从离散到连续演化模型：漂移扩散和复制子动力学的统一方法
From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics
摘要：
我们研究了大人口极限的Moran过程，假设弱选择，并为不同的缩放。根据特定的缩放选择，我们得到一个连续的模型，可以突出遗传漂移（中性进化）或自然选择;对于一个精确的缩放，这两种效果都存在。对于考虑遗传漂移的缩放，连续模型由奇异扩散方程给出，连同已经存在于离散水平的两个守恒律。对于只考虑自然选择的标度，我们得到了一个双曲奇异方程，它嵌入了复制动力学，只满足一个守恒律。推导分为两个步骤：一个正式的，候选人的极限模型，并严格的概率密度的收敛性证明。还提出了关于固定概率的其他结果。
英文摘要：
We study the large population limit of the Moran process, assuming weak-selection, and for different scalings. Depending on the particular choice of scalings, we obtain a continuous model that may highlight the genetic-drift (neutral evolution) or natural selection; for one precise scaling, both effects are present. For the scalings that take the genetic-drift into account, the continuous model is given by a singular diffusion equation, together with two conservation laws that are already present at the discrete level. For scalings that take into account only natural selection, we obtain a hyperbolic singular equation that embeds the Replicator Dynamics and satisfies only one conservation law. The derivation is made in two steps: a formal one, where the candidate limit model is obtained, and a rigorous one, where convergence of the probability density is proved. Additional results on the fixation probabilities are also presented.
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