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Honeybees have an irreplaceable position in agricultural production and the stabilization of natural ecosystems. Unfortunately, honeybee populations have been declining globally. Parasites, diseases, poor nutrition, pesticides, and climate changes contribute greatly to the global crisis of honeybee colony losses. Mathematical models have been used to provide useful insights on potential factors and important processes for improving the survival rate of colonies. In this review, we present various mathematical tractable models from different aspects: 1) simple bee-only models with features such as age segmentation, food collection, and nutrient absorption; 2) models of bees with other species such as parasites and/or pathogens; and 3) models of bees affected by pesticide exposure. We aim to review those mathematical models to emphasize the power of mathematical modeling in helping us understand honeybee population dynamics and its related ecological communities. We also provide a review of computational models such as VARROAPOP and BEEHAVE that describe the bee population dynamics in environments that include factors such as temperature, rainfall, light, distance and quality of food, and their effects on colony growth and survival. In addition, we propose a future outlook on important directions regarding mathematical modeling of honeybees. We particularly encourage collaborations between mathematicians and biologists so that mathematical models could be more useful through validation with experimental data. 

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. 

Abstract Varroa destructor is an ectoparasitic mite of immature and adult honey bees that can transmit several single-stranded RNA viruses to its host. Varroa reproduce in brood cells, and mite populations increase as colonies produce brood in spring and summer. Mite numbers also can sharply rise, particularly in the fall, by the migration of varroa into hives on foragers. Colonies with high levels of varroa and viruses often die over the winter. Feeding colonies pollen might keep virus levels low and improve survival because of the positive effects of pollen on immunity and colony growth. We compared varroa and virus levels and overwinter survival in colonies with (fed) and without (unfed) supplemental pollen. We also measured the frequency of capturing foragers with mites (FWM) at colony entrances to determine its relationship to varroa and virus levels. Colonies fed supplemental pollen were larger than unfed colonies and survived longer. Varroa populations and levels of Deformed wing virus (DWV) rose throughout the season, and were similar between fed and unfed colonies. The growth of varroa populations was correlated with FWM in fed and unfed colonies, and significantly affected DWV levels. Increasing frequencies of FWM and the effects on varroa populations might reduce the positive influence of supplemental pollen on immune function. However, pollen feeding can stimulate colony growth and this can improve colony survival.