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

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 $$S^{(n)}$$ 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 $$S^{(n)}$$ 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, $$\varphi $$ φ , by a fractional linear one that has the same survival probability $$S^\infty $$ S ∞ and yields the same rate of convergence of $$S^{(n)}$$ S ( n ) to $$S^\infty $$ S ∞ as $$\varphi $$ 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 $$S^\infty $$ 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 2 s for $$S^\infty $$ S ∞ , as well as less well-known results on sharp bounds for $$S^\infty $$ S ∞ . We apply them to explore when bounds for $$S^{(n)}$$ S ( n ) exist for a family of generalized Poisson distributions. Numerical results demonstrate the accuracy of our and of previously derived bounds for $$S^\infty $$ S ∞ and $$S^{(n)}$$ S ( n ) . Finally, we treat an application of these results to determine the response of a quantitative trait to prolonged directional selection.