## 4 Model Validations and Simulations
### Data and Parameter Estimations  
Based on the proof of Theorem 2, it is interesting to point out that \(N=\phi(X)\)(i.e., (19)) is a decreasing function in \(X\) when \(\frac{d_{g}}{d_{w}}<1\) and \(\frac{\delta_{g}}{d_{w}}>1\). This is in line with data collected from 65 colonies (see Figure 2). There are 65 open circles in Figure 2 which correspond to the data collected from three resources: (1) laboratory data from [7], (2) field data from [13], and (3) laboratory data collected collected in the Liebig Laboratory at Arizona State University. Figure 2 suggests that the colony size \(N\) decreases in a nonlinear fashion when the reproductive ratio \(X\) increases, which portrays in the similar patterns by \(N=\phi(X)\) when the scaling exponent of the metabolic rate of gamergates is larger than that of the non-reproductive workers and the mortality rate of gamergates is smaller than that of the non-reproductive workers per Theorem 2, i.e., \(\delta_{n}<\delta_{g}\) and \(d_{g}<d_{w}\), respectively. To further our understanding of the data and the colony size \(N\) as a function of the reproductive ratio \(X\), i.e., \(N=\phi(X)\), we let the average egg-laying rate of a gamergate being \(e=2.42\), the mortality of gamergates being \(d_{g}=0.00028\), and the mortality of workers being \(d_{w}=0.0025\) per literature and laboratory data. Although, the metabolic rate parameters \(\beta_{n},\beta_{g}\) and \(\delta_{n},\delta_{g}\) are not known, by taking advantage of data from 65 colonies, the best approximation of those metabolic rate parameters are \(\beta_{g}=33\), \(\delta_{g}=0.99\), \(\delta_{n}=0.261\), and \(\beta_{n}=74.6\) (solid curve in Figure 2). Under those parameters' values, the nonlinear hierarchy model (9) and (10) approaches to a median colony size \(N^{*}=60\) with the reproductive ratio being \(X^{*}=0.1114\). In the case that \(\beta_{n}=35.2\) and keeping the other parameters' values unchanged, the nonlinear hierarchy model (9) and (10) approaches a small colony size of about \(N^{*}=37\) with the reproductive ratio being \(X^{*}=0.0732\) (shown as the short-dashed curve in Figure 2). In the case that \(\beta_{n}=145.2\) and keeping the other parameters' values unchanged, the nonlinear hierarchy model (9) and (10) approaches a large colony size of about \(N^{*}=90\) with the reproductive ratio being \(X^{*}=0.16\) (shown as the long-dashed curve in Figure 2). Our model (9) and (10) employs the assumption that the social status, i.e., becoming gamergate or non-reproductive worker, is determined by the energy cost which is measured through the concept of the ecological metabolism theory with the form of \(\beta_{i}N_{i}^{\beta}\) whose values of \(\beta_{i}\) and \(\delta_{i}\) are not available in the literature. The data and the related fitting shown in Figure 2 provide a baseline on the related metabolic rate parameters \(\beta_{i}\) and \(\delta_{i}\); and the three curves suggest that the larger the values of the normalized metabolic rates for workers \(\beta_{n}\), the larger the colony size \(N^{*}\) and the reproductive ratio \(X^{*}\) (also see Figure 3b). Now we use time series data of two colonies (labeled F44 and SAFC12) to provide parameter estimations. As we only have three data points for each colony (Table 2), we fix \(e=2.42\), \(d_{g}=0.00028\), \(d_{w}=0.0025\), and \(\beta_{g}=33\) and estimate the values of \(\beta_{n}\) and \(\delta_{i},i=n,g\) for each colony. The results are shown in Table 3. In Figure 4a, the fitted parameters for Colony F44 are \(\beta_{n}=92.77,\delta_{g}=1.61\), and \(\delta_{n}=0.37\) with the colony reaching a size of \(N^{*}=51\)  
Figure 2: Colony size \(N\) as a function its reproductive ratio \(X\). Open circles are data collected from 65 colonies.