Comparison of theoretical predictions for *τ*, the expected time for the beneficial mutants to establish, to simulation results. These plots are for the case *K* = 2, with double-mutants having selective advantage *s* = 0.1. The crosses are the simulation results, while the solid curves show the theoretical predictions. The different parameter regimes are indicated on the plots. In (**a**) and (**b**), population size *N* is varied while the other parameters are held constant. In (**a**), *μ*_{0} = 10^{−5}, *μ*_{1} = 10^{−4}, and *δ*_{1} = 2×10^{−4}, so that the single-mutants are effectively neutral. In (**b**), *μ*_{0} = 10^{−5}, *μ*_{1} = 10^{−6}, and *δ*_{1} = 7 × 10^{−3}, so that the single-mutants are strongly deleterious for intermediate population sizes. In (**c**), the selective disadvantage *δ*_{1} of the single-mutants is varied, with *N* = 10^{5}, *μ*_{0} = 10^{−6}, and *μ*_{1} = 4 × 10^{−5}. Note that the tunneling-based theoretical prediction is accurate even for negative values of *δ*_{1}, corresponding to beneficial single-mutants. In (**d**), the rate of mutation from single- to double-mutants, *μ*_{1}, is varied, with *N* = 10^{5}, *μ*_{0} = 10^{−6}, and *δ*_{1} = 3 × 10^{−3}.

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