Wolbachia-induced unidirectional cytoplasmic incompatibility and the stability of infection polymorphism in parapatric host populations

Wolbachia are intracellular, maternally inherited bacteria that are widespread among arthropods and commonly induce a reproductive incompatibility between infected male and uninfected female hosts known as unidirectional cytoplasmic incompatibility (CI). If infected and uninfected populations occur parapatrically, CI acts as a post-zygotic isolation barrier. We investigate the stability of such infection polymorphisms in a mathematical model with two populations linked by migration. We determine critical migration rates below which infected and uninfected populations can coexist. Analytical solutions of the critical migration rate are presented for mainland-island models. These serve as lower estimations for a more general model with two-way migration. The critical migration rate is positive if either Wolbachia causes a fecundity reduction in infected female hosts or its transmission is incomplete, and is highest for intermediate levels of CI. We discuss our results with respect to local adaptations of the Wolbachia host, speciation, and pest control.