  This work is a systematic study of discrete Markov chains that are used to
describe the evolution of a two-types population. Motivated by results valid
for the well-known Moran (M) and Wright-Fisher (WF) processes, we define a
general class of Markov chains models which we term the Kimura class. It
comprises the majority of the models used in population genetics, and we show
that many well-known results valid for M and WF processes are still valid in
this class. In all Kimura processes, a mutant gene will either fixate or become
extinct, and we present a necessary and sufficient condition for such processes
to have the probability of fixation strictly increasing in the initial
frequency of mutants. This condition implies that there are WF processes with
decreasing fixation probability --- in contradistinction to M processes which
always have strictly increasing fixation probability. As a by-product, we show
that an increasing fixation probability defines uniquely an M or WF process
which realises it, and that any fixation probability with no state having
trivial fixation can be realised by at least some WF process. These results are
extended to a subclass of processes that are suitable for describing
time-inhomogeneous dynamics. We also discuss the traditional identification of
frequency dependent fitnesses and pay-offs, extensively used in evolutionary
game theory, the role of weak selection when the population is finite, and the
relations between jumps in evolutionary processes and frequency dependent
fitnesses.
