## 4 Model Validations and Simulations
### Simulations and Bifurcations  
Our simulations suggest that the system (9) and (10) always has a unique interior equilibrium \((N^{*},N_{S}^{*})\) which appears to be globally stable, i.e., the colony would reach its stable reproductive ratio for a fixed set of parameter values. Observations from Figure 3 indicate that the metabolic parameters are crucial for the colony size \(N\) and the reproductive ratio \(X\). The interior equilibrium \((N^{*},X^{*})\) depends on the metabolic parameters: Figure 3a,c shows that the larger the normalization measure of the gamergates \(\beta_{g}\) and the hypometric relation factor of gamergates \(\delta_{g}\), the smaller the colony size \(N\) and the reproductive ratio \(X\). Figure 3b,d shows that the larger the normalization measure of the workers \(\beta_{g}\) and the hypometric relation factor of workers \(\delta_{n}\), the larger the colony size \(N\) and the reproductive ratio \(X\). Figure 3 suggests that metabolic parameters of workers have opposite effects of gamergates' metabolic parameters. To have a better picture of how \(\beta_{i}\) and \(\delta_{i}\) impact the colony size \(N^{*}\) and the related reproductive ratio \(X^{*}=\frac{N_{S}^{*}}{N^{*}}\), we highlight the curve of \(N=\phi(X)\) in red when \(e=2.42\), \(d_{g}=0.00028\), \(d_{w}=0.0025\),\(\beta_{g}=33\), \(\beta_{n}=76.4\), \(\delta_{g}=0.99\), and \(\delta_{n}=0.261\) which is the best approximation shown in Table 1. Four subfigures in Figure 3 suggest that the hypometric relation factor of metabolic rate \(\delta_{i}\) is more sensitive than the normalization measure of \(\beta_{i}\). The bifurcation diagrams shown in Figure 5 were implemented to determine the effects of parameters, such as mortality and the egg laying rate, in shaping the hierarchy structure and demographics of the colony, i.e., \(N\) and \(X\), respectively. In Figure 5a, as the parameter \(d_{g}\) is increased in the range \((0.001;1)\) while fixing \(e=2.42\), \(d_{w}=0.0025\), \(\beta_{g}=33\), \(\beta_{n}=74.6\), \(\delta_{g}=0.99\), and \(\delta_{n}=0.261\), the colony size \(N^{*}\) decreases and the reproductive ratio \(X^{*}\) increases. Similarly, as the parameter \(d_{w}\) increases in the range \((0.0025;1)\) while fixing \(e=2.42\), \(d_{g}=0.00028\), \(\beta_{g}=33\), \(\beta_{n}=74.6\), \(\delta_{g}=0.99\), and \(\delta_{n}=0.261\) (Figure 5b), the colony size \(N^{*}\) decreases and the reproductive ratio \(X^{*}\) increases. In Figure 5c, as the parameter \(\beta_{g}\) increases in the range \((20;50)\) while fixing \(e=2.42\), \(d_{g}=0.00028\), \(d_{w}=0.0025\), \(\beta_{n}=74.6\), \(\delta_{g}=0.99\), and \(\delta_{n}=0.261\), both the colony size \(N^{*}\) and the reproductive ratio \(X^{*}\) decrease. In contrast, in Figure 5d, as the parameter \(\beta_{n}\) increases across the range \((35;150)\) while fixing \(e=2.42\), \(d_{g}=0.00028\), \(d_{w}=0.0025\), \(\beta_{g}=33\), \(\delta_{g}=0.99\), and \(\delta_{n}=0.261\), both the colony size \(N^{*}\) and the reproductive ratio \(X^{*}\) increase. In Figure 5e, as the hypometric relation factor in gamergates \((\delta_{g})\) increases across the range \((0;0.5)\) while fixing \(e=2.42\), \(d_{g}=0.00028\), \(d_{w}=0.0025\), \(\beta_{g}=33\), \(\beta_{n}=74.6\), and \(\delta_{n}=0.261\), the colony size \(N^{*}\) and the reproductive ratio \(X^{*}\) decrease. In Figure 5f, as the hypometric relation factor in non-reproductive workers \((\delta_{n})\) increases from 0 to 0.5 while fixing \(e=2.42\), \(d_{g}=0.00028\), \(d_{w}=0.0025\), \(\beta_{g}=33\), \(\beta_{n}=74.6\), and \(\delta_{g}=0.99\), the colony size \(N^{*}\) and the reproductive ratio \(X^{*}\) increase. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Colony** & \(\beta_{n}\) & \(\delta_{g}\) & \(\delta_{n}\) & \(N^{*}\) & \(Ng^{*}\) & \(X^{*}\) \\ \hline F44 & 92.77 & 1.61 & 0.37 & 51 & 5 & 0.098 \\ SAFC12 & 121.15 & 1.59 & 0.67 & 97 & 15 & 0.155 \\ \hline \end{tabular}
\end{table}