At Most Two Periodic Solutions for a Switching Mosquito Population Suppression Model

We fill a gap concerning a dynamical description for a switching mosquito population suppression model proposed in Yu and Li (J Differ Equ 269:6193-6215, 2020), where a constant amount c of sterile mosquitoes is released after a waiting period T larger than the sexual lifespan (T) over bar of the released male mosquitoes. The release amount thresholds g*, c* with g* < c* and the waiting period threshold T* were found, and it was proved that the origin is locally asymptotically stable in D = {(c, T) : g* < c < c*, T < T*}. 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.