Honeybees are masters of regulating their temperatures collectively, even in the absence of a hive. A reproductive swarm consisting of a queen and about half a colony's workers will leave their hive to find a new home. Prior to settling on a permanent home, the honeybees form a stationary swarm, where the bees cling onto one another from a roof-type structure, completely exposed to the elements. Because bees are so sensitive to extremes of heat and cold, it is essential that the swarm has ways to control its temperature. We present a mathematical model to study how honeybees thermoregulate by adjusting their movement and metabolic heat output. We introduce a system of coupled partial differential equations and integral equations to describe the swarm temperature, density, and size, along with a corresponding numerical scheme. We then relax the assumption of spherical symmetry and extend the model by studying non-spherical swarms.
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