Modeling mosquito control by an impulsive reaction-diffusion mosquito model with periodic evolution domain

In this paper, we construct and analyze a reaction-diffusion hybrid model incorporating impulsive control and periodic evolution domain in Wolbachia-infected and wild mosquito populations to investigate the joint impact of spatial diffusion, impulsive control and the evolution of a domain on the control of mosquitoes. The explicit formulas of the two ecological reproduction indexes RU and RI are introduced, which relate to the evolution rate of domain and the impulsive function. Then we establish the threshold-type dynamics for the impulsive problem in terms of RU and RI by the method of upper and lower solutions, which present the extinction of mosquitoes, and competition and the coexistence of Wolbachia-infected and wild mosquitoes under impulsive control and periodic evolution domain. Numerically, we perform simulations to certify the analytical results, determine the importance of parameters on the persistence and extinction of mosquito populations, and expound how impulsive control, the evolution domain and fitness effect caused by Wolbachia affect the mosquito population evolution.