You keep one colony in a cutaway Langstroth hive from 1 March to the next 1 March. Workers and brood follow differential equations driven by day length, temperature, nectar flow and the room the bees have. The brood nest is held near 35 C by a heat balance with the weather. Your decisions: when to add supers (empty comb must be drawn out of wax, which costs honey), when to harvest (unripe honey does not count, and you must leave stores) and whether the colony will survive winter. Keys: Space pause, S add super, H harvest, V view. Drag to orbit, pinch or wheel to zoom. Bees are drawn larger than life so you can see them.
The colony's RK4 stepper is run on dW/dt = r W (1 - W/K) and compared with the closed form W(t) = K / (1 + (K-W0)/W0 e-rt); a forward-Euler run is shown for contrast.
A lumped nest with heat capacity C, loss conductance UA and bee heat output Q relaxes to T = Ta + Q/UA with time constant C/UA. The explicit integration is compared with the closed form and with a bisection solve of Q - UA (T - Ta) = 0.
(1) 100 days of a frozen-population cluster at a fixed air temperature: simulated honey use against the closed-form W m0 + (UA (Tset-Ta) - Qfree)/(E). (2) A full managed year: honey in = honey out + stores change, to rounding.
Population. Eggs and young larvae (about 9 days), capped pupae (12 days) and adult workers form a chain. The queen lays L = Lmax · f(day length) · f(forage) · f(brood temperature) · (1 - B/Bcap) eggs per day; dB1/dt = L - B1/9, dB2/dt = B1/9 - B2/12, dW/dt = sT B2/12 - μW. Brood cooled below about 33 C survives less well (sT). Worker death rate switches between long-lived winter bees (about 160 days) and short-lived summer bees (about 38 days) as day length changes. The brood cap is a simple nurse-bee limit; real colonies are less tidy.
Thermoregulation. The nest loses heat UA (T - Ta). Bees need Q = UA (Tset - Ta) and can supply at most about 3 mW per bee; the steady state is T = Ta + Q/UA (Experiment B). Part of the heat is free from ordinary metabolism; the rest is burned honey at 13.4 MJ per kg. A weak colony in cold weather cannot hold 35 C and the brood chills. In winter the bees form a cluster with a surface near 14 C whose loss scales as the surface area, UA ∝ W2/3. Insulation scales UA.
Nectar to honey. Foragers (about 30 % of adults) bring I = (F q)(flow)/(F q + flow) kg per day, a saturating form that never exceeds either the foragers' capacity or the bloom. Bloom curves, weather and rain days are deterministic and seeded (same seed, same year). Honey is also used for maintenance (about 5 to 12 mg per bee per day) and heating. New supers hold nothing until bees have drawn wax; drawing costs honey (about 0.35 kg per kg of new capacity here, a rough figure). When the hive is nearly full in spring or summer the colony becomes congested and swarms.
Harvest and winter. Recently collected nectar is unripe, so the model counts only honey older than about 6 days of intake. The winter check asks whether stores last until 1 March with at least 4,500 workers left. The forecast runs the same model forward without further action.
Simplifications. One lumped nest; no disease, pests, drones, queen failure or queen age; no chemical or medical advice is modelled; forage is a smooth curve, not a landscape; wax, pollen and water are folded into the honey budget; the numbers are plausible for a temperate (UK-like) climate, not calibrated to a specific apiary. Bees and frames are drawn larger and more widely spaced than life. This is a teaching model, not management advice.