The mean time for absorption in a process of genetic type

Abstract

We consider the genetic model introduced by Moran [3] of a haploid population of fixed sizeMwith two genotypesAandafor which the possibility of selection is allowed. In this model an individual is randomly chosen to die and is replaced by a new individual whose probability of beingadepends on the selective advantages of the two genotypes and on the number ofaindividuals before the birth-death event. The probability of eventual elimination of the genotypea, both with and without selection, has been found by Moran [3], while Watterson [4] has found the mean time for absorption and the variance in the case where no selection is allowed. We derive here the mean time and the variance in the case where selection is allowed, thus extending Watterson's result. A diffusion approximation is available for the mean time; it is shown that this gives a very close approximation to the exact value. Comparison is made with the non-overlapping generation model due to Wright [5], and finally some numerical results are exhibited.