Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics
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| Format: | Preprint |
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2025
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| _version_ | 1866911499186864128 |
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| author | Bürger, Reinhard |
| author_facet | Bürger, Reinhard |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_21403 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics Bürger, Reinhard Probability Populations and Evolution Primary: 60J80, Secondary: 92D10, 92D15, 60J85 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. |
| title | Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics |
| topic | Probability Populations and Evolution Primary: 60J80, Secondary: 92D10, 92D15, 60J85 |
| url | https://arxiv.org/abs/2503.21403 |