  We study existence and stability of stationary solutions of a system of
semilinear parabolic partial differential equations that occurs in population
genetics. It describes the evolution of gamete frequencies in a geographically
structured population of migrating individuals in a bounded habitat. Fitness of
individuals is determined additively by two recombining, diallelic genetic loci
that are subject to spatially varying selection. Migration is modeled by
diffusion. Of most interest are spatially non-constant stationary solutions,
so-called clines. In a two-locus cline all four gametes are present in the
population, i.e., it is an internal stationary solution. We provide conditions
for existence and linear stability of a two-locus cline if recombination is
either sufficiently weak or sufficiently strong relative to selection and
diffusion. For strong recombination, we also prove uniqueness and global
asymptotic stability. For arbitrary recombination, we determine the stability
properties of the monomorphic equilibria, which represent fixation of a single
gamete.
