I update  my manuscripts in arXiv and submit to journal.  the manuscript was doing numerical simulation of the evolutionary fate of mutant gene at duplicate loci. Diffusion method was used in the mutation diffusion in the natural population, and Ito’s stochastic difference equation was employed to approximating  the  4-dimension Kolmognov backwark  equation. For more detail, please see my manuscripts bellow:

Numerical Studies of the Evolutionary Rate of Mutant Allele at Duplicate Loci

Yupeng Cun

Gene duplications are one of major primary driving forces for evolutionary novelty. We took population genetics models of genes duplicate to study how evolutionary forces acting during the fixation of mutant allele at duplicate loci. We study the fixation time of mutant allele at duplicate loci under double null recessive model (DNR) and haploinsufficient model (HI). And we also investigate how selection coefficients with other evolutionary force influence the fixation frequency of mutant allele at duplicate loci. Our results suggest that the selection plays a role in the evolutionary fate of duplicate genes, and tight linkage would help the mutant allele preserved at duplicate loci. Our theoretical simulation agree with the genomics data analysis result well, that selection, rather than drift, plays a important role in the establishment of duplicate loci, and recombination have a great opportunity to be acted upon selection.