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By encompassing the whole continuum between allopatric and sympatric scenarios, parapatric speciation includes many potential scenarios for the evolution of new species. Here, we investigate how a genetic barrier to gene flow, that relies on a single postzygotic genetic incompatibility, may further evolve under ongoing migration. We consider a continent island model with three loci involved in pairwise Dobzhansky-Muller incompatibilities (DMIs). Using an analytic approach, we derive the conditions for invasion of a new mutation and its consequences for the strength and stability of the initial genetic barrier. Our results show that the accumulation of genetic incompatibilities in the presence of gene flow is under strong selective constraints. In particular, preexisting incompatibilities do not always facilitate the invasion of further barrier genes. If new mutations do invade, they will often weaken or destroy the barrier rather than strengthening it. We conclude that migration is highly effective at disrupting the so-called "snowball effect", the accelerated accumulation of DMIs that has been described for allopatric populations en route to reproductive isolation.

The Limits to Parapatric Speciation II: Strengthening a Preexisting Genetic Barrier to Gene Flow in Parapatry