## 2 Model Formulation  
Motivated by the work of queenless colonies of the Ponerine ant, _Harpegathos saltator_[7], we derive our model on its shared hierarchy. A _Harpegathos saltator_ ant's life cycle consists of four stages: egg, larva, pupa, and adult. Fertilized eggs become female ants (queens and workers), whereas unfertilized eggs develop into male ants. The period from egg to adult can last over 79 days [26]. The average _Harpegathos_ worker lives from 200 to 400 days in the lab, while the ecological factors in the field reduce this life span [27]. In contrast, a reproductive queen can live for more than 5 years in the laboratory [28]; however, it is estimated that she may only live for as long as 1.79 years in the field [29]. Single individual queens establish new colonies as foundresses. After constructing a nest, a queen then produces and cares for eggs, which grow into larvae, pupae, and finally workers. In this species, a foundress hunts above ground to feed her first offspring. This continues until adult workers take over brood care, nest construction, and foraging. However, when the queen seneseces and/or dies, she is replaced by gametes (reproductive workers) who partition their reproduction among one another evenly [13; 30]. Most gametes lose their status through senescence and death, although some may revert to worker-like behavior and physiology in the laboratory [31]. While queens and workers of \(H\). _Saltator_ show morphological differences, non-reproductive workers also exhibit distinct behaviors such as rapid locomotion, hunting and handling of live arthropods, and low posture that differ from gamergates who show slow locomotion, high posture, and egg-laying [32]. Since any adult worker can become a gamergate, all can be categorized as potential gamergates during the establishment of a new reproductive hierarchy [33]. Over time, the shared nonlinear hierarchy is established through potential chemical cues and social interactions, such as policing, antennal dueling, and dominance biting. These interactions eventually divide the colony into two classes based on reproduction: established gamergates and non-reproductive workers [7]. Established gamergates are defined as gamergates who maintain their social position in the hierarchy. This specific biological background provides us with a template for mathematical modeling. We denote \(N_{g}(t)\) as the number of established gamergates, \(N_{w}(t)\) denotes the number of non-reproductive workers, and \(N(t)=N_{g}(t)+N_{w}(t)\) represents the total population size of female adult ants (colony size) at time \(t\) (days). We develop a mechanistic model based on nonlinear social interactions and metabolic theory to explore how life-history parameters and the related metabolic costs shape the hierarchy structure of a social insect colony. Ecological assumptions of our generalized model are as follows:
* **Modeling Relative Cost**: Assuming that gamergates are the only members of the colony who actively reproduce, the cost for all gamergates of remaining in their status in the colony is measured by \(E_{g}\) (Kcal) which consists of the energy cost of laying eggs as well as the energy to perform behaviors that maintain the social hierarchy such as antennal dueling and dominance behaviors. According to the ecological metabolism theory, the formulation of \(E_{g}\) is modeled by \[E_{g}=\beta_{g}N_{g}^{\delta_{g}},\] where \(\beta_{g}\) is a normalization constant (measured in microwatts per gram \(\mu\)W/gr). This coefficient \(\beta_{g}\) corresponds to the variation in metabolic rates across gamergates in the group. Then, \(\delta_{g}\) is a scaling exponent, which describes how fast the cost/metabolic rate changes with respect to the biomass of the group of gamergates [34]. If \(\delta_{g}<1\), then the metabolic rate of the group grows slower than the biomass. Instead, if \(\delta_{g}>1\) the metabolic rate of the group grows faster than the biomass.