Is cascade reinforcement likely when sympatric and allopatric populations exchange migrants?

When partially reproductively isolated species come back into secondary contact, these taxa may diverge in mating preferences and sexual cues to avoid maladaptive hybridization, a process known as reinforcement. This phenomenon often leads to reproductive character displacement (RCD) between sympatric and allopatric populations of reinforcing species that differ in their exposure to hybridization. Recent discussions have reinvigorated the idea that RCD may give rise to additional speciation between conspecific sympatric and allopatric populations, dubbing the concept "cascade reinforcement." Despite some empirical studies supporting cascade reinforcement, we still know very little about the conditions for its evolution. In the present article, we address this question by developing an individual-based population genetic model that explicitly simulates cascade reinforcement when one of the hybridizing species is split into sympatric and allopatric populations. Our results show that when sympatric and allopatric populations reside in the same environment and only differ in their exposure to maladaptive hybridization, migration between them generally inhibits the evolution of cascade by spreading the reinforcement alleles from sympatry into allopatry and erasing RCD. Under these conditions, cascade reinforcement only evolved when migration rate between sympatric and allopatric populations was very low. This indicates that stabilizing sexual selection in allopatry is generally ineffective in preventing the spread of reinforcement alleles. Only when sympatric and allopatric populations experienced divergent ecological selection did cascade reinforcement evolve in the presence of substantial migration. These predictions clarify the conditions for cascade reinforcement and facilitate our understanding of existing cases in nature.