Frank, S. A. 1998. Dynamics of cytoplasmic incompatibility with multiple Wolbachia infections. Journal of Theoretical Biology 192:213-218.

Wolbachia infections occur in many arthropods. These matrilineally inherited bacteria cause cytoplasmic incompatibility, in which a cross produces no offspring when between an infected male and an uninfected female. Some populations harbour multiple Wolbachia strains. Females fail to produce offspring when crossed to a male with a strain that the female lacks. Prior theoretical work showed that a panmictic population cannot maintain polymorphism for different strains when each female carries only a single strain. A few authors suggested that doubly infected females can stabilize multistrain polymorphism, but conditions for invasion and location of stable equilibria were not analyzed in detail. For two strains, I describe the conditions under which a multiply infected class can spread. Spread of the doubly infected type stabilizes polymorphism of the singly infected classes. This analysis also suggests an interesting extension to higher multiplicity of infection. For an arbitrary number of strains, $N$, a panmictic population cannot maintain different classes with $N-1$ infections unless the class with $N$ infections is also present. This pyramid of polymorphism may explain the puzzling diversity of incompatibility types observed in some Culex mosquitos. Multiple infection also has interesting consequences for the dynamics of spatial variation and reproductive isolation.