Figure 1

The red, blue, black, and green curves correspond to the eggs, larvae, pupae, adults classes, respectively. The survivorship functions are \(s_{i}(x)=a_{i}k_{i}/(k_{i}+a_{i}x)\), \(i=1,2,4\), with parameter values \(a_{1}=0.5\), \(a_{2}=0.4\), \(a_{4}=0.8\), \(k_{1}=400\), \(k_{2}=300\), \(k_{4}=800\), \(\gamma _{1}=0.3\), \(\gamma _{2}=0.4\), \(\gamma _{3}=0.5\), and \(s_{3}=0.6\). The initial conditions are given by \(E(0)=L(0)=P(0)=A(0)=50\). If \(b=9\), the inherent net reproductive number \(\Re _{0}=0.9370<1\), mosquito-free equilibrium \(E_{0}\) is globally asymptotically stable. Solution approach \(E_{0}\) as \(t\rightarrow \infty \), shown in the left figure, where \(\gamma _{1}=\gamma _{2}=\gamma _{3}=1\), with the same function forms and some parameters. As \(b=3\), the inherent net reproductive number \(\Re _{0}(1,1,1)=1.8>1\), \(E_{1}=(30,70,82,210)\) is globally asymptotically stable. Solutions approach \(E_{1}\), as \(t\rightarrow \infty \), shown in the right figure