关于决定性突变选择方程的概率观点：通过个体血统的动态、平衡和祖先
A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent
摘要：
我们重新考虑了两种类型的决定性单倍体突变选择方程。这是一个常微分方程，它描述了无限大的种群中的类型分布（在时间上向前）。本文建立了祖先（随机）结构固有的确定性模型。在第一步中，我们获得了一个代表性的确定性方程的解决方案（特别是，其平衡）在一个祖先的过程中被称为杀死的祖先选择图。这种表示允许人们从谱系的角度理解与误差阈值现象相关的分叉。其次，我们利用剪枝的下视祖先选择图来构造祖先类型分布，并研究了它在平衡态下的性质。我们还提供了一个替代的分段确定性的马尔可夫过程方面的表征。在整个过程中，强调的是潜在的二重性以及明确的结果。
英文摘要：
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 equation's 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.
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