arXiv2026
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 $S^\infty$ and yields the same rate of convergence of $S^{(n)}$ to $S^\infty$ 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 $S^\infty$, 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 $S^\infty$, as well as less well-known results on sharp bounds for $S^\infty$. 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 $S^\infty$ and $S^{(n)}$. Finally, as an application we determine the response of a quantitative trait caused by new beneficial mutations to prolonged directional selection.