## 1 Introduction
He found that a carry-over effect may explain why there are differences in predicted hierarchies in contrast with observed hierarchies; when individual differences in the group members are significant, they lead to a winner-loser/carry-over effect (successive victories lead to more victories and vice versa). Social dominance hierarchies are widespread with most attention on linear and despotic hierarchies. However, little is known about the mechanisms that produce nonlinear structures, especially shared hierarchies [7]. To our best knowledge, the available work on this topic has been done by [PERSON] et al. [7]. In this study, agent-based models, combined with experimental data on the ant species _H. Saltator_, were used to predict the emergence of complex animal hierarchies including shared hierarchies. Their computer simulations show that a linear hierarchy emerges when a typical winner-loser interaction (dominance biting of _H. Saltator_) is present; a despotic hierarchy emerges when a policing interaction is added; and a shared hierarchy emerges with the addition of a winner-winner interaction that results in a positive outcome for both interactors (antennal dueling). The model developed by [PERSON] et al. [7] provides a baseline for us to build mathematical tractable models that can explore the impacts of the colony size and metabolic rates on the establishment of shared hierarchies. A metabolic theory of ecology suggests that increased group size reduces overall metabolic costs, thereby increasing the efficiency of the group [19; 20]. Studies [21; 22; 23; 24] show that colony size (the group size of worker ants) affects parameters across different taxa, (e.g., metabolic rates), such that these parameters scale with colony size. For instance, the number of reproductive individuals in multiple ant species has been shown to be a function of the number of workers elevated to an exponent less than one. In addition, the number of reproductives per capita decreases with the size of mature colonies across some species [23]. _Temnothorax_ ants, in the laboratory, show a colony size scaling as lower per capita brood production and lower energy consumption occur at larger colony sizes [24]. In the ant _P. occidentalis_, the biomass of reproduction does not scale with colony size, but the probability of producing a reproductive individual increases with colony size [25]. This scaling phenomenon is not considered in the approach proposed by [PERSON] et al. [7] since their computational results are based on fixed colony size. So, there is a need to develop a simple mathematical model to explore the impacts of colony size and the related metabolic theory of ecology on shaping shared social hierarchy. Motivated by the ant species _Harpegathos saltator_ and the work by [PERSON] et al. [7], we develop a mechanistic-based but mathematical tractable model to understand how the shared hierarchy inside a colony, reflected by the ratio of reproductive workers to all workers, impacts the population dynamics and structure of social insect colonies. Our proposed model incorporates nonlinear social interactions and the related metabolic rate influenced by hierarchy establishment. We expect our proposed model provides insight into the role of colony size in shaping shared hierarchy. The paper is organized as follows: In Section 2, we start with the biological background of the particular ant species \(H\). _Saltator_ which motivates our work, and then we formulate a mathematical model based on ecological assumptions that are supported by data and literature. In Section 3, we perform mathematical analyses of the proposed model. In Section 4, we compare model simulations to data on colony size and reproductive ratio for laboratory-observed \(H\). _Saltator_ colonies. We also perform parameter estimation and bifurcation analysis for the proposed model. Finally, in Section 5, we summarize our results and elaborate on the implications of our model and potential future work.