Reçu le 3 mai 1994; accepté le 4 mai 1995) to generation. The dynamics of the genetic variability is described by recurrence equations, from one generation to the next, for the covariance matrix of the multinormal vector corresponding to the allelic effects. These equations admit only one equilibrium and global convergence toward that equilibrium is checked by numerical iterations. In this article we present a Gaussian polygenic model in which the selection forces (viability, preferential

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... , fertility) act generally on the vector of allelic effects, these being subject to mutation, segregation and recombination. In order to study the convergence of variability, we establish a criterion for the convergence of the iterates of transformations on semi-positive definite matrices under the condition that there exists a unique fixed point. This criterion is essentially a concavity property which, combined with a monotonicity property, has previously been used by Karlin to demonstrate the convergence of variability in Gaussian phenotypic models without recombination or segregation. We show that the monotonicity property does not have to be assumed and we give a direct proof of the convergence result under slightly weakened hypotheses. Finally, we show that this result applies to Gaussian polygenic models without differences between the sexes or without linkage between the loci. The robustness of the results without these hypotheses and the rate of convergence are studied by numerical iterations. polygenic model / Gaussian model / genetic variability / covariance matrix / concavity