Can reinforcement occur with a learned trait?

We use birdsong as a case study to ask whether reinforcement can occur via the spread of a genetically determined female preference for a socially inherited (learned) male trait. We envision secondary contact between two neighboring populations with different song dialects. An individual's ability to learn song is confined by a genetic predisposition: if predispositions are strong, there will be no phenotypic overlap in song between populations, whereas weak predispositions allow phenotypic overlap, or "mixed" song. To determine if reinforcement has occurred, we consider if an allele for within-population female mating preference, based on song, can spread, and whether population specific songs can concurrently be maintained at equilibrium. We model several scenarios, including costs to mating preferences, mating preferences in hybrids, and hybrids having the ability to learn pure songs. We find that when weak predispositions are fixed within a population reinforcement based on song cannot occur. However, when some individuals have strong predispositions, restricting phenotypic overlap between populations in the trait, reinforcement is only slightly inhibited from a purely genetic model. Generalizing beyond the example of song, we conclude that socially learned signals will tend to prohibit reinforcement, but it may still occur if some individuals acquire trait phenotypes genetically.