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We review recent progress on ancestral processes related to mutation-selection models, both in the deterministic and the stochastic setting. We mainly rely on two concepts, namely, the killed ancestral selection graph and the pruned lookdown ancestral selection graph. The killed ancestral selection graph gives a representation of the type of a random individual from a stationary population, based upon the individuals potential ancestry back until the mutations that define the individuals type. The pruned lookdown ancestral selection graph allows one to trace the ancestry of individuals from a stationary distribution back into the distant past, thus leading to the stationary distribution of ancestral types. We illustrate the results by applying them to a prototype model for the error threshold phenomenon.
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We consider the mutation--selection differential equation with pairwise interaction (or, equivalently, the diploid mutation--selection equation) 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 duality. 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 tripod 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 long-term 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.
We reconsider the deterministic haploid mutation-selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (random) structures inherent in this deterministic model. In a first step, we obtain a representation of the deterministic equations solution (and, in particular, of its equilibrium) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error threshold phenomenon from a genealogical point of view. Next, we characterise the ancestral type distribution by means of the pruned lookdown ancestral selection graph and study its properties at equilibrium. We also provide an alternative characterisation in terms of a piecewise-deterministic Markov process. Throughout, emphasis is on the underlying dualities as well as on explicit results.
We study ancestral structures for the two-type Moran model with two-way mutation and frequency-dependent selection that follows the nonlinear dominance or fittest-type-wins scheme. Both schemes lead, in distribution, to the same type-frequency process. Reasoning through the mutation structure on the ancestral selection graph (ASG), we derive processes suitable to determine the type distribution of the present and ancestral population, leading to, respectively, the killed and pruned lookdown ASG. To this end, we establish factorial moment dualities to the Moran model and a relative thereof, respectively. Finally, we extend the results to the diffusion limit.
Here we will use results of Cox, Durrett, and Perkins for voter model perturbations to study spatial evolutionary games on $Z^d$, $dge 3$ when the interaction kernel is finite range, symmetric, and has covariance matrix $sigma^2I$. The games we consider have payoff matrices of the form ${bf 1} + wG$ where ${bf 1}$ is matrix of all 1s and $w$ is small and positive. Since our population size $N=infty$, we call our selection small rather than weak which usually means $w =O(1/N)$. The key to studying these games is the fact that when the dynamics are suitably rescaled in space and time they convergence to solutions of a reaction diffusion equation (RDE). Inspired by work of Ohtsuki and Nowak and Tarnita et al we show that the reaction term is the replicator equation for a modified game matrix and the modifications of the game matrix depend on the interaction kernel only through the values of two or three simple probabilities for an associated coalescing random walk. Two strategy games lead to an RDE with a cubic nonlinearity, so we can describe the phase diagram completely. Three strategy games lead to a pair of coupled RDE, but using an idea from our earlier work, we are able to show that if there is a repelling function for the replicator equation for the modified game, then there is coexistence in the spatial game when selection is small. This enables us to prove coexistence in the spatial model in a wide variety of examples where the replicator equation of the modified game has an attracting equilibrium with all components positive. Using this result we are able to analyze the behavior of four evolutionary games that have recently been used in cancer modeling.
$Lambda$-Wright--Fisher processes provide a robust framework to describe the type-frequency evolution of an infinite neutral population. We add a polynomial drift to the corresponding stochastic differential equation to incorporate frequency-dependent selection. A decomposition of the drift allows us to approximate the solution of the stochastic differential equation by a sequence of Moran models. The genealogical structure underlying the Moran model leads in the large population limit to a generalisation of the ancestral selection graph of Krone and Neuhauser. Building on this object, we construct a continuous-time Markov chain and relate it to the forward process via a new form of duality, which we call Bernstein duality. We adapt classical methods based on the moment duality to determine the time to absorption and criteria for the accessibility of the boundaries; this extends a recent result by Gonzalez Casanova and Span`o. An intriguing feature of the construction is that the same forward process is compatible with multiple backward models. In this context we introduce suitable notions for minimality among the ancestral processes and characterise the corresponding parameter sets. In this way we recover classic ancestral structures as minimal ones.
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