Mean curvature versus normality: a comparison of two approximations of Fisher's geometrical model

Fisher's geometrical model amounts to a description of mutation and selection for individuals characterised by a number of quantitative traits. In the present work the fitness landscape is not assumed to be spherically symmetric, hence different points, i.e. phenotypes, on a surface of constant fitness generally have different curvatures. We investigate two different approximations of Fisher's geometrical model that have appeared in the literature. One approximation uses the average curvature of the fitness surface at the parental phenotype. The other approach is based on a normal approximation of a distribution associated with new mutations. Analytical results and simulations are used to compare the accuracy of the two approximations.