Wolbachia Invasion in Mosquitoes with Incomplete CI, Imperfect Maternal Transmission and Maturation Delay.

The mechanism of cytoplasmic incompatibility (CI) is important in the study of Wolbachia invasion in wild mosquitoes. Su et al. (Bull Math Biol 84(9):95, 2022) proposed a delay differential equation model by relating the CI effect to maturation delay. In this paper, we investigate the dynamics of this model by allowing the same density-dependent death rate and distinct density-independent death rates. Through analyzing the existence and stability of equilibria, we obtain the parameter conditions for Wolbachia successful invasion if the maternal transmission is perfect. While if the maternal transmission is imperfect, we give the ranges of parameters to ensure failure invasion, successful invasion and partially suppressing, respectively. Meanwhile, numerical simulations indicate that the system may exhibit monostable and bistable dynamics when parameters vary. Particularly, in the bistable situation an unstable separatrix, like a line, exists when choosing constant functions as initial values; and the maturation delay affects this separatrix in an interesting way.