Perturbation analysis of a two-locus model with directional selection and recombination

A population genetic two-locus model with additive, directional selection and recombination is considered. It is assumed that recombination is weaker than selection; i.e., the recombination parameter r is smaller than the selection coefficients. This assumption is appropriate for describing the effects of two-locus selection at the molecular level. The model is formulated in terms of ordinary differential equations (ODEs) for the gamete frequencies x = (x1, x2, x3, x4), defined on the simplex S4. The ODEs are analyzed using first a regular perturbation technique. However, this approach yields satisfactory results only if r is very small relative to the selection coefficients and if the initial values x(0) are in the interior part of S4. To cope with this problem, a novel two-scale perturbation method is proposed which rests on the theory of averaging of vectorfields. It is demonstrated that the zeroth-order solution of this two-scale approach approximates the numerical solution of the model well, even if recombination rate is on the order of the selection coefficients.