Two-locus clines on the real line with a step environment

The shape of allele-frequency clines maintained by migration–selection balance depends not only on the properties of migration and selection, but also on the dominance relations among alleles and on linkage to other loci under selection. We investigate a two-locus model in which two diallelic, recombining loci are subject to selection caused by an abrupt environmental change. The habitat is one-dimensional and unbounded, selection at each locus is modeled by step functions such that in one region one allele at each locus is advantageous and in the other deleterious. We admit an environmentally independent, intermediate degree of dominance at both loci, including complete dominance. First, we derive an explicit expression for the single-locus cline with dominance, thus generalizing classical results by Haldane (1948). We show that the slope of the cline in the center (at the step) or, equivalently, the width of the cline, is independent of the degree of dominance. Second, under the assumption of strong recombination relative to selection and migration, the first-order approximations of the allele-frequency clines at each of the loci and of the linkage disequilibrium are derived. This may be interpreted as the quasi-linkage-equilibrium approximation of the two-locus cline. Explicit asymptotic expressions for the clines are deduced as x→±∞. For equivalent loci, explicit expressions for the whole clines are derived. The influence of dominance and of linkage on the slope of the cline in the center and on a global measure of steepness are investigated. This global measure reflects the influence of dominance. Finally, the accuracy of the approximations and the dependence of the shape of the two-locus cline on the full range of recombination rates is explored by numerical integration of the underlying system of partial differential equations.