Honeybees play an important role in sustaining the ecosystem . The rapid decline of their population have sparked a great concern worldwide. Field and theoretical studies have shown that the infestation of the parasitic Varroa mite could be one of the main reasons that lead to colony collapsing . In order to understand how mites affect population dynamics of honeybees and the healthy status of colony , we propose and study a brood-adultbee-mite model with time lag from brood to adult bee . Noting that the temporal dynamics of honeybee colony varies with respect to changes in seasonality , we validate the model and perform parameter estimations under both constant and fluctuating seasonality scenarios. We found : (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 adult has the ability to promote a mite-free colony as time lag is small or at an intermediate level; (b) Small brood' s infestation rate could stabilize all populations at the unique interior equilibrium under constant seasonality while driving the mite to die out when changes in seasonality is taken into account; (c) Large brood' s infestation rate can destabilize the  dynamics leading to colony collapse depending on initial population size under constant and non-constant seasonal model ; (d) Results from our sensitivity analysis indicate that the queen's egg-laying may have the greatest effect on colony  size. The death rate of the brood and the colony size at which brood survivability is  half maximal were also shown to be highly sensitive with an inverse correlation to the colony population size. Our results provide the dynamics generated by seasonality where the mite population dies out leaving healthy colony while all population goes extinct without seasonality .
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 .
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