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The honeybee plays an extremely important role in ecosystem stability and diversity and in the production of bee pollinated crops. Honey bees and other pollinators are under threat from the combined effects of nutritional stress, parasitism, pesticides, and climate change that impact the timing, duration, and variability of seasonal events. To understand how parasitism and seasonality influence honey bee colonies separately and interactively, we developed a non-autonomous nonlinear honeybee-parasite interaction differential equation model that incorporates seasonality into the egg-laying rate of the queen. Our theoretical results show that parasitism negatively impacts the honey bee population either by decreasing colony size or destabilizing population dynamics through supercritical or subcritical Hopf-bifurcations depending on conditions. Our bifurcation analysis and simulations suggest that seasonality alone may have positive or negative impacts on the survival of honey bee colonies. More specifically, our study indicates that (1) the timing of the maximum egg-laying rate seems to determine when seasonality has positive or negative impacts; and (2) when the period of seasonality is large it can lead to the colony collapsing. Our study further suggests that the synergistic influences of parasitism and seasonality can lead to complicated dynamics that may positively and negatively impact the honey bee colony's survival. Our work partially uncovers the intrinsic effects of climate change and parasites, which potentially provide essential insights into how best to maintain or improve a honey bee colony's health.

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.

Western honeybees (Apis Mellifera) serve extremely important roles in our ecosystem and economics as they are responsible for pollinating $ 215 billion dollars annually over the world. Unfortunately, honeybee population and their colonies have been declined dramatically. The purpose of this article is to explore how we should model honeybee population with age structure and validate the model using empirical data so that we can identify different factors that lead to the survival and healthy of the honeybee colony. Our theoretical study combined with simulations and data validation suggests that the proper age structure incorporated in the model and seasonality are important for modeling honeybee population. Specifically, our work implies that the model assuming that (1) the adult bees are survived from the {egg population} rather than the brood population; and (2) seasonality in the queen egg laying rate, give the better fit than other honeybee models. The related theoretical and numerical analysis of the most fit model indicate that (a) the survival of honeybee colonies requires a large queen egg-laying rate and smaller values of the other life history parameter values in addition to proper initial condition; (b) both brood and adult bee populations are increasing with respect to the increase in the {egg-laying rate} and the decreasing in other parameter values; and (c) seasonality may promote/suppress the survival of the honeybee colony.

We propose a honeybee-mite-virus model that incorporates (1) parasitic interactions between honeybees and the Varroa mites; (2) five virus transmission terms between honeybees and mites at different stages of Varroa mites: from honeybees to honeybees, from adult honeybees to phoretic mites, from honeybee brood to reproductive mites, from reproductive mites to honeybee brood, and from honeybees to phoretic mites; and (3) Allee effects in the honeybee population generated by its internal organization such as division of labor. We provide completed local and global analysis for the full system and its subsystems. Our analytical and numerical results allow us have a better understanding of the synergistic effects of parasitism and virus infections on honeybee population dynamics and its persistence. Interesting findings from our work include: (a) Due to Allee effects experienced by the honeybee population, initial conditions are essential for the survival of the colony. (b) Low adult honeybee to brood ratios have destabilizing effects on the system, generate fluctuated dynamics, and potentially lead to a \emph{catastrophic event} where both honeybees and mites suddenly become extinct. This catastrophic event could be potentially linked to Colony Collapse Disorder (CCD) of honeybee colonies. (c) Virus infections may have stabilizing effects on the system, and could make disease more persistent in the presence of parasitic mites. Our model illustrates how the synergy between the parasitic mites and virus infections consequently generates rich dynamics including multiple attractors where all species can coexist or go extinct depending on initial conditions. Our findings may provide important insights on honeybee diseases and parasites and how to best control them., Comment: 40 pages, 5 figures