Within hybrid zones that are maintained by a balance between selection and dispersal, linkage
disequilibrium is generated by the mixing of divergent populations. This linkage disequilibrium
causes selection on each locus to act on all other loci, thereby steepening clines, and generating a
barrier to gene flow. Diffusion models predict simple relations between the strength of linkage
disequilibrium and the dispersal rate, σ, and between the barrier to gene flow, B, and the reduction
in mean fitness, W¯. The aim of this paper is to test the accuracy of these predictions by
comparison with an exact deterministic model of unlinked loci (r = 0·5). Disruptive selection acts
on the proportion of alleles from the parental populations (p,
q): W = exp[−S(4 pq)β],
such that the least fit genotype has fitness e−S. Where
β [Lt ] 1, fitness is reduced for a wide range of intermediate genotypes; where
β [Gt ] 1, fitness is only reduced for those genotypes close to p = 0·5.
Even with strong epistasis, linkage disequilibria are close to
rij, where p′i,
p′j are the gradients in allele frequency at loci
i, j. The barrier to gene flow, which is reflected in the
steepening of neutral clines, is given by
where r¯, the harmonic mean recombination rate between the neural and selected loci, is here 0·5.
This is a close approximation for weak selection, but underestimates B for strong selection. The
barrier is stronger for small β, because hybrid fitness is then reduced over a wider range of p. The
widths of the selected clines are harder to predict: though simple approximations are accurate for
β = 1, they become inaccurate for extreme β because, then, fitness changes sharply with p.
Estimates of gene number, made from neutral clines on the assumption that selection acts against
heterozygotes, are accurate for weak selection when β = 1; however, for strong selection, gene
number is overestimated. For β > 1, gene number is systematically overestimated and, conversely,
when β < 1, it is underestimated.