Continuous and discrete dynamical systems for the declines of honeybee colonies

Abstract

This work deals with two mathematical models on the declines of honeybee colonies. Starting from the model proposed eight years ago by Khoury, Meyerscough, and Barron, (KMB model), we kept the eclosion function and changed the recruitment function to design a new model for social parasitism in honeybees. The KMB model is characterized by the fact that there exists a critical value of the foragers' death rate above which there is colony collapse disorder in the sense that the “trivial” equilibrium point is globally asymptotically stable. We design a nonstandard finite difference (NSFD) scheme that preserves this property. It is established that in the social parasitic (SP) model, the colony decays exponentially to zero irrespective of the value of foragers' death rate. An NSFD scheme is constructed for the SP model. The faster decline in the SP setting is demonstrated theoretically for the NSFD scheme. Numerical simulations are provided to confirm that the colony declines faster in the SP setting than in the KMB model.