Module 3

Thermoregulation & Energetics

Flight muscle thermogenesis, hive temperature control, metabolic rate scaling, and winter cluster thermodynamics

Honeybees are endothermic insects capable of maintaining their brood nest at 35 ± 0.5°C year-round — a thermoregulatory precision rivaling the best artificial incubators. This remarkable feat requires the highest mass-specific metabolic rates in the animal kingdom during flight and sophisticated collective heating and cooling behaviors that emerge from simple individual thermoregulatory rules.

3.1 Flight Muscle Thermogenesis

The honeybee's flight apparatus is dominated by the indirect flight muscles (IFMs), which constitute approximately 30% of total body mass — making them the single largest organ system. Unlike most insect flight muscles, bee IFMs are asynchronous (also called fibrillar or myogenic): a single nerve impulse triggers multiple mechanical contractions through a process known as stretch activation.

Asynchronous Muscle Mechanics

In synchronous flight muscles (found in dragonflies), each contraction requires a separate nerve impulse. This limits wingbeat frequency to approximately 100 Hz. In asynchronous muscles, the mechanical resonance of the thorax-wing system amplifies each neural signal. The thorax acts as a click mechanism with two stable states:

When the dorsoventral muscles (DVMs) contract, the thorax "clicks" down, stretching the dorsolongitudinal muscles (DLMs). This stretch activates the DLMs, which contract and click the thorax up, stretching the DVMs again. The cycle repeats at wingbeat frequency:

\[ f_{\text{wingbeat}} = \frac{1}{2\pi}\sqrt{\frac{k_{\text{thorax}}}{m_{\text{eff}}}} \approx 200\text{--}250\text{ Hz} \]

This resonant frequency far exceeds what neural firing rates alone could achieve. The Ca²⁺ cycling in asynchronous muscles is slower than in synchronous muscles because Ca²⁺ only needs to maintain a tonic level for stretch activation rather than triggering each individual twitch. The neural firing rate is approximately 5–15 Hz, while the mechanical oscillation runs at 200+ Hz — a mechanical gain of ~20x.

Pre-Flight Warm-Up

Honeybees cannot initiate flight until thoracic temperature reaches approximately 35°C. At lower temperatures, the asynchronous click mechanism fails because muscle stiffness is too high and the resonant frequency shifts below the threshold for aerodynamic force production.

To warm up, bees employ shivering thermogenesis: the DVM and DLM muscle groups contract simultaneously rather than antagonistically. This isometric contraction prevents wing movement (no net click) while converting metabolic energy directly to heat. The wings may vibrate slightly but do not execute full strokes.

Heat Production Derivation

The total metabolic power during flight muscle activity can be partitioned:

\[ P_{\text{metabolic}} = P_{\text{mechanical}} + P_{\text{heat}} \]

The mechanical efficiency of insect flight muscle is:

\[ \eta_{\text{mech}} = \frac{P_{\text{mechanical}}}{P_{\text{metabolic}}} \approx 0.10\text{--}0.25 \]

Therefore the heat fraction is:

\[ \eta_{\text{heat}} = 1 - \eta_{\text{mech}} \approx 0.75\text{--}0.90 \]

The heat production rate is thus:

\[ Q = \eta_{\text{heat}} \cdot P_{\text{metabolic}} = (1 - \eta_{\text{mech}}) \cdot \dot{V}_{O_2} \cdot \frac{\Delta H_c}{V_{O_2,\text{STP}}} \]

where \(\dot{V}_{O_2}\) is the oxygen consumption rate (~30 mL O\(_2\)/g muscle/h during flight) and \(\Delta H_c \approx 21.1\) kJ/L O\(_2\) for carbohydrate metabolism.

During shivering, \(\eta_{\text{mech}} \approx 0\) since no useful mechanical work is performed, so \(\eta_{\text{heat}} \approx 1.0\). The bee's thoracic temperature rises at a rate determined by the thermal balance:

\[ m_{\text{thorax}} c_p \frac{dT_{\text{thorax}}}{dt} = P_{\text{metabolic}} - h_{\text{eff}} A_{\text{thorax}} (T_{\text{thorax}} - T_{\text{amb}}) \]

With typical values (\(m_{\text{thorax}} \approx 30\) mg, \(c_p \approx 3.4\) J/(g·K)), the warm-up rate is approximately 5–10°C/min, meaning a bee at 15°C ambient can prepare for flight in 2–4 minutes.

Countercurrent Heat Exchanger

The petiole (waist) connecting thorax and abdomen contains a countercurrent heat exchange system in the aorta. During flight, this allows the thorax to remain at 40°C+ while the abdomen stays near ambient temperature (~25°C). During cold stress, the heat exchanger can be bypassed by increasing hemolymph flow rate, allowing heat to reach the abdomen for brood warming.

3.2 Hive Temperature Control

The honeybee brood nest is maintained at 35.0 ± 0.5°C— more precisely regulated than most laboratory incubators. Even 1°C deviation during pupal development produces detectable behavioral deficits in adult bees. This extraordinary precision is achieved without any central controller; instead, individual bees respond to local temperature cues, and collective behavior emerges from these simple rules.

Heating Mechanisms

1. Incubation Behavior

Heater bees press their thorax against capped brood cells and engage flight muscles isometrically (shivering). A single heater bee can raise the temperature of adjacent cells by 2–3°C. Heater bees preferentially occupy empty cells adjacent to brood, warming cells from the sides rather than above. Infrared imaging reveals that individual heater bees can sustain thoracic temperatures of 42–44°C for 30+ minutes.

2. Clustering

When ambient temperature drops, bees form tighter clusters on the brood, reducing the surface area available for heat loss. The cluster acts as a living insulation layer, with bees on the outer shell maintaining lower body temperatures (15–20°C) to reduce the temperature gradient and heat flux to the environment.

Cooling Mechanisms

1. Fanning

Fanner bees position themselves at the hive entrance and use their wings to create directed airflow. Remarkably, fanners organize into chains that create a coherent air circulation pattern: fresh air enters through one side of the entrance while warm, moist air exits the other side. The convective cooling power of fanning chains can reach 5–10 W depending on colony size and ambient conditions.

2. Evaporative Cooling

When fanning alone is insufficient (ambient temperature above ~35°C), water foragers collect water and deposit thin films on comb surfaces and cell rims. Fanner bees then accelerate evaporation. The latent heat of water vaporization (\(L_v = 2.45\) MJ/kg) provides approximately 680 W·h per liter of water evaporated. On hot days, a colony may collect and evaporate 1–2 liters of water.

Lumped-Parameter Heat Equation

The hive brood nest temperature can be modeled using a lumped-parameter energy balance:

\[ C \frac{dT}{dt} = Q_{\text{bees}} - hA(T - T_{\text{amb}}) - L_v \dot{m}_{\text{water}} \]

where:

\(C\) = thermal capacity of hive + bees + comb + honey (typically 5,000–15,000 J/K)

\(T\) = brood nest temperature (K)

\(Q_{\text{bees}}\) = total metabolic heat from all bees in the hive (W), depends on number of heater bees active

\(h\) = effective heat transfer coefficient (W/(m\(^2\)·K)), includes conduction through hive walls + convection

\(A\) = effective surface area of the hive (~0.5–1.0 m\(^2\))

\(L_v\) = latent heat of vaporization of water (2.45 × 10\(^6\) J/kg)

\(\dot{m}_{\text{water}}\) = mass flow rate of water evaporated (kg/s)

At steady state (\(dT/dt = 0\)), the heat production must balance losses:

\[ Q_{\text{bees}} = hA(T_{\text{set}} - T_{\text{amb}}) + L_v \dot{m}_{\text{water}} \]

The thermoregulatory feedback is implemented through individual bee behavior. Each bee possesses thermoreceptors in the antennal flagellum with a remarkable resolution of approximately 0.25°C. When a bee detects a local temperature deviation from the setpoint, it responds probabilistically:

\[ P_{\text{heat}} = \frac{1}{1 + e^{\alpha(T_{\text{local}} - T_{\text{set}})}} \qquad P_{\text{fan}} = \frac{1}{1 + e^{-\alpha(T_{\text{local}} - T_{\text{set}})}} \]

where \(\alpha \approx 2\text{--}4\) °C\(^{-1}\) determines the sharpness of the behavioral switch.

The collective effect of hundreds of bees each making independent probabilistic decisions produces the remarkably stable macroscopic temperature. This is a beautiful example of distributed feedback control without any central thermostat.

3.3 Metabolic Rate Scaling

The honeybee in flight achieves the highest mass-specific metabolic rate of any animal. Flight muscle generates approximately 500–600 W/kg of muscle tissue, or about 100–120 W/kg of total body mass. For comparison, a human sprinter generates roughly 15–20 W/kg.

Fuel Selection

Honeybees are unusual among flying insects in that they fuel flight almost exclusively with carbohydrates (sugars from honey or nectar). The respiratory quotient (RQ) during flight is:

\[ RQ = \frac{\dot{V}_{CO_2}}{\dot{V}_{O_2}} \approx 1.0 \]

This RQ of 1.0 indicates pure carbohydrate oxidation:

\[ C_6H_{12}O_6 + 6O_2 \longrightarrow 6CO_2 + 6H_2O \quad \Delta H = -2803 \text{ kJ/mol} \]

The advantage of carbohydrate fuel is speed of ATP production. Glycolysis + oxidative phosphorylation from glucose yields ATP at roughly 2x the rate of fatty acid beta-oxidation, critical for the extreme power demands of flight. The trade-off is lower energy density (17 kJ/g for sugar vs 39 kJ/g for fat), which limits flight range but is compensated by the bee's foraging strategy of making multiple short trips from the hive.

Metabolic Allometry

Resting metabolic rate scales allometrically across insects:

\[ P_{\text{rest}} = P_0 \left(\frac{m}{m_0}\right)^{0.75} \]

During flight, the metabolic rate increases by a factor of 50–100x above resting, which is the highest factorial aerobic scope known for any animal:

\[ \text{Factorial Aerobic Scope} = \frac{P_{\text{flight}}}{P_{\text{rest}}} \approx 50\text{--}100 \]

Optimal Nectar Concentration

Not all nectar is equally valuable. The optimal sugar concentration for bee foraging is approximately 50% w/w. This represents a trade-off between two competing factors:

Energy Density

Energy per unit volume increases linearly with sugar concentration:\(\varepsilon = c \cdot \Delta H_{\text{sugar}} / M_{\text{sugar}}\)

Viscosity

Viscosity increases exponentially with sugar concentration, making high-concentration nectar extremely difficult to drink through the proboscis.

To derive the optimal concentration, we model nectar uptake through the proboscis as Poiseuille flow through a tube of radius \(a\) and length \(L\):

\[ \dot{V} = \frac{\pi a^4 \Delta P}{8 \mu(c) L} \]

The energy intake rate is the product of volumetric flow rate and energy density:

\[ \dot{E} = \dot{V} \cdot \varepsilon(c) = \frac{\pi a^4 \Delta P}{8 L} \cdot \frac{c \cdot \Delta H}{M_s \cdot \mu(c)} \]

The viscosity of sugar solution follows an empirical relation:

\[ \mu(c) = \mu_0 \exp(\beta c) \]

where \(\mu_0 \approx 1\) mPa·s (water), \(\beta \approx 5\text{--}7\), and \(c\) is mass fraction.

To find the optimal concentration, we maximize \(\dot{E}\) with respect to \(c\):

\[ \frac{d\dot{E}}{dc} = 0 \quad \Longrightarrow \quad \frac{d}{dc}\left[\frac{c}{\mu(c)}\right] = 0 \]

\[ \frac{d}{dc}\left[c \cdot e^{-\beta c}\right] = e^{-\beta c}(1 - \beta c) = 0 \]

\[ c_{\text{opt}} = \frac{1}{\beta} \approx 0.15\text{--}0.20 \text{ (mass fraction for pure Poiseuille)} \]

However, real bees lick rather than suck at high viscosities, and the tongue has surface texture that enhances uptake. When these corrections are included, along with the metabolic cost of flight to and from the flower, the practical optimum shifts to approximately c = 0.40–0.60 (40–60% w/w), consistent with behavioral preference experiments.

3.4 Winter Cluster Thermodynamics

During winter, when ambient temperatures can drop well below 0°C, a colony of 10,000–30,000 bees forms a roughly spherical cluster. This cluster is a self-organizing, self-heating structure that maintains survivable conditions for months without any external energy input beyond stored honey.

Cluster Structure

The winter cluster has a distinct two-zone structure:

Outer Shell (Mantle)

Temperature: ~10–15°C. Bees are relatively inactive, packed tightly with inter-bee spaces minimized. Acts as living insulation. Thickness: 2–4 cm. Effective thermal conductivity: ~0.02–0.04 W/(m·K), comparable to commercial insulation materials.

Inner Core

Temperature: ~33–36°C. Bees are active, consuming honey and generating heat through shivering thermogenesis. If brood is present (late winter), the core temperature is maintained more precisely at 35°C.

Spherical Shell Heat Transfer

Modeling the cluster as a spherical shell with inner radius \(r_i\) (core) and outer radius \(r_o\) (cluster surface), the steady-state heat conduction through the mantle is:

\[ \dot{Q} = \frac{4\pi k \, r_i \, r_o \, \Delta T}{r_o - r_i} \]

where \(\Delta T = T_{\text{core}} - T_{\text{surface}}\) and \(k\) is the effective thermal conductivity of the mantle.

Derivation: For steady-state radial conduction in a spherical shell, Fourier's law gives:

\[ \dot{Q} = -kA(r)\frac{dT}{dr} = -k(4\pi r^2)\frac{dT}{dr} \]

Since \(\dot{Q}\) is constant through each spherical surface (steady state), we separate and integrate:

\[ \dot{Q} \int_{r_i}^{r_o} \frac{dr}{r^2} = -4\pi k \int_{T_i}^{T_o} dT \]

\[ \dot{Q} \left[-\frac{1}{r}\right]_{r_i}^{r_o} = -4\pi k (T_o - T_i) \]

\[ \dot{Q} \left(\frac{1}{r_i} - \frac{1}{r_o}\right) = 4\pi k (T_i - T_o) \]

\[ \dot{Q} = \frac{4\pi k \, r_i \, r_o \,(T_{\text{core}} - T_{\text{surface}})}{r_o - r_i} \]

Adaptive Cluster Behavior

The cluster exhibits several adaptive behaviors:

Contraction

As ambient temperature drops, the cluster contracts (smaller \(r_o\)), reducing surface area and thus heat loss. Cluster volume can decrease by 50% between 10°C and -10°C ambient.

Rotation

Bees continuously rotate between the cold outer shell and warm inner core. No individual bee remains on the cold periphery indefinitely. The rotation period is approximately 1–2 hours.

Metabolic Modulation

Heat production in the core increases as ambient temperature drops. The colony's honey consumption rate approximately doubles for each 10°C drop in ambient temperature (\(Q_{10} \approx 2\) for thermoregulatory cost).

The total winter honey consumption for a typical colony is 15–25 kg, representing an energy budget of:

\[ E_{\text{winter}} = m_{\text{honey}} \times \Delta H_{\text{honey}} \approx 20 \text{ kg} \times 13.4 \text{ MJ/kg} \approx 268 \text{ MJ} \]

This is equivalent to roughly 75 kWh — enough to run a small space heater for 3 days continuously.

3.5 Hive Thermoregulation Diagram

Cross-section of the hive showing the thermal management strategies employed during summer cooling and winter heating, including temperature gradients, fanning chains, water carriers, and the winter cluster structure.

3.6 Simulation: Hive Temperature Dynamics

This simulation models three aspects of honeybee thermoregulation: daily hive temperature regulation with heating and cooling responses, winter cluster radius as a function of ambient temperature, and the metabolic cost of thermoregulation across seasons.

Hive Temperature Dynamics & Winter Cluster Thermodynamics

Python

script.py186 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fig, axes = plt.subplots(2, 2, figsize=(14, 12))

# ======= Panel 1: Daily Hive Temperature Regulation =======
ax1 = axes[0, 0]

hours = np.linspace(0, 48, 2000)
dt = hours[1] - hours[0]

# Ambient temperature: sinusoidal daily cycle
T_amb_base = 25.0
T_amb_amp = 10.0
T_amb = T_amb_base + T_amb_amp * np.sin(2 * np.pi * (hours - 6) / 24)

T_set = 35.0
C_hive = 10000.0   # J/K thermal capacity
hA = 2.0            # W/K total heat loss coefficient
Lv = 2.45e6         # J/kg latent heat
alpha = 3.0         # feedback sharpness (1/K)

T_hive = np.zeros_like(hours)
T_hive[0] = T_set
Q_bees_arr = np.zeros_like(hours)
Q_evap_arr = np.zeros_like(hours)

for i in range(1, len(hours)):
    dT = T_hive[i-1] - T_set
    # Heating probability (collective)
    P_heat = 1.0 / (1.0 + np.exp(alpha * dT))
    Q_heat = P_heat * 40.0  # Max heating power 40 W

# Cooling: fanning + evaporation
    P_cool = 1.0 / (1.0 + np.exp(-alpha * dT))
    m_dot_water = P_cool * 0.0001  # kg/s max water evaporation
    Q_evap = Lv * m_dot_water

Q_loss = hA * (T_hive[i-1] - T_amb[i])
    dTdt = (Q_heat - Q_loss - Q_evap) / C_hive
    T_hive[i] = T_hive[i-1] + dTdt * dt * 3600

Q_bees_arr[i] = Q_heat
    Q_evap_arr[i] = Q_evap

ax1.plot(hours, T_amb, color='#60a5fa', linewidth=1.5, label='Ambient', alpha=0.7)
ax1.plot(hours, T_hive, color='#f59e0b', linewidth=2.5, label='Hive (brood nest)')
ax1.axhline(T_set, color='#fde047', linestyle='--', alpha=0.4, linewidth=1)
ax1.axhspan(T_set - 0.5, T_set + 0.5, alpha=0.1, color='#22c55e')
ax1.text(1, T_set + 1.0, 'Setpoint 35.0 +/- 0.5 C', color='#86efac', fontsize=9)
ax1.set_xlabel('Time (hours)', fontsize=11, color='white')
ax1.set_ylabel('Temperature (C)', fontsize=11, color='white')
ax1.set_title('Daily Hive Temperature Regulation', fontsize=13, color='#f59e0b', fontweight='bold')
ax1.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#f59e0b', labelcolor='white')
ax1.set_facecolor('#0a0a0a')
ax1.tick_params(colors='white')
ax1.grid(True, alpha=0.15, color='white')
ax1.set_ylim(10, 45)

# ======= Panel 2: Winter Cluster Radius vs Ambient Temp =======
ax2 = axes[0, 1]

T_ambient_range = np.linspace(-20, 15, 200)
N_bees = 20000
V_bee = 0.15e-6   # m^3 per bee (approximate)
V_total = N_bees * V_bee

# Cluster contracts with colder temps
# Packing fraction increases from ~0.5 (loose) to ~0.8 (tight)
packing = 0.5 + 0.3 / (1 + np.exp(0.3 * (T_ambient_range + 5)))
r_outer = (3 * V_total / (4 * np.pi * packing))**(1.0/3.0) * 100  # cm

# Core radius stays roughly constant
r_inner = 0.5 * np.min(r_outer) * np.ones_like(T_ambient_range)  # cm

ax2.plot(T_ambient_range, r_outer, color='#3b82f6', linewidth=2.5, label='Outer radius (cluster)')
ax2.fill_between(T_ambient_range, r_inner, r_outer, alpha=0.15, color='#3b82f6')
ax2.plot(T_ambient_range, r_inner, color='#f59e0b', linewidth=2, linestyle='--', label='Inner radius (core)')

ax2.set_xlabel('Ambient Temperature (C)', fontsize=11, color='white')
ax2.set_ylabel('Cluster Radius (cm)', fontsize=11, color='white')
ax2.set_title('Winter Cluster Size vs. Ambient Temperature', fontsize=13, color='#f59e0b', fontweight='bold')
ax2.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#f59e0b', labelcolor='white')
ax2.set_facecolor('#0a0a0a')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.15, color='white')

ax2.annotate('Cluster contracts\nin cold', xy=(-15, r_outer[25]), xytext=(-10, r_outer[25] + 3),
            arrowprops=dict(arrowstyle='->', color='#93c5fd'), fontsize=9, color='#93c5fd')

# ======= Panel 3: Metabolic Cost of Thermoregulation =======
ax3 = axes[1, 0]

T_amb_year = np.linspace(-15, 40, 300)
Q_thermo = np.zeros_like(T_amb_year)

for i, Ta in enumerate(T_amb_year):
    if Ta < T_set:
        # Heating: linear increase with temperature deficit
        Q_thermo[i] = hA * (T_set - Ta)  # W
    else:
        # Cooling: evaporative cost
        Q_thermo[i] = 0.5 * (Ta - T_set)**2  # quadratic increase

# Convert to honey consumption (g/day)
# 1 g honey = 13.4 kJ, efficiency ~0.3
honey_rate = Q_thermo * 86400 / (13400 * 0.3)  # g/day

ax3.plot(T_amb_year, honey_rate, color='#f59e0b', linewidth=2.5)
ax3.fill_between(T_amb_year, 0, honey_rate, alpha=0.15, color='#f59e0b')

# Mark thermoneutral zone
ax3.axvspan(33, 36, alpha=0.2, color='#22c55e')
ax3.text(34.5, max(honey_rate) * 0.9, 'TNZ', color='#86efac', fontsize=10, ha='center')

ax3.set_xlabel('Ambient Temperature (C)', fontsize=11, color='white')
ax3.set_ylabel('Honey Consumption (g/day)', fontsize=11, color='white')
ax3.set_title('Metabolic Cost of Thermoregulation', fontsize=13, color='#f59e0b', fontweight='bold')
ax3.set_facecolor('#0a0a0a')
ax3.tick_params(colors='white')
ax3.grid(True, alpha=0.15, color='white')

# Annotate winter cost
ax3.annotate('Winter:\nhigh heating cost', xy=(-10, honey_rate[30]), xytext=(5, honey_rate[30] * 0.8),
            arrowprops=dict(arrowstyle='->', color='#93c5fd'), fontsize=9, color='#93c5fd')

# ======= Panel 4: Heat Loss vs Shell Thickness =======
ax4 = axes[1, 1]

shell_thickness = np.linspace(0.5, 8, 200)  # cm
r_i_fixed = 6.0   # cm (core radius)
r_o_vals = r_i_fixed + shell_thickness  # cm

k_mantle = 0.03  # W/(m*K) effective
T_core = 35.0
T_surface = 10.0
dT = T_core - T_surface

# Heat loss: Q = 4*pi*k*r_i*r_o*dT / (r_o - r_i)
r_i_m = r_i_fixed / 100
r_o_m = r_o_vals / 100
shell_m = shell_thickness / 100

Q_loss = 4 * np.pi * k_mantle * r_i_m * r_o_m * dT / shell_m

ax4.plot(shell_thickness, Q_loss, color='#ef4444', linewidth=2.5, label='Spherical shell model')

# Planar approximation for comparison
A_sphere = 4 * np.pi * ((r_i_m + r_o_m) / 2)**2
Q_planar = k_mantle * A_sphere * dT / shell_m
ax4.plot(shell_thickness, Q_planar, color='#a78bfa', linewidth=1.5, linestyle='--', label='Planar approximation')

ax4.set_xlabel('Shell Thickness (cm)', fontsize=11, color='white')
ax4.set_ylabel('Heat Loss (W)', fontsize=11, color='white')
ax4.set_title('Cluster Heat Loss vs. Mantle Thickness', fontsize=13, color='#f59e0b', fontweight='bold')
ax4.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#f59e0b', labelcolor='white')
ax4.set_facecolor('#0a0a0a')
ax4.tick_params(colors='white')
ax4.grid(True, alpha=0.15, color='white')
ax4.set_ylim(0, max(Q_loss) * 1.1)

ax4.annotate('Thicker mantle =\nless heat loss', xy=(5, Q_loss[140]), xytext=(6, Q_loss[60]),
            arrowprops=dict(arrowstyle='->', color='#fca5a5'), fontsize=9, color='#fca5a5')

fig.patch.set_facecolor('#0a0a0a')
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()

print("=== Honeybee Thermoregulation Simulation ===")
print(f"\nHive thermal capacity: {C_hive:.0f} J/K")
print(f"Heat loss coefficient (hA): {hA:.1f} W/K")
print(f"Brood nest setpoint: {T_set:.1f} C (+/- 0.5 C)")
print(f"\nWinter cluster (N = {N_bees} bees):")
print(f"  At -15 C: outer radius = {r_outer[10]:.1f} cm")
print(f"  At +10 C: outer radius = {r_outer[-20]:.1f} cm")
print(f"  Contraction ratio: {r_outer[-20]/r_outer[10]:.2f}")
print(f"\nMantle thermal conductivity: {k_mantle} W/(m*K)")
print(f"  (comparable to fiberglass insulation: 0.04 W/(m*K))")
print(f"\nHeat loss at 3 cm mantle: {Q_loss[80]:.1f} W")
print(f"Heat loss at 6 cm mantle: {Q_loss[170]:.1f} W")
print(f"\nHoney consumption at -10 C: {honey_rate[30]:.0f} g/day")
print(f"Honey consumption at 20 C: {honey_rate[190]:.0f} g/day")
print(f"Honey consumption at 35 C (TNZ): {honey_rate[270]:.0f} g/day")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

References

Heinrich, B. (1993). The Hot-Blooded Insects: Strategies and Mechanisms of Thermoregulation. Harvard University Press.
Stabentheiner, A., Preuss, H., Hrassnigg, N. & Crailsheim, K. (2003). Endothermic heat production in honeybee winter clusters. Journal of Experimental Biology, 206, 353–358.
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Harrison, J.F. & Roberts, S.P. (2000). Flight respiration and energetics. Annual Review of Physiology, 62, 179–205.
Kim, W., Gilet, T. & Bush, J.W.M. (2011). Optimal concentrations in nectar feeding. Proceedings of the National Academy of Sciences, 108(40), 16618–16621.
Tautz, J. (2008). The Buzz about Bees: Biology of a Superorganism. Springer-Verlag.
Ocko, S.A. & Mahadevan, L. (2014). Collective thermoregulation in bee clusters. Journal of the Royal Society Interface, 11(91), 20131033.

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