A Periodic Delay Differential Equation Model for Mosquito Suppression Based on Beverton–Holt-Type of Birth

Releasing Wolbachia-infected male mosquitoes to suppress wild females through cytoplasmic incompatibility (CI) has proven to be a safe and effective method for mosquito control. We study a release strategy that males are released with a constant amount c at time \(t_k=kT\) $t_k=kT$ , \(k=0, 1, 2, \ldots \) $k=0,1,2,\dots$ . The determination of T is related to the sexually active lifespan of released males \(\bar{T}\) $\overline{T}$ . By introducing the early lethal effect of CI, we formulate a delay differential equation model for mosquito suppression based on Beverton-Holt-type of birth. After explicitly expressing two thresholds \(c^*\) $c^{\ast }$ and \(T^*\) $T^{\ast }$ , we prove that when \(c\in (0, c^*)\) $c\in (0,c^{\ast })$ and \(T\in (\bar{T}, T^*)\) $T\in (\overline{T},T^{\ast })$ , the model displays the bistable dynamics, where mosquito population either declines to zero, or approaches a positive periodic state. The rigorous mathematical proof of the exact number of periodic solutions and their stability analyses is obtained by proving that every solution is sandwiched between two “good” solutions, whose initial functions are chosen to be the solutions of the model without delay.