On the adapted equations in various dyploid model and Hardy-Weinburg equilibrium in a triploid model

Abstract

For a locus with two alleles ($I^A$ and $I^B$), the frequencies of the alleles are represented by $$ p=f(I^A)= \frac {2N_{AA} +N_{AB}} {2N} , \ q=f(I^B)= \frac {2N_{BB} +N_{AB}}{2N} $$ where $N_{AA},~N_{AB}$ and $N_{BB}$ are the numbers of $I^A I^A ,~I^A I^B$ and $I^B I^B$ respectively and $N$ is the total number of populations. The frequencies of the genotypes expected are calculated by using $p^2 ,~2pq$ and $q^2$. Choi defined the density and operator for the value of the frequency of one gene and found the adapted partial differential equation as a follow-up for the frequency of alleles and applied this adapted partial differential equation to several diploid model \cite {Choi1}. In this paper, we find adapted equations for the model for selection against recessive homozygotes and in case that the alley frequency changes after one generation of selection when there is no dominance. Also we consider the triploid model with three alleles $I^A,I^B$ and $i$ and determine whether six genotypes observed are in Hardy-Weinburg for equilibrium.