"The divergence of songs in the new population away from those in the progenitor population would only be prevented if these processes were balanced by repeated immigration and subsequent breeding: song flow"

This quote by Peter R. Grant is referring to the process of speciation, or the formation of new species. In this case, the quote is referring to the divergence of tunes in a brand-new population away from those in the progenitor population. Grant is recommending that in order for the divergence of tunes to be avoided, there need to be a balance of migration and breeding. Migration is the procedure of individuals from the progenitor population going into the brand-new population, and breeding is the process of those individuals mating with people already in the new population. This balance of migration and reproducing would make sure that the tunes of the new population remain comparable to those of the progenitor population, hence avoiding the divergence of songs.