## Appendix A Proofs of Theorems and Corollaries
(2) \(\psi_{T*}-\psi_{T1}=\frac{b_{p}(\zeta-r)(R+(1+\zeta))}{2r(r-R\zeta)}+\frac{E(r- \zeta)}{2(r-R\zeta)}>0\) and \(\psi_{T2}-\psi_{T*}=-\frac{(\zeta-r)((\eta+\xi)r+(1+(\eta-\xi))R)p}{2r(r-R \zeta)}-\frac{E(r-\zeta)}{2(r-R\zeta)}>0\), i.e., \(\psi_{T*}\in[\psi_{T1},\psi_{T2}]\) hold in the case of \(E>b_{p}(1+\xi+\frac{R}{r})\) and \(E<b_{p}(\eta+\xi+\frac{(1-(\eta-1)\zeta)R}{r})\). Therefore, in the case of \(E>b_{p}(1+\xi+\frac{R}{r})+b_{p}\frac{(\eta-1)(r-\zeta)(1+R)}{2r}\) and \(E<b_{p}(\eta+\xi+\frac{(1-(\eta-1)\zeta)R}{r})-b_{p}\frac{(\eta-1)(r-\zeta)(1 +R)}{2r}\), \(\psi_{T*}\in[\psi_{T1},\psi_{T2}]\) obviously holds because \(b_{p}\frac{(\eta-1)(r-\zeta)(1+R)}{2r}>0\). Furthermore, for all \(x\) such as \(x<\psi_{T1}\), \((Dh_{T1}-1)(x-\psi_{T1})>0\) holds because \((Dh_{T1}-1)<0\) and \((x-\psi_{T1})<0\) for all \(x\) such as \(x<\psi_{T1}\). \(\frac{E}{1+R}+(Dh_{T1}-1)\psi_{T1}=\frac{b_{p}(r-\zeta)(\eta-1)}{2r}>0\) holds in the case of \(E>b_{p}(1+\xi+\frac{R}{r})+b_{p}\frac{(\eta-1)(r-\zeta)(1+R)}{2r}\). Therefore, for all \(x\) such as \(x<\psi_{T1}\), the following is the case. \[h_{T}(x)-x = (Dh_{T1}-1)(x-\psi_{T1})+\frac{E}{1+R}\] \[+ (Dh_{T1}-1)\psi_{T1}>0.\]  
In a similar manner as the above, for all \(x\) such as \(x>\psi_{T2}\), \((1-Dh_{T3})(x-\psi_{T2})>0\) holds because \((1-Dh_{T3})>0\) and \((x-\psi_{T2})>0\) for all \(x\) such as \(x>\psi_{T2}\). \(-\frac{E}{1+R}+(1-Dh_{T3})\psi_{T2}=\frac{b_{p}(r-\xi)(\eta-1)}{2r}>0\) holds in the case of \(E<b_{p}(\eta+\xi+\frac{(1-(\eta-1)\zeta)R}{r})-b_{p}\frac{(\eta-1)(r-\zeta)(1 +R)}{2r}\). Therefore, for all \(x\) such as \(x>\psi_{T2}\), the following is the case. \[x-h_{T}(x) = (1-Dh_{T3})(x-\psi_{T2})-\frac{E}{1+R}\] \[+ (1-Dh_{T3})\psi_{T2}>0.\]  
It is easy to check that \(\psi_{T*}=h_{T2}(\psi_{T*})\) and \(Dh_{T2}<-1\). Thus the 1-D map: \(\psi(\tau)=h_{T}(\psi(\tau-1))\) has a unique unstable fixed point \(\psi_{T*}(>0)\) in the interval \([\psi_{T1},\psi_{T2}]\). In the case of \(k=N\): It is easy to check the following properties: (1) \(\psi_{N*}>0\) holds because \(r-\zeta>0\) and \(r-R\zeta>0\). (2) \(\psi_{N*}-\psi_{N1}=\frac{b_{p}(r-\zeta)(r(1-\xi)-R\zeta)}{2r(r-R\zeta)}+\frac {E(r-\zeta)}{2(r-R\zeta)}>0\) and \(\psi_{N2}-\psi_{N*}=\frac{(r-\zeta)(\zeta r+\eta(r-R\zeta))b_{p}}{2r(r-R\zeta )}-\frac{E(r-\zeta)}{2(r-R\zeta)}>0\), i.e., \(\psi_{N*}\in[\psi_{N1},\psi_{N2}]\) hold in the case of \(E>b_{p}(\xi+\frac{R\zeta-r}{r})\) and \(E<b_{p}(\xi-\frac{\eta(R\zeta-r)}{r})\). Furthermore, for all \(x\) such as \(x<\psi_{N1}\), \((Dh_{N1}-1)(x-\psi_{N1})>0\) holds because \((Dh_{N1}-1)<0\) and \((x-\psi_{N1})<0\) for all \(x\) such as \(x<\psi_{N1}\). \(\frac{E}{1+R}+(Dh_{N1}-  
It is easy to check that \(\psi_{N*}=h_{N2}(\psi_{N*})\) and \(Dh_{N2}<-1\). Thus, the 1-D map \(\psi(\tau)=h_{N}(\psi(\tau-1))\) has a unique unstable fixed point \(\psi_{N*}(>0)\) in the interval \([\psi_{N1},\psi_{N2}]\). Item 1 is proved. Next, under the conditions that Theorem 2 holds, the following is the case. In the case of \(k=T\), it is easy to check the following properties. \((1)\)\(0<Dh_{T1}<1\), \(Dh_{T2}<-1\), and \(0<Dh_{T3}<1\) hold from Corollary 1. (2) \(h_{T}(\psi_{T1})>\psi_{T2}\) because \(h_{T}(\psi_{T1})-\psi_{T2}>0\) holds in the case of \(E>b_{p}(1+\xi+\frac{R}{r})+b_{p}\frac{(\eta-1)(r-\zeta)(1+R)}{2r}\). \((3)\)\(\psi_{T1}>h_{T}(\psi_{T2})\) because \(\psi_{T1}-h_{T}(\psi_{T2})>0\) holds in the case of \(E<b_{p}(\eta+\xi+\frac{(1-(\eta-1)\zeta)R}{r})-b_{p}\frac{(\eta-1)(r-\zeta)(1 +R)}{2r}\). In the case of \(k=N\), it is easy to check the following properties: (1) \(0<Dh_{N1}<1\), \(Dh_{N2}<-1\), and \(0<Dh_{N3}<1\) hold from Corollary 1. (2) \(h_{N}(\psi_{N1})>\psi_{N2}\) because \(h_{N}(\psi_{N1})-\psi_{N2}>0\) holds in the case of \(E>b_{p}(\xi+\frac{R\xi-r}{r})\). (3) \(\psi_{N1}>h_{N}(\psi_{N2})\) because \(\psi_{N1}-h_{N}(\psi_{2})>0\) holds in the case of \(E<b_{p}(\xi-\frac{\eta(K-r)}{r})\).