We consider three classical models of biological evolution: (i) the Moran process, an example of a reducible Markov Chain; (ii) the Kimura Equation, a particular case of a degenerated Fokker-Planck Diffusion; (iii) the Replicator Equation, a paradigm in Evolutionary Game Theory. While these approaches are not completely equivalent, they are intimately connected, since (ii) is the diffusion approximation of (i), and (iii) is obtained from (ii) in an appropriate limit. It is well known that the Replicator Dynamics for two strategies is a gradient flow with respect to the celebrated Shahshahani distance. We reformulate the Moran process and the Kimura Equation as gradient flows and in the sequel we discuss conditions such that the associated gradient structures converge: (i) to (ii), and (ii) to (iii). This provides a geometric characterisation of these evolutionary processes and provides a reformulation of the above examples as time minimisation of free energy functionals.

我们考虑了三种经典的生物进化模型：（i）Moran过程，可简化的马尔可夫链的一个例子；（ii）Kimura方程，退化的Fokker-Planck扩散的一种特殊情况；（iii）复制子方程，这是进化博弈论的范例。尽管这些方法并不完全等效，但它们紧密相连，因为（ii）是（i）的扩散近似，并且（iii）是在适当的限制下从（ii）获得的。众所周知，两种策略的复制器动力学是相对于著名的Shahshahani距离的梯度流。我们将Moran过程和Kimura方程重新定义为梯度流，并在后续部分中讨论使相关的梯度结构收敛的条件：（i）至（ii），以及（ii）至（iii）。