Branching with selection and mutation I: Mutant fitness of Fr\'echet type

We investigate two stochastic models of a growing population subject to selection and mutation. In our models each individual carries a fitness which determines its mean offspring number. Many of these offspring inherit their parent's fitness, but some are mutants and obtain a fitness randomly sampled from a distribution in the domain of attraction of the Fr\'echet distribution. We give a rigorous proof for the precise rate of superexponential growth of these stochastic processes and support the argument by a heuristic and numerical study of the mechanism underlying this growth.