Swarm Timing and the Price of a Swarm: A Congestion Crossing, an Inspection-Interval Bound, and a Honey Cost in Kilograms

A prime swarm is the most expensive routine event in a managed apiary, yet beekeepers usually discuss its timing and its cost in separate conversations, in qualitative terms. This paper joins the two with elementary arithmetic. A spring colony is modelled from 1 March with exponential growth in adult numbers and in the laying rate, and two congestion measures are tracked in parallel: adult bees per litre of brood-box volume and the occupied fraction of brood-box cells. Under the stated assumptions the bees-per-litre threshold is crossed first, at 36.48 days after 1 March, slightly ahead of the cell-occupancy crossing. From queen-cell development times the longest inspection gap that still catches queen cells before sealing is derived as 7 days. A swarm is then priced: the laying gap is 18 days, corresponding to 27000 bees that are never laid, and the total honey cost is 38.46 kg in a strong-flow scenario and 10.58 kg in a weak-flow scenario. This study is simulated, not measured; every threshold, rate and fraction is a stated assumption, and the results are the arithmetic consequences of those assumptions rather than observations of real colonies.