论文标题
从谱系的角度求解迁移重组方程
Solving the migration-recombination equation from a genealogical point of view
论文作者
论文摘要
我们考虑了离散的时间迁移 - 分组方程，这是一种确定性的非线性动力学系统，描述了在大量设定定律中，迁移和重组在迁移和重组下的遗传类型分布的演变。我们将这种动态（时间向前）与马尔可夫链相关联，即标记的分区过程，在时间上向后。这样，我们获得了迁移重组方程解的随机表示。结果，仅根据马尔可夫链的过渡矩阵的功能，获得了非线性动力学的明确解。最后，我们研究了马尔可夫链的限制和准限制行为，该行为立即访问了动力学系统的渐近行为。我们终于在连续的时间绘制了类似情况。
We consider the discrete-time migration-recombination equation, a deterministic, nonlinear dynamical system that describes the evolution of the genetic type distribution of a population evolving under migration and recombination in a law of large numbers setting. We relate this dynamics (forward in time) to a Markov chain, namely a labelled partitioning process, backward in time. This way, we obtain a stochastic representation of the solution of the migration-recombination equation. As a consequence, one obtains an explicit solution of the nonlinear dynamics, simply in terms of powers of the transition matrix of the Markov chain. Finally, we investigate the limiting and quasi-limiting behaviour of the Markov chain, which gives immediate access to the asymptotic behaviour of the dynamical system. We finally sketch the analogous situation in continuous time.
