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. The limiting and quasi-limiting behaviour of the Markov chain are investigated, which gives immediate access to the asymptotic behaviour of the dynamical system. We finally sketch the analogous situation in continuous time.

我们考虑离散时间迁移-重组方程，这是一个确定性的非线性动力系统，它描述了在大数定律环境中迁移和重组下进化的种群的遗传类型分布的进化。我们将这种动态（在时间上向前）与马尔可夫链联系起来，即一个标记的分区过程，在时间上向后。通过这种方式，我们获得了迁移-重组方程解的随机表示。因此，我们可以简单地根据马尔可夫链的转移矩阵的幂获得非线性动力学的显式解。研究了马尔可夫链的限制和准限制行为，这可以立即访问动力系统的渐近行为。