# Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics

## TL;DR

This paper provides new bounds for the survival probability of beneficial mutations in population genetics using Galton-Watson processes.

## Contribution

A novel method to derive explicit upper and lower bounds for survival probabilities in supercritical Galton-Watson processes.

## Key findings

- An upper bound for offspring distributions like Poisson, binomial, and negative binomial is proven.
- The method characterizes when it yields bounds or approximations for distributions with at most three offspring.
- Numerical results confirm the accuracy of derived bounds for survival probabilities.

## Abstract

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 \documentclass[12pt]{minimal}
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\begin{document}$$S^{(n)}$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$S^{(n)}$$\end{document}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, \documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}φ, by a fractional linear one that has the same survival probability \documentclass[12pt]{minimal}
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\begin{document}$$S^\infty $$\end{document}S∞ and yields the same rate of convergence of \documentclass[12pt]{minimal}
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\begin{document}$$S^{(n)}$$\end{document}S(n) to \documentclass[12pt]{minimal}
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\begin{document}$$S^\infty $$\end{document}S∞ as \documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$S^\infty $$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$S^\infty $$\end{document}S∞, as well as less well-known results on sharp bounds for \documentclass[12pt]{minimal}
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\begin{document}$$S^\infty $$\end{document}S∞. We apply them to explore when bounds for \documentclass[12pt]{minimal}
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\begin{document}$$S^{(n)}$$\end{document}S(n) exist for a family of generalized Poisson distributions. Numerical results demonstrate the accuracy of our and of previously derived bounds for \documentclass[12pt]{minimal}
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\begin{document}$$S^\infty $$\end{document}S∞ and \documentclass[12pt]{minimal}
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\begin{document}$$S^{(n)}$$\end{document}S(n). Finally, we treat an application of these results to determine the response of a quantitative trait to prolonged directional selection.

The online version contains supplementary material available at 10.1007/s00285-026-02349-7.