In his study of a kin selection model for the evolution of workers' behavior in an incipiently social insect, Craig (1979) found that the haplodiploid mode of sex determination, combined with a female-biased sex ratio, cannot accelerate an evolutionary trend toward eusociality. This seems to contradict to Hamilton's (1964) theory. It is my intention to prove that Craig's result is due to the dependence of relative reproductive success in each sex on the sex ratio of a population. The oviposition of unfertilized eggs by workers is indispensable, in primitive stages in the evolution of eusociality, for maintaining the relative reproductive success of a female. Assuming that workers control the sex ratios, we can distinguish the following three evolutionary phases in accordance with the ratio N2 N10 (the ratio of the number of the queen's second brood to that of the progeny laid by workers derived from the same nest). During Phase 1 in which N2 N10 < 1 2, the reproductive successes of a male and a female are equal, and Hamilton's rule holds. In Phase 2 in which 1 2 < N2 N10 < 3 2, the queen produces females, and workers oviposit males. As N2 N10 increases, the sex ratio in the whole population becomes female-biased and relative reproductive success of a male increases. In Phase 3 in which N2/N10 > 3 2, the queen lays some male eggs in addition to female eggs. The sociality threshold ( B C)crit the minimum value of the benefit/cost ratio leading to the evolution of altruism, is 2 3on Phase 1. It rises threefold as N2/N10 increases in Phase 2, and, in Phase 3, it is twice as high as that in a diploid species. The anatomical and physiological characteristics of workers must have been so developed that they are efficient in helping activities after the beginning of eusociality. The evolutionary process before the beginning of helping behavior is also discussed. The egg laying by unmated females, which stay their mother's nest, seems to have been an important preadaptation for the evolution of eusociality in Hymenoptera.
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