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Pressure of parasites that are short-lived and rapid-evolving compared to the hosts they attack could be an evolutionary factor sufficiently general to account for sex wherever it exists. To be such a factor, parasites must show virulences specific to differing genotypes. Models are set up on this basis (one-locus diploid-selection and two-locus haploid-selection) in which the rapid demographic reactivity of parasite strains to abundance of susceptible hosts becomes represented in a single frequency-dependent fitness function which applies to every host genotype. It is shown that with frequency dependence sufficiently intense such models generate cycles, and that in certain states of cycling sexual species easily obtain higher long-term geometric mean fitness than any competing monotypic asexual species or mixture of such. In the successful cycle of the two-locus model, it is seen that both population size and gene frequencies may be steady while only oscillating linkage disequilibrium reflects the intense selection by parasites. High levels of recombination work best. Fecundity in the models can be low and no incidence of competition of siblings or other relatives is required.