Suzanne Sumner and Kelly Brown
Mathematical Models of Honey Bee Populations: Rapid Population Decline

Recent reports of honey bee colony deaths worldwide [VanEngelsdorp et al, 2009] have prompted interest in mathematical models to study the decrease of bees within a colony. Some factors contributing to the colony losses include Varroa mites, viruses and brood diseases, pesticides, inadequate nutrition, climate and seasonal changes, and the stresses of moving colonies for crop pollination. A new condition, Colony Collapse Disorder (CCD), describes mass colony deaths with no clear cause, and CCD features empty hives with dead brood and very few adult bees, yet adequate food stores, all signs of rapid depopulation. No one agent is thought to cause CCD [Watanabe, 2008], and lacking specific evidence, CCD is blamed on a combination of the multiple stressors listed above. Khoury, Myerscough and Barron [2011] derived a single-colony model with two differential equations describing how the hive bee and forager bee populations interact. Labor tasks among honey bees differ by age: the younger hive bees H perform maintenance tasks within the hive and the older forager bees F perform more hazardous tasks outside the hive, such as collecting nectar, pollen, or water. The hive bee population H changes at a rate dependent on their emergence rate from pupae and the transition rate to foraging. The forager bee population F changes at a rate dependent on the transition rate from hive bee status and their death rate. Khoury et al [2011] consider the death rate of the hive bees to be negligible. Brown and Sumner have extended this model to refine some of the parameters and to include a hive bee death rate. Factors such as pesticide-contaminated pollen in food stores or Varroa mites would adversely affect hive bee mortality. The model predicts the existence of a stable positive equilibrium in which both hive bees and forager bees persist when forager death rates are low. Past a threshold level when forager death rates are high, colony failure is inevitable as the hive bee and forager bee numbers are driven to zero.

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