Stability in a Periodic Model for Genetically Altered Mosquitos

An extension of a discrete-time population dynamics model for wild and genetically altered mosquitoes that takes into account the effect of seasonal variability is presented. The seasonal variability is introduced into the model by allowing the birth and survival functions to have periodic parameters. We consider the stability of periodic solutions of the ratio dynamics and show that under some conditions the ratio dynamics has a globally asymptotically stable periodic solution. Using the ratio dynamics we decouple the system into two Ricker equations with periodic parameters. The stability of the periodic solutions of the original system is given by the stability of the uncoupled system. We propose a conjecture about the stability of the periodic solutions of the Ricker equation with periodic parameters, and for the special case of period-two parameters we analytically show the conjecture is true. For this case we show that the stability region in parameter space obtained from the conjecture is larger than a previously proposed stability region. The period-three case is investigated numerically.