Neutral selection plays an important role in the
evolution of populations, having a finite population size *n *[1]
*.* To demonstrate the neutral selection features
explicitly, let's consider the pure neutral evolution game, which
is defined as follows:

; |
1. There is a population of black and white balls,
the total number of the balls in a population is equal to
*n .* |

; |
2. The evolution consists of consequent generations.
Each generation consists of two steps. At the first step
we duplicate each ball, conserving its color: a black
ball has two black offsprings, a white ball has two white
ones. At the second step we randomly remove from a
population exactly half of the balls with equal
probability for black and white "species",
independently of their color. |

We say that the population is in *l *-state, if the
numbers of black and white balls at a considered generation are
equal to *l *and *n-l*, respectively. We can
characterize the evolution by the probability *P*_{lm}*
*of a transition from *l *-state to *m *-state
during one generation. Using a straightforward combinatorial
consideration, we can calculate the values of *P*_{lm}
:

The matrix *P*_{lm} determines the
random Markovian process, which can be considered as an example
of a simple stochastic genetic process [2]. Using the general
methods of analysis of such processes [2], we can deduce that:

1) the considered process always converges to one of two
absorbing states, namely, to* *0-state (all balls are
white), either to* n*-state* *(all balls are
black);

2) at large *n* the characteristic number of
generations* T*_{n} , needed to converge
to the either absorbing state, is equal to 2*n *:

Thus, although this evolution is purely neutral (black and white
balls have equal chances to survive), nevertheless only one
species is selected. The value *T*_{n }characterizes
the neutral selection rate, it is used in our estimations (Quasispecies , Estimation
of the evolution rate).

**References:**

1. *M. Kimura. *"The neutral theory
of molecular evolution". Cambridge Un-ty Press. 1983.

2. *S. Karlin. *"A first course in
stochastic processes". Academic Press. New York, London,
1968.

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