evolution of endosymbiont density in doubly infected host species

Multiple infection of individual hosts with several species or strains of maternally inherited endosymbionts is commonly observed in animals, especially insects. Here, we address theoretically the effect of co-infection on the optimal density of the endosymbionts in doubly infected hosts. Our analysis is based on the observation that a maternally inherited double infection is only stable if doubly infected females produce more doubly infected daughters than singly infected or uninfected females produce daughters. We consider both a general model and a model involving two endosymbionts inducing bidirectional cytoplasmic incompatibility (CI). We demonstrate that the optimal replication rate of endosymbionts in doubly infected hosts can be expected to be similar to or below the optimal replication rate in singly infected hosts. This is in contrast to some theoretical predictions for horizontally transmitted parasites and stems from the two strains of endosymbionts having coupled fitness. We discuss our results with respect to recent empirical results on endosymbiont densities, the evolution of CI-inducing bacteria and, more generally, the evolution of cooperation through direct fitness benefits.