Author Summary Changes to an organism's environment may have such adverse effects on fitness that the population begins to decline in size. To survive, the population must adapt before it goes extinct. Such “evolutionary rescue” is characterized by a U-shaped curve: population size declines and then recovers as a beneficial allele increases in number. Here we describe this U-shaped curve mathematically when the rescuing allele starts out rare. We obtain several results. First, we calculate when evolutionary rescue is more likely to come from new mutation than from the standing genetic variation. Second, by describing the entire U-shaped curve mathematically, we derive the time until the average rescued population begins to rebound in size as well as the smallest average population size experienced before rescue. We also find that evolutionary rescue from new mutation takes longer and involves a smaller minimum population size than rescue from the standing genetic variation.
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