Equilibrium Frequency of Endosymbionts in Multiple Infections Based on the Balance between Vertical Transmission and Cytoplasmic Incompatibility

Cytoplasmic incompatibility (CI)-inducing endosymbiotic bacteria, such as Wolbachia and Cardinium, have been well studied through field data and validations on the basis of numerical simulations. However, the analytically derived equilibrium frequency of multiple infections has not yet been determined, although the equilibrium for cases of single infection has been reported. In this study, we considered the difference equation for endosymbionts using three parameters: the probability of the failure of vertical transmission ([Formula: see text]), CI strength ([Formula: see text]), and the level of host inbreeding ([Formula: see text]). To analyze this model, we particularly focused on [Formula: see text], i.e., the frequency of host individuals completely infected with all [Formula: see text]-bacterial strains in the population. [Formula: see text], [Formula: see text] at the equilibrium state, was analytically calculated in the cases where [Formula: see text] and [Formula: see text] is any arbitrary value. We found that [Formula: see text] can be described using two parameters: [Formula: see text] and [Formula: see text], which is identical to [Formula: see text]. [Formula: see text] has a larger value in a system with a smaller [Formula: see text]. In addition, [Formula: see text] determines the maximum number of strains that infect a single host. Our results revealed the following: i) three parameters can be reduced to a single parameter, i.e., [Formula: see text] and ii) the threshold of the maximum number of infections is defined by [Formula: see text], which prevents additional invasions by endosymbionts.