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Supplementary material
Consider two populations of Wolbachia that are bidirectionally incompatible and exist parapatrically with migration in each direction at rate m per unit time. Define wij to be the frequency of Wolbachia of type i in region j. We assume each Wolbachia may reduce its host fecundity by (1-fi) but that transmission is perfect, rescue of same-type sperm is complete, and of other-type sperm zero. Now consider a Mendelian-inherited gene that allows females with i-type Wolbachia to use sperm from males infected with j-type Wolbachia with probability hi. We make the simplifying assumption of haploid genetics and define xij as the frequency of individuals with the nuclear gene and Wolbachia of type i in region j. With these assumptions the system can be described by a system of eight equations
where the fj is the total output of patch j.
We adapted this model to approximate the interactions between Pel and Bei colonies of C. quinquefasciatus. We assume initially population 2 is at x22=1 and population 1 w11=1. We set f1 = f2 = 0, h1 = 0.9 h2 = 0.02 and assume m = 0.09. We begin the simulation by introducing x11 at a frequency of 0.01. We also modified the above equation to allow P(wP) to use sperm from a P(wB) male at low frequency (0.2) as observed in the laboratory, though this has only a small effect on the dynamics.