Figure 2

The function forms and some parameters are given in Fig. 1. When the breeding of the mosquito population is seasonal with 2-cycle, \(b(2t)=\widehat{b}\), \(b(2t+1)=0\). As \(\widehat{b}=2.5\), the inherent net reproductive number \(\widehat{\Re }_{0}=0.8333<1\), \(\widehat{E}_{0}\) is globally asymptotically stable. Solutions approach \(\widehat{E}_{0}\) as \(t\rightarrow \infty \), shown in the left figure. As \(\widehat{b}=4\), \(\widehat{\Re }_{0}=1.3333>1\), and a stable 2-cycle appears, which is globally asymptotically stable, shown in the right figure