A migration-selection model in genetic engineering

We investigate a migration-selection system arising from CRISPR-Cas9 genetic engineering, which describes the evolution of the frequencies of a wild allele O, a drive allele D, and a brake allele B. The purpose is to see whether the drive allele D can persist in the population and whether its spread can be limited or stopped by the brake allele B when necessary. We give a complete classification of the dynamics of this system when there is no migration. We further show that migration may cause complex spatiotemporal patterns by demonstrating the existence of spatially inhomogeneous periodic solutions and steady state solutions.