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 g ̃ ( x ) in the selection term, where x denotes position, is a single step at the origin: g ̃ ( x ) = − α < 0 if x < 0 , and g ̃ ( 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 ∗ ∗ = 2 α / ( 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 ) .