The ancestral selection graph for a $Λ$-asymmetric Moran model

TL;DR

This work develops a two-type $\Lambda$-Moran model with asymmetric reproduction measures and constructs a $\Lambda$-asymmetric ancestral selection graph (ASG) using a monotone coupling, yielding a pathwise duality between the forward frequency dynamics and its ancestral process. It establishes scaling limits to a jump-diffusion frequency process and a limiting ancestral process with a moment dual, linking population genetics to optimal transport via the coupling. The paper generalises to finite measures, analyzes coupling non-uniqueness, and provides a Griffiths-type representation that leads to a semi-explicit formula for the fixation probability of the inferior type. These results extend classical ASG theory to skewed offspring distributions and offer scalable descriptions beyond the Wright–Fisher regime, with clear connections to duality and transport theory.

Abstract

Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanims, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure $Λ$ for each of the two types. Our approach consists of constructing a $Λ$-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Under certain conditions, that we call the "partial order of adaptation", we can couple these measures. This allows us to construct the central object of this paper, the $Λ-$asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time $Λ$-asymmetric Moran model with its ancestral process. Interestingly, the construction also provides a connection to the theory of optimal transport. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.

Figures (4)

• Figure 1: A realisation of the $\Lambda$-asymmetric frequency process. Arrows point from the reproducing individual to their offspring. In the first reproductive event, a type $\oplus$ reproduces, in the second one a type $\ominus$ individual. The role of selection in this construction will be discussed later in Section \ref{['sect:adaptation-selection']}, cf. also Figure \ref{['fig:frequency_Lambda']}.
• Figure 2: The realisation of the $\Lambda$-asymmetric frequency process from Figure \ref{['fig:frequency']} now in terms of the coupling construction. Neutral arrows are black, selective arrows grey. Individuals of type $\oplus$ may reproduce through any arrow, individuals of type $\ominus$ only through neutral arrows. Therefore some of the arrows and the last reproductive event weren't present in Figure \ref{['fig:frequency']}, where the measures applied to determine the reproduction depended on the type of the reproducing individuals.
• Figure 3: On the left: The ancestral line process of a sample of three individuals (marked by an asterisk), using the realisation of the $\Lambda$-asymmetric ancestral selection graph from Figure \ref{['fig:frequency_Lambda']} and going backward in time (right to left). Lines of potential ancestors are colored in black. At a reproductive event, lines from individuals hit by a black (neutral) arrow are alway merged with the reproducing line at the origin of the arrow, where for those hit by a grey (selective) arrow, both lines are continued. On the right: The ancestry of one individual. The type of this individual is $\oplus$ if one potential ancestor has type $\oplus$, and $\ominus$ otherwise.
• Figure 4: The three possible types of transitions of the ancestral process. On the left, the reproducing individuals belongs to the current sample(thick black lines), there is one neutral (black) and two selctive (grey) arrows. The line of the individual at the tip of the neutral arrow is discarded, resp. merges with the reproducing line, while the lines hit by a selective arrow remain, as well as those lines not affected by the reproduction event. In the middle, the reproducing individuals doesn't belong to the current sample, there are two neutral and one selctive arrows. The line of the reproducing individual is thus added, the lines of the individuals at the end of a neutral arrow are discarded, resp. merge with the reproducing line. On the right, only selective arrows are present, where the incoming lines are kept, but the reproducing line is added. We therefore see a branching.

Theorems & Definitions (26)

• Definition 2.1: $\Lambda$-asymmetric Moran model
• Lemma 3.1: Coupling lemma
• proof
• Definition 3.2: $\Lambda-$asymmetric ancestral selection graph, ASG
• Proposition 3.3
• Definition 4.1
• Definition 4.2
• Proposition 4.3
• proof
• proof : Proof of Proposition \ref{['prop:frequency_equal']}