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Theory and Modern Applications

Figure 3 | Advances in Difference Equations

Figure 3

From: A discrete-time mathematical model of stage-structured mosquito populations

Figure 3

The functions \(s_{1}(E)\) and \(s_{4}(A)\) still have the Beverton–Holt form such that \(s_{i}(x)=a_{i}k_{i}/(k_{i}+a _{i}x)\), \(i=1,4\), but \(s_{2}(L)\) has the Ricker-type nonlinearity such that \(s_{2}(L)=a_{2}e^{-L/k_{2}}\) and \(E(0)=L(0)=P(0)=A(0)=10\), other parameters are given in Fig. 1. As \(b=8\), the inherent net reproductive number \({\Re }_{0}=0.8329<1\), trivial fixed point \(E_{0}\) is globally asymptotically stable. Solutions approach \(E_{0}\) as \(t\rightarrow \infty \), shown in the left figure. As \(b=13\), \({\Re }_{0}=1.3534>1\), \(E_{0}\) becomes unstable and \(E_{1}=(4.29,5.71,20,114.2)\), which is globally asymptotically stable, shown in the right figure

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