Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4502343 | Theoretical Population Biology | 2014 | 12 Pages |
Abstract
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 gÌ(x) in the selection term, where x denotes position, is a single step at the origin: gÌ(x)=âα<0 if x<0, and gÌ(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 βââ=2α/(1âα). For α>1, a unique, asymptotically stable cline exists if and only if β is less than the critical value βâ; then a smaller, unique, unstable equilibrium also exists whenever β<βâ. Two coupled, nonlinear polynomial equations uniquely determine βâ. Explicit solutions are derived for each of the above equilibria. If β>0 and a cline exists, some polymorphism is maintained even at x=±â. Both the preceding result and the existence of an unstable equilibrium when α>1 and 0<β<βâ differ qualitatively from the classical case (β=0).
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Authors
Thomas Nagylaki, Kai Zeng,