  Honeybees play an important role in the production of many agricultural crops
and in sustaining plant diversity in undisturbed ecosystems. The rapid decline
of honeybee populations have sparked great concern worldwide. Previous studies
have shown that the parasitic Varroa mite could be the main reason for colony
losses. In order to understand how mites affect population dynamics of
honeybees and a colony health, we propose a brood-adult bee-mite model in which
the time lag from brood to adult is taken into account. Noting that the
dynamics of a honeybee colony varies with respect to season, we validate the
model and perform parameter estimations under both constant and fluctuating
seasonality scenarios. Our analytical and numerical studies reveal the
following: (a) In the presence of parasite mites, the large time lag from brood
to adult could destabilize population dynamics and drive the colony to
collapse; but the small natural mortality of the adult population can promote a
mite-free colony when time lag is small or at an intermediate level; (b) Small
brood' infestation rates could stabilize all populations at the unique interior
equilibrium under constant seasonality while may drive the mite population to
die out when seasonality is taken into account; (c) High brood' infestation
rates can destabilize the colony dynamics leading to population collapse
depending on initial population size under constant and seasonal conditions;
(d) Results from sensitivity analysis indicate the queen's egg-laying may have
the greatest effect on colony population size. The brood death rate and the
colony size at which brood survivability is the half maximal were also shown to
be highly sensitive with an inverse correlation to the colony population size.
Our results provide insights on the effects of seasonality on the dynamics.
