GENETIC STEADY-STATE UNDER BLUP SELECTION FOR AN INFINITE AND HOMOGENEOUS POPULATION WITH DISCRETE GENERATIONS

A matrix derivation is proposed to analytically calculate the asymptot ic genetic variance-covariance matrix under BLUP selection according t o the initial genetic parameters in a large population with discrete g enerations. The asymptotic genetic evolution of a homogeneous populati on with discrete generations is calculated for a selection operating o n an index including all information (pedigree and records) from a non -inbred and unselected base population (BLUP selection) or on an index restricted to records of a few ancestral generations. Under the first hypothesis, the prediction error variance of the selection index is i ndependent of selection and is calculated from the genetic parameters of the base population. Under the second hypothesis, the prediction er ror variance depends on selection. Furthermore, records of several gen erations of ancestors of the candidates for selection must be used to maintain a constant prediction error variance over time. The number of ancestral generations needed depends on the population structure and on the occurrence of fixed effects. Without fixed effects to estimate, accounting for two generations of ancestors is sufficient to estimate the asymptotic prediction error variance. The amassing of information from an unselected base population proves to be important in order no t to overestimate the asymptotic genetic gains and not to underestimat e the asymptotic genetic variances.