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 ¯ 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. [ABSTRACT FROM AUTHOR]