CONVERGENCE OF VARIABILITY IN GAUSSIAN PO LYGENIC MODELS

In the Gaussian polygenic models developed by Lande, the genetic contr ibution to the variability of a phenotypic character is given by a sum of allelic effects at several loci subject to mutations. The environm ental contribution to that variability is the result of a random devia tion, independent of the allelic effects, with a zero mean and a const ant variance from generation to generation. The dynamics of the geneti c variability is described by recurrence equations, from one generatio n to the next, for the covariance matrix of the multinormal vector cor responding to the allelic effects. These equations admit only one equi librium and global convergence toward that equilibrium is checked by n umerical iterations. In this article we present a Gaussian polygenic m odel in which the selection forces (viability, preferential mating, fe rtility) act generally on the vector of allelic effects, these being s ubject to mutation, segregation and recombination. In order to study t he convergence of variability, we establish a criterion for the conver gence of the iterates of transformations on. semi-positive definite ma trices tinder the condition that there exists a unique fixed point. Th is criterion is essentially a concavity property which, combined with a monotonicity property, has previously been wed by Karlin to demonstr ate the convergence of variability in, Gaussian, phenotypic models wit hout recombination or segregation. We show that the monotonicity prope rty does not have to be assumed and we give a direct proof of the conv ergence result under slightly weakened hypotheses. Finally, we show th at this result applies to Gaussian polygenic models without difference s between the sexes or without linkage between the loci. The robustnes s of the results without these hypotheses and the rate of convergence are studied by numerical iterations.