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We study the evolution of the population genealogy in the classic neutral Moran Model of finite size and in discrete time. The stochastic transformations that shape a Moran population can be realized directly on its genealogy and give rise to a process with a state space consisting of the finite set of Yule trees of a certain size. We derive a number of properties of this process, and show that they are in agreement with existing results on the infinite-population limit of the Moran Model. Most importantly, this process admits time reversal, which gives rise to another tree-valued Markov Chain and allows for a thorough investigation of the Most Recent Common Ancestor process.
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We define the Sampled Moran Genealogy Process, a continuous-time Markov process on the space of genealogies with the demography of the classical Moran process, sampled through time. To do so, we begin by defining the Moran Genealogy Process using a novel representation. We then extend this process to include sampling through time. We derive exact conditional and marginal probability distributions for the sampled process under a stationarity assumption, and an exact expression for the likelihood of any sequence of genealogies it generates. This leads to some interesting observations pertinent to existing phylodynamic methods in the literature. Throughout, our proofs are original and make use of strictly forward-in-time calculations and are exact for all population sizes and sampling processes.
We consider t
We reconsider the Moran model in continuous time with population size $N$, two allelic types, and selection. We introduce a new particle representation, which we call the labelled Moran model, and which has the same distribution of type frequencies as the original Moran model, provided the initial values are chosen appropriately. In the new model, individuals are labelled $1,2, dots, N$; neutral resampling events may take place between arbitrary labels, whereas selective events only occur in the direction of increasing labels. With the help of elementary methods only, we not only recover fixation probabilities, but also obtain detailed insight into the number and nature of the selective events that play a role in the fixation process forward in time.
The distributions of the times to the first common ancestor t_mrca is numerically studied for an ecological population model, the extended Moran model. This model has a fixed population size N. The number of descendants is drawn from a beta distribution Beta(alpha, 2-alpha) for various choices of alpha. This includes also the classical Moran model (alpha->0) as well as the uniform distribution (alpha=1). Using a statistical mechanics-based large-deviation approach, the distributions can be studied over an extended range of the support, down to probabilities like 10^{-70}, which allowed us to study the change of the tails of the distribution when varying the value of alpha in [0,2]. We find exponential distributions p(t_mrca)~ delta^{t_mrca} in all cases, with systematically varying values for the base delta. Only for the cases alpha=0 and alpha=1, analytical results are known, i.e., delta=exp(-2/N^2) and delta=2/3, respectively. We recover these values, confirming the validity of our approach. Finally, we also study the correlations between t_mrca and the number of descendants.
Evolutionary dynamics has been classically studied for homogeneous populations, but now there is a growing interest in the non-homogenous case. One of the most important models has been proposed by Lieberman, Hauert and Nowak, adapting to a weighted directed graph the classical process described by Moran. The Markov chain associated with the graph can be modified by erasing all non-trivial loops in its state space, obtaining the so-called Embedded Markov chain (EMC). The fixation probability remains unchanged, but the expected time to absorption (fixation or extinction) is reduced. In this paper, we shall use this idea to compute asymptotically the average fixation probability for complete bipartite graphs. To this end, we firstly review some recent results on evolutionary dynamics on graphs trying to clarify some points. We also revisit the Star Theorem proved by Lieberman, Hauert and Nowak for the star graphs. Theoretically, EMC techniques allow fast computation of the fixation probability, but in practice this is not always true. Thus, in the last part of the paper, we compare this algorithm with the standard Monte Carlo method for some kind of complex networks.
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