The effect of multiple inseminations on the evolution of social behaviors in diploid and haplo-diploid organisms

Using a classical population genetic model, the necessary conditions for the spread of genes that determine social behaviors and the rate of spread of these genes are derived. The influence of 1, 2, 3, or k inseminations per female on these conditions is investigated for both diploid and haplodiploid organisms. These results are then extended to a population in which there are arbitrary variations among females in their numbers of mates. These results do not depend upon assuming equal paternity by all inseminating males; the effects of sperm competition and unequal paternity are also derived. The rates and conditions for social evolution in these groups of complex composition are discussed in relation to Hamilton's rule.For all models, the total change in gene frequency, Δq, is partitioned into two components: (1) $\text{Δq}{}_{\mathrm{I}}$, the change in gene frequency caused by selection within groups; this component is always negative, illustrating that individual selection always operates against the evolution of social behaviors; and (2) Δq_G, the change in gene frequency caused by selection between groups; this component is generally positive. Hamilton's rule is shown to specify the necessary and sufficient conditions for $\text{Δq}{}_{\mathrm{G}}\text{> |}\text{Δq}{}_{\mathrm{I}}\text{|}$, that is, for selection among kin groups to over-ride individual selection within kin groups.