The opportunity for polymorphism and genic variation in social hymenoptera with worker-produced males

Abstract

In many eusocial Hymenoptera, for example, bumble bees, a proportion, ψ of the males are produced by the workers. The conditions for there to be at least one stable polymorphic equilibrium were found to be, w11v1 – ½ψv2(w11 – wl2) w22v2 – ½ψv1(w22 – w12); where w11, w12 w22 are respectively the fitnesses of the female genotypes A1A1, A1A2, A2A2 and v1, v2 are the fitnesses of the male genotypes A1 and A2. The equilibrium gene frequencies are given by the root(s) of a cubic polynomial. The conditions given above were shown to be both sufficient and necessary to find all stable equilibria for three regular selection models analysed in detail: (1) Codominance and (2) Dominance with opposite selection in the sexes (3) Overdominance with identical selection in the sexes. The effect of worker-produced males on the probability of a polymorphism and the expected genic variation was investigated for these three models. Compared to other Hymenoptera, species with worker-produced males will have a reduced frequency of polymorphisms and lower mean genic diversity at loci with dominant and overdominant fitnesses. In contrast at loci with codominant fitnesses the probability of a polymorphism and the genic diversity will be higher if the selective differentials are large (0–1·0), with smaller selective differentials (0–0·125, 0–0·01) a reduction will again occur. Therefore the overall level of genic diversity depends on the proportion of codominant loci and the selection intensity.