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He imagined an asexual mutant arising in a sexual population, half of which comprises males that cannot themselves produce offspring.
With female-only offspring, the asexual lineage doubles its representation in the population each generation, all else being equal.
This, however, conflates sex and reproduction which are two separate events.
The "two-fold cost of sex" may more accurately be described as the cost of anisogamy. There are numerous species which are sexual and do not have this problem because they do not produce males or females.
It is widely believed that a disadvantage of sexual reproduction is that a sexually reproducing organism will only be able to pass on 50% of its genes to each offspring.
This is a consequence of the fact that gametes from sexually reproducing species are haploid.
Since the chance of a seed's success in occupying the patch depends upon its genotype, and a parent cannot anticipate which genotype is most successful, each parent will send many seeds, creating competition between siblings.
Natural selection therefore favours parents which can produce a variety of offspring (see lottery principle).
In fact, sex is invariably associated with organisms that produce a few large offspring, whereas organisms producing small offspring frequently engage in parthenogenesis [asexual reproduction].
Sexual reproduction must offer significant fitness advantages to a species because despite the two-fold cost of sex, it dominates among multicellular forms of life, implying that the fitness of offspring produced outweighs the costs.
Sexual reproduction derives from recombination, where parent genotypes are reorganized and shared with the offspring.
This implies that an asexual population has an intrinsic capacity to grow more rapidly with each generation.
The cost was first described in mathematical terms by John Maynard Smith.