Wolbachia invasion dynamics of a random mosquito population model with imperfect maternal transmission and incomplete CI

In this work, we formulate a random Wolbachia invasion model incorporating the effects of imperfect maternal transmission and incomplete cytoplasmic incompatibility (CI). Under constant environments, we obtain the following results: Firstly, the complete invasion equilibrium of Wolbachia does not exist, and thus the population replacement is not achievable in the case of imperfect maternal transmission; Secondly, imperfect maternal transmission or incomplete CI may obliterate bistability and backward bifurcation, which leads to the failure of Wolbachia invasion, no matter how many infected mosquitoes would be released; Thirdly, the threshold number of the infected mosquitoes to be released would increase with the decrease of the maternal transmission rate or the intensity of CI effect. In random environments, we investigate in detail the Wolbachia invasion dynamics of the random mosquito population model and establish the initial release threshold of infected mosquitoes for successful invasion of Wolbachia into the wild mosquito population. In particular, the existence and stability of invariant probability measures for the establishment and extinction of Wolbachia are determined.