We studied invasion and survival of weakly beneficial mutations that arise in linkage to an established migration-selection polymorphism. Our focus was on a continent--island model of migration, with selection at two biallelic loci for adaptation to the island environment. The new mutation arises on the focal locus, whereas the established migration-selection polymorphism exists at the background locus. Employing a two-type branching process, we derived numerical approximations to the invasion probability on average, as well as conditional on initial occurrence of the mutation on the beneficial or deleterious background. Assuming that the branching-process is slightly supercritical, we derived analytical approximations that are accurate for weak evolutionary forces. Interestingly, the invasion probability is maximised at a non-zero recombination rate if the selection coefficient of the focal mutation is above a critical value, for which we found an explicit expression. Moreover, if a proportion of migrants carries a beneficial background allele, the mutation is less likely to invade, because it has to compete against a fitter resident population. We also derived approximations to the sojourn-time density and the mean extinction time using a diffusion approximation. For this, we had to assume quasi-linkage equilibrium (QLE), i.e. that recombination is strong relative to migration and selection. We found that linked selection may increase the survival time by several orders of magnitude, even when QLE applies. By altering the time scale of stochastic loss, linked selection can therefore affect the dynamics at the focal locus to an extent that is of evolutionary importance, especially for populations of small to moderate size. The background locus essentially acts as a barrier to gene flow, and this effect spills over to partially linked loci. Our analytical approximation to the sojourn-time density directly leads to a compact expression for the effective migration rate experienced at a partially linked site. This formula applies to both weakly beneficial as well as neutral focal mutations. Using the concept of the effective migration rate, we also quantified the long-term effects on neutral variation embedded in a genome with arbitrarily many sites at migration--selection balance. Patterns of neutral diversity change qualitatively and quantitatively as the position of the neutral locus is moved along the chromosome. Our results are relevant for inference about local adaptation from population-genomic data, which are rapidly becoming available. Moreover, our findings have implications for our understanding of so-called islands of differentiation and the evolution of the genetic architecture of polygenic local adaptation.

population genetics; polygenic local adaptation; stochastic processes; selection; migration; linkage; branching process; diffusion approximation; invasion probability; extinction time