Apiary Space and Forage (2D): Plan, Flyway Section and Foraging Range

What is simulated

A row of hives in a garden with a neighbour behind the boundary fence. Two questions are answered with real arithmetic: do the departing bees climb above head height before the fence, and is there enough forage within flying range to feed the colonies?

Site footprint

The working room is a strip around each hive: width = (n - 1)(s + W) + W + 2 cside and depth = boundary + D + cback with W = D = 0.6 m and 0.9 m of clearance at the sides and back. The page checks that the formula equals the area of the union of the strips.

Flyway

A bee leaves the entrance at 0.3 m and climbs at angle θ. A barrier at 1.1 m forces it above the barrier height h_b: h(b) = max(hb, h0 + tanθ db) + tanθ (b - db). A bee below 1.8 m at the fence crosses at face height.

How far bees fly

A load of 40 mg nectar with sugar fraction c holds E = 40 c x 16 J/mg. A round trip of length 2d takes τ = 2d / v in the air (v = 7 m/s, flight power 40 mW) plus a handling time th = 40 mg / rate + 60 s. The net energy rate is p(d) = (E - P τ) / (τ + th). Foragers keep a patch while p ≥ f p(0) (f = 0.5), which gives τ* = E (1 - f) / (P + f E / th) and a radius d* = v τ* / 2.

Forage balance

Supply S = y a π d*2 (honey-equivalent yield y per hectare of forage, forage share a of the land) is shared by all colonies in range; one colony needs about 80 kg of honey equivalent a year. The colony count the forage can feed is S / 80.

Simplifications

• Flight energy, speed, load and the 50 % acceptance threshold are round order-of-magnitude values, not measurements.
• Foragers are spread evenly over the disc; real foraging follows the best patches found by dancing scouts.
• Yield and forage share are user inputs; real nectar flow varies through the season.
• The climb angle is constant and wind, hedges and the bees' own height preference are ignored.
• Spacing between stands is a management choice here; the drifting of bees between nearby hives is not modelled.

Checks

Four boxes: union area of the working strips against the footprint formula, a 1 mm stepping of a bee path against the flyway formula, a bisection for the foraging radius against its closed form, and a Monte Carlo integration of the foraging disc and forage land.