The Information Dynamics of the Diffusion Processes of Population Genetics

Abstract

With no prior knowledge assumed, the Wright-Fisher-Kimura diffusion process on gene frequency space is introduced and discussed from the geometric symmetries point of view. This derives from consideration of the covariance matrix of this diffusion as dynamical information analogous to Mahalonobis‘ static covariance matrix for informational discrimination of data in statistics. This model allows one to obtain Antonelli‘s ray solution in closed form for all dimensions of the gene frequency space, and to study the more difficult Felsenstein natural selection diffusion in higher dimensions. Of particular interest are the effects of positive and negative curvature in various regions of the frequency space. For the first time, several computer generated 3-dimensional geometries are presented in the collection of figures for the Felsenstein process of 3 alleles. Regions of negative curvature exhibit weak chaos in natural selection, while those of positive curvature exhibit meandering paths. An open problem is stated at the end.