In eusocial colonies the sexually reproducing pair is associated with a large caste of non-reproducing offspring workers. The evolution of such colonies have traditionally been explained by Hamiltonian models of inclusive fitness (Hamilton, 1963, 1964), relabelled kin selection by Maynard Smith (1964) . These models are essentially group selection models (Hamilton, 1975; Sober and Wilson, 1998) that aim at explaining why offspring choose to raise sibs instead of raising their own offspring. Assuming that the haplodiploid genome is more ancestral to hymenoptera than eusociality, Hamilton (1964) noted that hymenoptera can be predisposed to eusociality because of the high relatedness among sisters of haplodiploids. Trivers and Hare (1976) showed that sex allocation theory is central for this mechanism to work; haplodiploids are predisposed to eusociality only if the sex ratio of the sexual caste of the ancestor is female biased. Female biased sex ratios and, thus, the potential evolution of eusociality might arise through split sex ratios (Grafen, 1986), where broods produced at different times (Seger, 1983) or conditions (Frank and Crespi, 1989) have different sex ratios.

But as co-operative and eusocial reproduction are widespread also in diploid species the high relatedness among sisters in haplodiploids may not be the factor that induces the evolution of eusocial colonies. And in Malthusian Relativity the eusocial colony evolves independently of the haplodiploid genome; in fact the theory even suggests that the haplodiploid genome may evolve as a consequence of a particular mating strategy in hymenoptera-like eusocial colonies. Here the eusocial colony evolves because the interacting, reproducing, and sexual units can enhance the interactive quality of the interacting unit (Witting, 2002).

Many of the eusocial colonies in insects can be seen as being defined by a large interacting unit that contains a single reproducing unit where a sexually reproducing pair receives help from a large caste of non-reproducing offspring workers. A large interacting unit with only one large reproducing unit will evolve by density dependent competitive interactions if the body mass is upward constrained relative to the energetic level of the organism. Then, both the population density and the level of interactive competition will be high inducing selection for a high interactive quality, and this is obtained by a large interacting unit with many non-replicating individuals that allocate time and energy to competitive interactions instead of replication. But such units will not necessarily evolve into eusocial colonies. If e.g. the offspring workers can be produced only asexually, then, the sexual units will evolve to unknown levels where the sexually reproducing female mate with a large number of male and all sexual individuals transfer only a tiny fraction of their hereditary material to the offspring. In these colonies there will be no offspring workers but only a single sexual female and a large caste of sexually reproducing males.

But if offspring workers can be produced also by sexual reproduction, then, we can expect that density dependent competitive interactions select for eusocial colonies. In this case sexually produced offspring workers can evolve because the interactive quality of the male is transferred to the sexually produced offspring workers. This generates a diminishing return where the interactive quality that can be gained by exchanging an offspring worker with a sexual male is a declining function of the number of males per female. It is only for the initial transition from asexual to pair-wise sexual reproduction that the interactive quality of the male can outbalance the cost of sexual reproduction. For all potential remaining transitions, the transfer of male interactive quality to sexually produced offspring workers implies that the interactive quality that can be gained by exchanging an offspring worker with an additional sexual male cannot outbalance the extra cost to sexual reproduction associated with that transition.

The conclusion is that sexually produced offspring workers and kin selection evolve as a result of large interacting units, instead of being the traits that induce the evolution of such units. Here kin selection and the sexually produced offspring workers evolve as a balance between the cost of sexual reproduction and the overall interactive quality of the interacting unit. In result, the expected number of sexually produced offspring workers per reproducing unit becomes a function of the level of interactive competition. The optimum is zero workers for low-energy organisms and high-energy organisms with body masses in evolutionary equilibrium, while it is one and up to an infinite number of workers for high-energy organisms with body masses at respectively the evolutionary steady state and the upward constrained equilibrium.

The fully evolved eusocial colonies are known mainly from social insects where they occur in one form in ants and bees, and in another form in termites. Ants and bees have a haplodiploid genome, their workers are the daughters of the queen, and there are typically three queens per sexual male. In termites however the genome is diploid, the workers are both the daughters and sons of the queen, and there are typically one king per queen. In Malthusian Relativity these differences can evolve from the distinction that ant/bee colonies are formed only the queen while the termite colony is formed by the sexually reproducing pair.

References

• Frank, S. A. & Crespi, B. J. (1989). Synergism between sib-rearing and sex ratio in Hymenoptera. Behavioural Ecology and Sociobiology 24, 155--162.
• Grafen, A. (1986). Split sex ratios and the evolutionary origins of eusociality. Journal of Theoretical Biology 122, 95--121.
• Hamilton, W. D. (1963). The evolution of altruistic behavior. The American Naturalist 97, 354--356.
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• Maynard Smith, J. (1964). Group selection and kin selection. Nature 201, 1145--1146.
• Seger, J. (1983). Partial bivoltinism may cause alternating sex-ratio biases that favour eusociality. Nature 301, 59--62.
• Sober, E. & Wilson, D. S. (1998). Unto others: The evolution and psychology of unselfish behavior. Cambridge: Harvard University Press.
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• Witting, L. (2002). From asexual to eusocial reproduction by multilevel selection by density dependent competitive interactions. Theoretical Population Biology 61, 171--195.