Conditions for protection of an allele in linear homogeneous stepping stone models

Abstract

The one-dimensional linear homogeneous stepping stone migration structure is an important model in that it represents a short-range migration extreme for geographically structured populations and also serves as the underlying discretespace model for much of the work on continuous-space clines. We examine conditions for the protection of an allele under a stepping stone migration structure by using a recursive method based on Sturm sequences. Necessary and sufficient conditions for protection of an allele are found for a generalized step-cline selection gradient, which is the selection scheme used in much of the early cline work. Sufficient conditions for the protection of an allele are also found for (i) an advantageous patch at the stepping stone boundary which is followed by an arbitrary selection gradient, and for (ii) an advantageous patch embedded within an otherwise arbitrary selection gradient. We let m be the migration rate between neighboring demes. If within the advantageous patch W Aa W aa = 1 + s , then for (i) s ⩾ m (1 − m) is sufficient to protect allele A, even if the patch consists of only a single deme, while for (ii) s ⩾ 2m (1 − 2m) guarantees that a single deme will protect A. If the advantageous patch consists of k demes, each with W Aa W aa = 1 + s, then s ⩾ ( π 2 4 ) m k 2 is sufficient for protection of A in (i), while s ⩾ π 2m k 2 is sufficient for protection of A in (ii). Sufficient conditions for a protected polymorphism are found, and a bound on the level of migration is determined, below which a protected polymorphism exists, as predicted from Karlin and McGregor's (1972. Theor. Pop. Biol. 3, 186–209, 210–238) small parameter results. Finally, our patch swamping conditions (protection of an allele given a single advantageous patch) are compared to Nagylaki's (1975. Genetics 80, 595–615) conditions for the existence of continuous-space clines under analogous selection schemes and are shown to be identical for the two specific cases examined. We also discuss extensions of some of the above results to circular stepping stone migration structures.