Effective sizes of livestock populations to prevent a decline in fitness

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Abstract
In livestock populations, fitness may decrease due to inbreeding depression or as a negatively correlated response to artificial selection. On the other hand, fitness may increase due to natural selection. In the absence of a correlated response due to artificial selection, the critical population size at which the increase due to natural selection and the decrease due to inbreeding depression balance each other is approximately D/2σwa 2, where D=the inbreeding depression of fitness with complete inbreeding, and σwa 2=the additive genetic variance of fitness. This simple expression agrees well with results from transmission probability matrix methods. If fitness declines as a correlated negative response to artificial selection, then a large increase in the critical effective population size is needed. However, if the negative response is larger than the response to natural selection, a reduction in fitness cannot be prevented. From these results it is concluded that a negative correlation between artificial and natural selection should be avoided. Effective sizes to prevent a decline in fitness are usually larger than those which maximize genetic gain of overall efficiency, i.e., the former is a more stringent restriction on effective size. In the examples presented, effective sizes ranged from 31 to 250 animals per generation.