Summary: We fill a gap concerning a dynamical description for a switching mosquito population suppression model proposed in [J. Yu and J. Li, J. Differ. Equations 269, No. 7, 6193–6215 (2020; Zbl 1444.34103)], where a constant amount \(c\) of sterile mosquitoes is released after a waiting period \(T\) larger than the sexual lifespan \({\bar{T}}\) of the released male mosquitoes. The release amount thresholds \(g^\ast, c^\ast\) with \(g^\ast<c^\ast\) and the waiting period threshold \(T^\ast\) were found, and it was proved that the origin is locally asymptotically stable in \(D=\{(c, T): g^\ast<c<c^\ast,\, T<T^\ast\}\). However, the periodic solutions as well as the global asymptotical stability of the origin remains unknown. By ingeniously finding a useful separatrix \(L\) which can divide \(D\) into two sub-regions \(D_1\) and \(D_2\), we show that the origin is globally asymptotically stable in \(D_1\), and the model admits exactly two periodic solutions in \(D_2\), with one stable, and the other unstable, and a unique periodic solution on \(L\), which is semi-stable, respectively. Numerical examples to illustrate our theoretical results are also provided.