论文标题
恒星图上的出生死亡和死亡率依赖频率过程的确切固定概率
Exact fixation probabilities for the Birth-Death and Death-Birth frequency-dependent Moran processes on the star graph
论文作者
论文摘要
扫帚和瑞奇塔[Proc。 R. Soc。 A（2008）464，2609--2627]找到了一个精确的解决方案，用于用于结构化种群的Moran过程的固定概率，其中所谓的恒星图由个体之间的相互作用结构，即一个中央顶点和$ n $叶，仅连接到中心。我们通过允许个人的健身来依赖人口频率，并允许繁殖和死亡抽取顺序的变化来概括他们的解决方案。在引用的论文中，Broom和Rychtá找考虑了出生死亡（BD）过程，在该过程中，在每个步骤中，每个人都首先绘制一个人进行复制，然后选择一个人死亡。在死亡 - 出生过程（DB）过程中，抽奖的顺序被颠倒了。可以看出，抽签的顺序对固定概率产生了很大的不同。我们的解决方案方法适用于BD和DB情况。不出所料，固定概率的确切公式很复杂。我们还将用一些示例来说明它们，并为固定概率的渐近行为提供结果，当时图中的叶子的数量$ n $倾向于无穷大。
Broom and Rychtář [Proc. R. Soc. A (2008) 464, 2609--2627] found an exact solution for the fixation probabilities of the Moran process for a structured population, in which the interaction structure among individuals is given by the so-called star graph, i.e. one central vertex and $n$ leaves, the leaves connecting only to the center. We generalize on their solution by allowing individuals' fitnesses to depend on the population frequency, and also by allowing a possible change in the order of reproduction and death draws. In their cited paper, Broom and Rychtář considered the birth-death (BD) process, in which at each time step an individual is first drawn for reproduction and then an individual is selected for death. In the death-birth (DB) process, the order of the draws is reversed. It may be seen that the order of the draws makes a big difference in the fixation probabilities. Our solution method applies to both the BD and the DB cases. As expected, the exact formulae for the fixation probabilities are complicated. We will also illustrate them with some examples and provide results on the asymptotic behavior of the fixation probabilities when the number $n$ of leaves in the graph tends to infinity.
