Clines with complete dominance and partial panmixia in an unbounded unidimensional habitat

In spatially structured populations, global panmixia can be viewed as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines maintained by migration and selection with complete dominance in an unbounded unidimensional habitat is investigated. The population density is uniform. Migration and selection are both weak; the former is homogeneous and symmetric; the latter is frequency independent. The spatial factor g̃(x) in the selection term, where x denotes position, is a single step at the origin: g̃(x)=-α<0 if x<0, and g̃(x)=1 if x>0. If α=1, there exists a globally asymptotically stable cline. For α<1, such a cline exists if and only if the scaled panmictic rate β is less than the critical value β(∗∗)=2α/(1-α). For α>1, a unique, asymptotically stable cline exists if and only if β is less than the critical value β(∗); then a smaller, unique, unstable equilibrium also exists whenever β<β(∗). Two coupled, nonlinear polynomial equations uniquely determine β(∗). Explicit solutions are derived for each of the above equilibria. If β>0 and a cline exists, some polymorphism is maintained even at x=±∞. Both the preceding result and the existence of an unstable equilibrium when α>1 and 0<β<β(∗) differ qualitatively from the classical case (β=0).