  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.
