I’m going to wave a magic wand and reduce the mutation rate to zero, instantly, in all species, and forever. Then I’m going to watch to see how long it takes for evolution to stop.
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Actually stopping mutations is a physical impossibility - hence the need for a magic wand. But if they were to stop, so would raw invention. But evolution would not. Not for a long time.
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And sometimes natural selection actively promotes the persistence of genetic variation. This can happen when there’s an advantage to having genes that are rare. Among guppies, for example, males with rare color patterns are much more likely to survive than those with common color patterns, presumably because predators get good at spotting the patterns they encounter often. In such situations, the rare type does well, begins to become common - and then becomes the victim of its own success and starts to do badly. In situations like this, the frequencies of different genes can rise and fall, cycling indefinitely.
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Among lifestyles that promote genetic diversity, far and away the most important is sex. Sex shuffles up genes, continually producing new gene combinations. (An important difference between sex and mutation is that sex can only create genetic novelty if it already exists in the population. If everyone is genetically identical, sex will have no effect.) Sex also - and this is important - decouples the fates of genes from one another.