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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msup> </mml:math> 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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>φ</mml:mi></mml:math> , by a fractional linear one that has the same survival probability <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mi>∞</mml:mi></mml:msup> </mml:math> and yields the same rate of convergence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msup> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mi>∞</mml:mi></mml:msup> </mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>φ</mml:mi></mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mi>∞</mml:mi></mml:msup> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mi>∞</mml:mi></mml:msup> </mml:math> , as well as less well-known results on sharp bounds for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mi>∞</mml:mi></mml:msup> </mml:math> . We apply them to explore when bounds for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msup> </mml:math> exist for a family of generalized Poisson distributions. Numerical results demonstrate the accuracy of our and of previously derived bounds for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mi>∞</mml:mi></mml:msup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:msup> </mml:math> . Finally, we treat an application of these results to determine the response of a quantitative trait to prolonged directional selection.