On a Genetics Model of Moran

Abstract

Wright (14), Feller (2) and others have proposed various stochastic models of genetics to study the fluctuations of gene frequency under the influence of mutation and selection. One of their simplest models has the following structure. There are a fixed number N of gametes each of which may be of two types a or A. The process X(n), n = 0, 1, 2, … which is assumed to have stationary transitions, is said to be in state j when there are j gametes of type a, and N − j of type A. Let γ1 denote the probability that immediately after formation an a gamete mutates into an A gamete and let γ2 denote the probability of an A gamete mutating into an a gamete. Each of the gametes of the next generation is independently formed by making a random selection from the gametes of the present generation. The probability of a particular gamete in the next generation being of type a is then and of type A where j represents the state of the process. Roughly speaking, the chance of a mating resulting in a gamete of a prescribed kind for the next generation is proportional to the fraction of these gametes present in this generation allowing for mutation effects.