We consider the mutation-selection differential equation with pairwise interaction and establish the corresponding ancestral process, which is a specific random tree and a variant of the ancestral selection graph. To make this object tractable, we prune the tree upon mutation , thus reducing it to its informative parts. The hierarchies inherent in the tree are encoded systematically via ternary trees with weighted leaves; this leads to the stratified ancestral selection graph. The latter is dual to the mutation-selection equation and provides a stochastic representation of its solution . It also allows to reveal the genealogical structures inherent in the bifurcations of the equilibria of the differential equation. Furthermore, we establish constructions, again based on the stratified ancestral selection graph, that allow to trace back the ancestral lines into the distant past and obtain explicit results about the ancestral population in the case of unidirectional mutation.
We consider the mutation-selection differential equation with pairwise interaction and establish the corresponding ancestral process, which is a random tree and a variant of the ancestral selection graph. The formal relation to the forward model is given via a duality relation. To make the tree tractable, we prune branches upon mutations , thus reducing it to its informative parts. The hierarchies inherent in the tree are encoded systematically via ternary trees with weighted leaves; this leads to the stratified ancestral selection graph. The latter also satisfies a duality relation with the mutation-selection equation . Each of the dualities provides a stochastic representation of the solution of the differential equation. This allows us to connect the equilibria and their bifurcations to the asymptotic behaviour of the ancestral process . Furthermore, with the help of the stratified ancestral selection graph, we obtain explicit results about the ancestral type distribution in the case of unidirectional mutation.
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It also allows to reveal the genealogical structures inherent in the bifurcations of the equilibria of the differential equation.
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we establish constructions, again based on the
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