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首页> 外文期刊>Journal of Mathematical Biology >A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent
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A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent

机译：关于确定性突变选择方程的概率视图：通过单个下降线的动态，均衡和祖先

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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 equilibria) 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.
机译：我们重新考虑具有两种类型的确定性单倍体突变选择方程。 这是一种常用方程，其描述了无限尺寸的群体中的类型分布（在时间）。 本文建立了该确定性模型固有的祖先（随机）结构。 在第一步中，我们在称为被杀死的祖先选择图的祖先过程方面获得确定性等式的解决方案（以及尤其是其均衡）的表示。 该表示允许人们从族裔观点来理解与误差阈值现象有关的分叉。 接下来，我们通过修剪的阴扫祖先选择图来表征祖先类型分布，并在均衡时研究其属性。 我们还通过分段确定的马尔可夫过程提供替代表征。 整个，重点是底层的二元以及明确的结果。

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