We introduce a Cannings model with directional selection via a paintbox construction and establish a strong duality with the line counting process of a new Cannings ancestral selection graph in discrete time. This duality also yields a formula for the fixation probability of the beneficial type. Haldane's formula states that for a single selectively advantageous individual in a population of haploid individuals of size N the probability of fixation is asymptotically (as N → ∞) equal to the selective advantage of haploids sN divided by half of the offspring variance. For a class of offspring distributions within Kingman attraction we prove this asymptotics for sequences sN obeying N −1 sN N −1/2 , which is a regime of "moderately weak selection". It turns out that for sN N −2/3 the Cannings ancestral selection graph is so close to the ancestral selection graph of a Moran model that a suitable coupling argument allows to play the problem back asymptotically to the fixation probability in the Moran model, which can be computed explicitly.

For a class of Cannings models we prove Haldane’s formula, $$\pi (s_N) \sim \frac{2s_N}{\rho ^2}$$ π ( s N ) ∼ 2 s N ρ 2 , for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for $$s_N \sim N^{-b}$$ s N ∼ N - b and $$0< b<1/2$$ 0 < b < 1 / 2 . Here, $$s_N$$ s N is the selective advantage of an individual carrying the beneficial type, and $$\rho ^2$$ ρ 2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.

Analysing survival or fixation probabilities for a beneficial allele is a prominent task in the field of theoretical population genetics. Haldane's asymptotics is an approximation for the fixation probability in the case of a single beneficial mutant with small selective advantage in a large population. In this thesis we analyse the interplay between genetic drift and directional selection and prove Haldane's asymptotics in different settings: For the fixation probability in Cannings models with moderate selection and for the survival probability of a slightly supercritical branching processes in a random environment. In Chapter 3 we introduce a class of Cannings models with selection that allow for a forward and backward construction. In particular, a Cannings ancestral selection process can be defined for this class of models, which counts the number of potential parents and is in sampling duality to the forward frequency process. By means of this duality the probability of fixation can be expressed through the expectation of the Cannings ancestral selection process in stationarity. A control of this expectation yields that the fixation probability fulfils Haldane's asymptotics in a regime of moderately weak selection (Thm. 8). In Chapter 4 we study the fixation probability of Cannings models in a regime of moderately strong selection. Here couplings of the frequency process of beneficial individuals with slightly supercritical Galton-Watson processes imply that the fixation probability is given by Haldane's asymptotics (Thm. 9). Lastly, in Chapter 5 we consider slightly supercritical branching processes in an independent and identically distributed random environment and study the probability of survival as the number of expected offspring tends from above to one. We show that only if variance and expectation of the random offspring mean are of the same order the random environment has a non-trivial influence on the probability of survival, which results in a modification of Haldane's asymptotics. Out of the critical parameter regime the population goes extinct or survives with a probability that fulfils Haldane's asymptotics (Thm. 10). The proof establishes an expression for the survival probability in terms of the shape function of the random offspring generating functions. This expression exhibits similarities to perpetuities known from a financial context. Consequently, we prove a limiting theorem for perpetuities with vanishing interest rates (Thm. 11).

Branching processes in a random environment are a natural generalisation of Galton-Watson processes. In this paper we analyse the asymptotic decay of the survival probability for a sequence of slightly supercritical branching processes in a random environment, where the offspring expectation converges from above to 1. We prove that Haldane's asymptotics, known from classical Galton-Watson processes, turns up again in the random environment case, provided that one stays away from the critical or subcritical regimes. A central building block is a connection to and a limit theorem for perpetuities with asymptotically vanishing interest rates.

Building on the spinal decomposition technique in [10] we prove a Yaglom limit law for the rescaled size of a nearly critical branching process in varying environment conditional on survival. In addition, our spinal approach allows us to prove convergence of the genealogical structure of the population at a fixed time horizon -where the sequence of trees are envisioned as a sequence of metric spaces -in the Gromov-Hausdorff-Prokorov (GHP) topology. We characterize the limiting metric space as a time-changed version of the Brownian coalescent point process [37].Beyond our specific model, we derive several general results allowing to go from spinal decompositions to convergence of random trees in the GHP topology. As a direct application, we show how this type of convergence naturally condenses the limit of several interesting genealogical quantities: the population size, the time to the most-recent common ancestor, the reduced tree and the tree generated by k uniformly sampled individuals. As in a recent article by the authors [10], we hope that our specific example illustrates a general methodology that could be applied to more complex branching processes.