arXiv 2106.162012021-06math.PR

选择和竞争下分支群体的进化谱系

Evolving genealogies for branching populations under selection and competition

Airam Blancas, Stephan Gufler, Sandra Kliem, Viet Chi Tran, Anton Wakolbinger

备注：: This version takes the suggestions of two anonymous referees into account, which led to an improvement in the presentation and to some corrections in the proofs in Section 5

摘要：

对于具有两类个体的连续状态分支过程，在选择和密度依赖竞争的条件下，我们刻画了总体规模、类型配置和谱系的联合演化为一类种群系统的唯一强解.我们的构造是在向下看的框架中实现的，并提供了Donnelly和Kurtz（1999）的两篇开创性论文中分别考虑的情况的综合和概括，即中性下的波动总体规模和恒定总体规模的选择。作为一个概念的核心，在我们的方法中，我们引入了选择性的向下看空间，这是从它的中性对应物通过一个状态依赖的“潜在的”选择/竞争事件，其速率与类型密度的演变相互作用变薄。在“主动”选择/竞争事件处的系谱距离矩阵的更新通过从选择性向下看空间的适当采样来获得。然后将上述系统的解映射为总体规模与对称型构型和谱系的联合演化，即标记距离矩阵分布。利用Kurtz的马氏映射定理，我们将后一个过程刻画为一个鞅问题的唯一解。为了透明起见，我们将演示文稿的主要部分限制为具有两种类型的原型示例，其中包含基本功能。在最后一节中，我们概述了一个扩展到多个类型的过程，包括突变。

英文摘要：

For a continuous state branching process with two types of individuals which are subject to selection and density dependent competition, we characterize the joint evolution of population size, type configurations and genealogies as the unique strong solution of a system of SDE's. Our construction is achieved in the lookdown framework and provides a synthesis as well as a generalization of cases considered separately in two seminal papers by Donnelly and Kurtz (1999), namely fluctuating population sizes under neutrality, and selection with constant population size. As a conceptual core in our approach we introduce the selective lookdown space which is obtained from its neutral counterpart through a state-dependent thinning of ``potential'' selection/competition events whose rates interact with the evolution of the type densities. The updates of the genealogical distance matrix at the ``active'' selection/competition events are obtained through an appropriate sampling from the selective lookdown space. The solution of the above mentioned system of SDE's is then mapped into the joint evolution of population size and symmetrized type configurations and genealogies, i.e. marked distance matrix distributions. By means of Kurtz' Markov mapping theorem, we characterize the latter process as the unique solution of a martingale problem. For the sake of transparency we restrict the main part of our presentation to a prototypical example with two types, which contains the essential features. In the final section we outline an extension to processes with multiple types including mutation.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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