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 – w_l2) < ½w₁₂(v₁ + v₁) > 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.