Oscillation threshold for a mosquito population suppression model with time delay

We consider a mosquito population suppression model with time delay. We show that, in the absence of sterile mosquitoes released, the model solutions oscillate with respect to its unique non-zero equilibrium. With the releases of sterile mosquitoes, we then determine an oscillation threshold, denoted by $\hat{b}$, for the constant release rate of the sterile mosquitoes such that all non-trivial positive solutions oscillate when the release rate of the sterile mosquitoes is less than $\hat{b}$, and the oscillation disappears as the release rate exceeds $\hat{b}$. We also provide some numerical simulations to validate our theoretical results.