees: a quantitative approach. J. Apic. Res. 2, 25–32. — cruising ≈ 7.5 m/s. BioNumbers 107550
• Western Honey Bee dimensions — dimensions.com. Page