Population models of genomic imprinting. II. Maternal and fertility selection

Under several hypotheses for the evolutionary origin of imprinting, genes with maternal and reproductive effects are more likely to be imprinted. We thus investigate the effect of genomic imprinting in single-locus diallelic models of maternal and fertility selection. First, the model proposed by Gavrilets for maternal selection is expanded to include the effects of genomic imprinting. This augmented model exhibits novel behavior for a single-locus model: long-period cycling between a pair of Hopf bifurcations, as well as two-cycling between conjoined pitchfork bifurcations. We also examine several special cases: complete inactivation of one allele and when the maternal and viability selection parameters are independent. Second, we extend the standard model of fertility selection to include the effects of imprinting. Imprinting destroys the "sex-symmetry" property of the standard model, dramatically increasing the number of degrees of freedom of the selection parameter set. Cycling in all these models is rare in parameter space.