The evolution of eusociality through kin selection was analyzed by simple population genetical models. Models were solved analytically with no approximation. The main results are (1) Sex ratio in reproductives in a colony of haplodiploid species does not affect the direction of evolution, contrary to the hypothesis of Trivers and Hare (1976). Female biased sex ratio increases the rate of evolution irrespective of its direction. (2) The only factor that determines the direction of evolution is the balance of benefit and cost of altruism of workers. (3) The value of ratio of benefit to cost of altruism of workers when the change of gene frequency of altruistic allele does not take place is unity in both haplodiploid and diploid species. There is no theoretical reason that the eusociality through kin selection evolves more easily in haplodiploidy than in diploidy, contrary to the hypotheses of Hamilton (1964) and Trivers and Hare (1976). (4) The larger the colony size is, the lower the rate of evolution is irrespective of its direction. It was concluded that discussion on the evolution of altruism which depended on only the values of the degrees of relatedness is misleading. The importance of life history structure, oviposition of workers and number of relating gene(s) in the evolution of eusociality were discussed.
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