CYTONUCLEAR THEORY FOR HAPLODIPLOID SPECIES AND X-LINKED GENES. II. STEPPING-STONE MODELS OF GENE FLOW AND APPLICATION TO A FIRE ANT HYBRID ZONE.

We develop cytonuclear, hybrid zone models for haplodiploid species or X-linked genes in diploid species using a stepping-stone framework of migration, in which migration rates vary with both direction and sex. The equilibrium clines for the allele frequencies, cytonuclear disequilibria, and frequencies of pure parental types are examined for species with diagnostic markers, under four important migration schemes: uniform migration of both sexes in both directions, greater migration of both sexes from one direction, greater migration of females, and greater migration of males. Of the three cytonuclear variables examined, the allele frequency clines are the most informative in differentiating among the various migration patterns. The cytonuclear disequilibria and the frequency of the pure parental types tend to be useful only in revealing directional asymmetries in migration. The extent of hybrid zone subdivision has quantitative but not qualitative effects on the distribution of cytonuclear variables, in that the allele frequency clines become more gradual, the cytonuclear disequilibria decrease in magnitude, and the frequencies of pure parentals decline with increasing subpopulation number. Also, the only major difference between the X-linked and haplodiploid frameworks is that a higher frequency of pure parentals is found when considering haplodiploids, in which male production does not require mating. The final important theoretical result is that censusing after migration yields greater disequilibria and parental frequencies than censusing after mating. We analyzed cytonuclear data from two transects from a naturally occurring hybrid zone between two haplodiploid fire ant species, Solenopsis invicta and S. richteri, using our stepping-stone framework. The frequency of S. invicta mtDNA exceeds the frequency of the S. invicta nuclear markers through much of this hybrid zone, indicating that sex differences in migration or selection may be occurring. Maximum-likelihood estimates for the migration rates are very high, due to an unexpectedly large number of pure parental types in the hybrid zone, and differ substantially between the two transects. Overall, our model does not provide a good fit, in part because the S. invicta-S. richteri hybrid zone has not yet reached equilibrium.