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Mendelian segregation: A choice between “order” and “chaos”. (English) Zbl 0733.92011

The following one locus multiallele model is considered: Let x and y be the frequency-vectors of the gametes in males and females and let M and F denote the (nonnegative) segregation matrices in males and females, respectively. Then the population state \((x',y')\) of the next generation is given by \[ x'=x\circ My+y\circ Mx,\quad y'=x\circ Fy+yFx, \] where \(\circ\) denotes the Schur-product. For the elements \(m_{ij}\) of M and \(f_{ij}\) of F it is assumed that \[ m_{ii}=f_{ii}=1/2,\quad m_{ij}+m_{ji}=f_{ij}+f_{ji}=1. \] First, the case that F and M are chosen at random for each generation is studied. Next, the matrices M and f are considered constant over the entire evolution of the population. In particular, it is assumed that if M and F are presented in the form \[ M=2^{-1}E+A,\quad F=2^{-1}E+B, \] the disparity matrices A and B are proportional.
The main result of both models is: Any departure from Mendelian segration pushes the evolution of the population to either a fixation of one allele or into a rather chaotic behaviour.
Reviewer: D.Dorninger (Wien)

MSC:

92D15 Problems related to evolution
92D10 Genetics and epigenetics
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