A model for heat transfer in a honey bee swarm

A swarm is a temporary structure formed when several thousand honey bees leave their hive and settle on some object such as the branch of a tree. They remain in this position until a suitable site for a new home is located by the scout bees. A continuum model based on heat conduction and heat generation is used to predict temperature profiles in swarms. Since internal convection is neglected, the model is applicable only at low values of the ambient temperature T a . Guided by the experimental observations of Heinrich (1981a–c, J. Exp. Biol. 91, 25–55; Science 212, 565–566; Sci. Am. 244, 147–160), the analysis is carried out mainly for non-spherical swarms. The effective thermal conductivity is estimated using the data of Heinrich (1981a, J. Exp. Biol. 91, 25–55) for dead bees. For T a = 5 and 9°C, results based on a modified version of the heat generation function due to Southwick (1991, The Behaviour and Physiology of Bees, pp. 28–47. C.A.B. International, London) are in reasonable agreement with measurements. Results obtained with the heat generation function of Myerscough (1993, J. Theor. Biol. 162, 381–393) are qualitatively similar to those obtained with Southwick's function, but the error is more in the former case. The results suggest that the bees near the periphery generate more heat than those near the core, in accord with the conjecture of Heinrich (1981c, Sci. Am. 244, 147–160). On the other hand, for T a = 5°C, the heat generation function of Omholt and Lønvik (1986, J. Theor. Biol. 120, 447–456) leads to a trivial steady state where the entire swarm is at the ambient temperature. Therefore an acceptable heat generation function must result in a steady state which is both non-trivial and stable with respect to small perturbations. Omholt and Lønvik's function satisfies the first requirement, but not the second. For T a = 15°C, there is a considerable difference between predicted and measured values, probably due to the neglect of internal convection in the model.