## 3 Mathematical Analysis
If \(\delta_{g}=\delta_{n}\), then the positive equilibrium \((N^{*},N^{*}_{g})\) exists if and only if \(X^{*}=\frac{N^{*}_{g}}{N^{*}}\in(0,1)\) solves the equation \(\frac{\beta_{n}}{\beta_{g}X^{N^{*}}+\beta_{n}}=f(X)\) and \(N^{*}=g(X^{*})\). Denote  
\[\psi(X)=-X^{2}(ed_{g}+2d_{g}-2d_{w})+2X(d_{g}-2d_{w})+2d_{w}. \tag{18}\]  
When \(ed_{g}+2d_{g}-2d_{w}\
eq 0\), i.e., \(\frac{d_{g}}{d_{w}}\
eq\frac{2}{2+e}\), \(\psi(X)\) has two real zeros:  
\[X_{1}=\frac{d_{g}-2d_{w}+\sqrt{2 ed_{g}d_{w}+d_{g}^{2}}}{ed_{g}+2d_{g}-2d_{w}}, \leavevmode\
obreak\ \leavevmode\
obreak\ X_{2}=\frac{d_{g}-2d_{w}-\sqrt{2 ed_{g}d_{w}+d_{g}^{2}}}{ed _{g}+2d_{g}-2d_{w}}.\]  
If \(\frac{d_{g}}{d_{w}}=\frac{2}{2+e}\), then \(\psi(X)\) has a unique positive root \(\frac{d_{w}}{2d_{w}-d_{g}^{2}}\). We still denote it as \(X_{1}\), i.e., \(X_{1}=\frac{d_{w}}{2d_{w}-d_{g}}=\frac{2+e}{2+2e}\). For each case, we have \(0<X_{1}<1\) (see the proof of Theorem 2 in Appendix A.2). Now, we show the existence of an interior equilibrium of the system (9) and (10). **Theorem 2**. : _Let \(\frac{d_{g}}{d_{w}}<1\)._  
1. _If_ \(\delta_{g}>\delta_{n}\)_, then there exists_ \(X^{c}\) _such that_ \(0<X^{c}<X_{1}<1\) _and_ \(\phi(X)=g(X)\) _has a unique solution in_ \((0,X^{c})\) _and no solution in_ \((X^{c},1)\)_._
2. _If_ \(\delta_{g}=\delta_{n}\)_,_ \(\phi(X)=g(X)\) _has a unique solution in_ \((0,X_{1})\) _and no solution in_ \((X_{1},1)\)_._
3. _If_ \(\delta_{g}<\delta_{n}\)_, then there exists_ \(X^{c}\) _such that_ \(0<X^{c}<X_{1}<1\) _and_ \(\phi(X)=g(X)\) _has no solution in_ \((X^{c},1)\)_. In_ \((0,X^{c})\)_, we have the following statements:_ 1. _If_ \(\delta_{g}<\delta_{n}<2\delta_{g}\)_,_ \(\phi(X)=g(X)\) _has at least one positive solution._ 2. _If_ \(2\delta_{g}<\delta_{n}\)_,_ \(\phi(X)=g(X)\) _has no positive solution provided that_ \(\beta_{n}/\beta_{g}\) _is small enough._ 3. _If_ \(2\delta_{g}=\delta_{n}\)_,_ \(\phi(X)=g(X)\) _has at least one positive solution provided that_ \(\frac{\beta_{n}}{\beta_{g}}>\left(\frac{e}{2d_{w}}\right)^{-d_{g}}\)_, while no positive solution provided that_ \(\beta_{n}/\beta_{g}\) _is small enough._ _If_ \(\phi(X^{*})=g(X^{*})\) _has a positive solution_ \(X^{*}\in(0,1)\)_, then the system (_9_) and (_10_) has an interior equilibrium_ \((N^{*},N^{*}_{S})\) _with_ \(X^{*}=\frac{N^{*}_{S}}{N^{*}}\) _and_ \(N^{*}=g(X^{*})\)_._  
Notes: Theorem 2 provides theoretical results regarding how the mortality ratio \(\frac{d_{g}}{d_{w}}\) and the nonlinear metabolic ratio \(\frac{\delta_{g}}{d_{w}}\) determine the existence of an interior equilibrium \((N^{*},N^{*}_{S})\) for the system (9) and (10). The results indicate that if \(\frac{d_{g}}{d_{w}}<1\) and \(\frac{\delta_{g}}{d_{w}}>1\), then the system (9) and (10) has a unique interior equilibrium \((N^{*},N^{*}_{S})\). The biological meaning of \(\frac{d_{g}}{d_{w}}<1\) and \(\frac{\delta_{g}}{d_{w}}>1\) is that the mortality of gamergates is smaller than that of the non-reproductive workers and the metabolic rate of gamergates is higher than that of the non-reproductive workers. The first condition is in line with literature reports [38; 39]. In addition, Theorem 2 also suggests the following:  
1. The values of \(X_{1}\) and \(X_{2}\) are determined by the average reproduction ability of gamergate \(e\) and the average mortality of gamergates and workers, which provide an upper bound for the reproductive ratio. 2. Note that \(X_{1}\) can be rewritten as \[X_{1}=\frac{1-\frac{2d_{w}}{d_{g}}+\sqrt{\frac{2 ed_{w}}{d_{g}}+1}}{e+2-\frac{2 d_{w}}{d_{g}}}.\] provided that \(\frac{d_{g}}{d_{w}}\
eq\frac{2}{2+e}\). Due to \[\frac{dX_{1}}{d\frac{d_{w}}{d_{g}}}=\frac{\left(\sqrt{2e\frac{d_{w}}{d_{g}}+1} -e-1\right)^{2}}{\sqrt{2e\frac{d_{w}}{d_{g}}+1}\Big{(}e+2-2\frac{d_{w}}{d_{g} }\Big{)}^{2}}>0,\]  
\(X_{1}\) is increasing as \(\frac{d_{w}}{d_{g}}\) increases. If \(\frac{d_{g}}{d_{w}}=\frac{2}{2+e}\), we also have that \(X_{1}=\frac{d_{w}}{2d_{w}-d_{g}}\) is increasing as \(\frac{d_{w}}{d_{g}}\) increases.