Much of the theory of population genetics is based on the assumption that populations are panmictic. While this may be nearly true in some instances, most populations of interest (particularly in man) exhibit some form of subdivision. Most often, spatial subdivision is observable, and population geneticists have studied subdivisions ranging in magnitude from villages to continents. Other types of subdivisions exist as well. Linguistic differences often define barriers to random mating, as do differences in social class, clan or caste membership, and religious affiliation. In addition, various cultural regulations, such as taboos or prescribed mating systems, result in subdivision.

The Genetic Structure of Subdivided Human Populations

Stochastic calculus can be used to provide a satisfactory theory of random processes on differentiable manifolds and, in particular, a description of Brownian motion on a Riemannian manifold which lends itself to constructions generalizing the classical development of smooth paths on a manifold. An introduction to this theory is given, and a survey is made of the relationship between curvature properties of the manifold and the asymptotic behaviour of the Brownian motion on the manifold. It is then explained how these results can be used to prove geometrical theorems concerning special classes of maps between manifolds.

Stochastic Differential Geometry: An Introduction

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The f unction and evolution of egg colour in birds

Nonlinear filtering and Riemannian scalar curvature, R

A new diffusion model for random genetic drift of a two-locus di-allelic system is proposed. The Christoffel velocity field and the intrinsic geometry of the diffusion is computed for the equilibrium surface. It is seen to be radically non-spherical and to depend explicitly on the recombination fraction. The model has not been shown to be a limit of discrete Markov chains. For large values of the recombination, the present model is radically different from that of Ohta and Kimura, which is an approximation to the discrete process of random mating in the limit as the value of the recombination fraction goes to zero. LINKAGE; TWO-LOCUS DI-ALLELIC SYSTEM; RECOMBINATION DIFFUSION; RIEMANNIAN GEOMETRY; SAMPLING PROCESS; LINKAGE DISEQUILIBRIUM; CURVATURE; GENE FREQUENCY SPACE; STRATONOVICH STOCHASTIC CALCULUS; AVERAGE FLUCTUATING FORCE; MAXIMUM LIKELIHOOD CURVES

The geometry of random drift III. Recombination and diffusion

It has been conjectured that a certain transformation of gene frequency space due to Fisher and Bhattacharyya will map the random genetic drift process, or its diffusion approximation, into one with radial symmetry. This paper proves rigorously that the Fisher-Bhattacharyya mapping does not do this. This implies that the initial state of an evolving ensemble can only be unbiasedly estimated from the means of a sample if we weight by the proper divergence times. If the ensemble is known not to have begun at the centroid of frequency space, the estimate of the initial state vector is not simply the arithmetic average, as symmetry analysis of the Christoffel velocity field shows.

The geometry of random drift ii. the symmetry of random genetic drift

The ray solution of Felsenstein's n-allele random selection diffusion process is given for small values of the selection parameters. This solution holds away from, but not near, the boundary of frequency space. The solution is possible only because the coefficients of the associated Jacobi field equations agree uniformly with those for the case of zero selection up to fourth powers in the selection parameters, whilst the covariance of the diffusion has only quadratic dependence. RANDOM SELECTION DIFFUSION; RAY SOLUTION; JACOBI FIELDS; RIEMANNIAN GEOMETRY; CURVATURE TENSOR; WFK PROCESS; CHRISTOFFEL VELOCITY FIELDS; WEAK CONVERGENCE OF FINITE MARKOV CHAINS

Jared Chapin

Papers

The geometry of random drift ii. the symmetry of random genetic drift

The geometry of random genetic drift. VI. A random selection diffusion model.