  This paper is based on the complete classification of evolutionary scenarios
for the Moran process with two strategies given by Taylor et al. (B. Math.
Biol. 66(6): 1621--1644, 2004). Their classification is based on whether each
strategy is a Nash equilibrium and whether the fixation probability for a
single individual of each strategy is larger or smaller than its value for
neutral evolution. We improve on this analysis by showing that each
evolutionary scenario is characterized by a definite graph shape for the
fixation probability function. A second class of results deals with the
behavior of the fixation probability when the population size tends to
infinity. We develop asymptotic formulae that approximate the fixation
probability in this limit and conclude that some of the evolutionary scenarios
cannot exist when the population size is large.
