A generalisation of Kingman’s model of selection and mutation has been made in a previous paper which assumes all mutation probabilities to be i.i.d.. The weak convergence of fitness distributions to a globally stable equilibrium was proved. The condensation occurs if almost surely a positive proportion of the population travels to and condensates on the largest fitness value due to the dominance of selection over mutation. A criterion of condensation was given which relies on the equilibrium whose explicit expression is however unknown. This paper tackles these problems based on the discovery of a matrix representation of the random model. An explicit expression of the equilibrium is obtained and the key quantity in the condensation criterion can be estimated. Moreover we examine how the design of randomness in Kingman’s model affects the fitness level of the equilibrium by comparisons between different models. The discovered facts are conjectured to hold in other more sophisticated models.

中文翻译：

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具有随机突变概率的金曼模型：收敛和凝聚 II

Kingman 的选择和变异模型在之前的一篇论文中进行了概括，该论文假设所有变异概率都是 iid。证明了适应度分布对全局稳定均衡的弱收敛性。如果几乎可以肯定，由于选择优于突变，正比例的种群迁移到最大适应度值并凝聚在最大适应度值上，就会发生凝聚。给出了一个凝聚的标准，它依赖于平衡，但其明确的表达是未知的。本文基于发现随机模型的矩阵表示来解决这些问题。得到了平衡的显式表达式，并且可以估计凝聚准则中的关键量。此外，我们通过不同模型之间的比较来研究金曼模型中的随机性设计如何影响均衡的适应度水平。推测发现的事实适用于其他更复杂的模型。