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Social hierarchies are ubiquitous in social groups such as human societies and social insect colonies; however, the factors that maintain these hierarchies are less clear. Motivated by the shared reproductive hierarchy of the ant species Harpegnathos saltator, we have developed simple compartmental nonlinear differential equations to explore how key life-history and metabolic rate parameters may impact and determine its colony size and the length of its shared hierarchy. Our modeling approach incorporates nonlinear social interactions and metabolic theory. The results from the proposed model, which were linked with limited data, show that: (1) the proportion of reproductive individuals decreases over colony growth; (2) an increase in mortality rates can diminish colony size but may also increase the proportion of reproductive individuals; and (3) the metabolic rates have a major impact in the colony size and structure of a shared hierarchy. 

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.