Bidirectional cytoplasmic incompatibility and the stable coexistence of two Wolbachia strains in parapatric host populations

Wolbachia are intracellular bacteria which are very widely distributed among arthropods. In many insect species Wolbachia are known to induce cytoplasmic mating incompatibility (CI). It has been suggested that Wolbachia could promote speciation in their hosts if parapatric host populations are infected with two different Wolbachia strains causing bidirectional mating incompatibilities. A necessary condition for this speciation scenario to work is that the two Wolbachia strains can stably coexist. The following study investigates this problem analysing a mathematical model with two host populations and migration between them. We show that the stability of bidirectional CI can be fully described in terms of a critical migration rate which is defined as the highest migration below which a stable coexistence of two Wolbachia strains is possible. For some special cases we could derive analytical solutions for the critical migration rate; for the general case estimations of the critical migration rate are given. Our main finding is that bidirectional CI can stably persist in the face of high migration and can be as high as over 15% per generation for CI levels observed in nature. These results have implications for the potential of Wolbachia to promote genetic divergence and speciation in their hosts.