ABSTRACT: We construct a mathematical model to quantify the loss of resilience in collapsing honey bee colonies due to the presence of a strong Allee effect. In the model, recruitment and mortality of adult bees have substantial social components, with recruitment enhanced and mortality reduced by additional adult bee numbers. The result is an Allee effect, a net per-individual rate of hive increase that increases as a function of adult bee numbers. The Allee effect creates a critical minimum size in adult bee numbers, below which mortality is greater than recruitment, with ensuing loss of viability of the hive. Under ordinary and favorable environmental circumstances, the critical size is low, and hives remain large, sending off viably-sized swarms (naturally or through beekeeping management) when hive numbers approach an upper stable equilibrium size (carrying capacity). However, both the lower critical size and the upper stable size depend on many parameters related to demographic rates and their enhancement by bee sociality. Any environmental factors that increase mortality, decrease recruitment, or interfere with the social moderation of these rates has the effect of exacerbating the Allee effect by increasing the lower critical size and substantially decreasing the upper stable size. As well, the basin of attraction to the upper stable size, defined by the model potential function, becomes narrower and shallower, indicating the loss of resilience as the hive becomes subjected to increased risk of falling below the critical size. Environmental effects of greater severity can cause the two equilibria to merge and the basin of attraction to the upper stable size to disappear, resulting in collapse of the hive from any initial size. The model suggests that multiple proximate causes, among them pesticides, mites, pathogens, and climate change, working singly or in combinations, could trigger hive collapse.
