Population genetic processes, such as the adaptation of a quantitative trait to directional selection, may occur on longer time scales than the sweep of a single advantageous mutation. To study such processes in finite populations, approximations for the time course of the distribution of a beneficial mutation were derived previously by branching process methods. The application to the evolution of a quantitative trait requires bounds for the probability of survival S ( n ) up to generation n of a single beneficial mutation. Here, we present a method to obtain a simple, analytically explicit, either upper or lower, bound for S ( n ) in a supercritical Galton-Watson process. We prove the existence of an upper bound for offspring distributions including Poisson, binomial, and negative binomial. They are constructed by bounding the given generating function, φ , by a fractional linear one that has the same survival probability S ∞ and yields the same rate of convergence of S ( n ) to S ∞ as φ . For distributions with at most three offspring, we characterize when this method yields an upper bound, a lower bound, or only an approximation. Because for many distributions it is difficult to get a handle on S ∞ , we derive an approximation by series expansion in s, where s is the selective advantage of the mutant. We briefly review well-known asymptotic results that generalize Haldane's approximation 2s for S ∞ , as well as less well-known results on sharp bounds for S ∞ . We apply them to explore when bounds for S ( n ) exist for a family of generalized Poisson distributions. Numerical results demonstrate the accuracy of our and of previously derived bounds for S ∞ and S ( n ) . Finally, we treat an application of these results to determine the response of a quantitative trait to prolonged directional selection.
Reinhard Bürger (2026) studied this question.