A quasispecies on a moving oasis

A population evolving in an inhomogeneous environment will adapt differently to different areas. We study the conditions under which such a population can maintain adaptations to a particular region when that region is not stationary, but can move. In particular, we consider a haploid population living near a moving favorable patch (“oasis”) in the middle of a large “desert.” At one genetic locus, individuals may have one of a few gene sequences that convey an advantage while in the oasis at the cost of a disadvantage in the desert. The distribution of genetic states in the population, possibly localized in genome space around the oasis-adapted genotypes, is known as a quasispecies. We find that the ratio of oasis-adapted individuals to desert-adapted ones exhibits sharp transitions at particular oasis velocities. We calculate an extinction velocity, and a switching velocity above which the dominance switches from the oasis-adapted genotype to the desert-adapted one. This switching velocity is analogous to the quasispecies mutational error threshold. Above this velocity, the population cannot maintain adaptations to the properties of the oasis.