The Model
This section is devoted to the presentation of the model. Let A be a finite
alphabet and let κ = card A be its cardinality. Let 1 be an integer. We
consider the space A of sequences of length over the alphabet A. Elements of
this space represent the chromosome of an haploid individual, or equivalently its
genotype. In our model, all the genes have the same set of alleles and each letter
of the alphabet A is a possible allele. Typical examples are A = { A, T, G, C }
to model standard DNA, or A = { 0, 1 } to deal with binary sequences. Generic
elements of A will be denoted by the letters u, v, w. We shall study a simple
model for the evolution of a finite population of chromosomes on the space A .
An essential feature of the model we consider is that the size of the population is
constant throughout the evolution. We denote by m the size of the population. A
population is an m–tuple of elements of A . Generic populations will be denoted
by the letters x, y, z. Thus a population x is a vector
x =


whose components are chromosomes. For i { 1,...,m }, we denote by
x(i, 1),...,x(i, )
the letters of the sequence x(i). This way a population x can be represented as an
x =

x(1, 1) · · · x(1, )
x(m, 1) · · · x(m, )

of size m × of elements of A, the i–th row being the i–th chromosome. The
evolution of the population will be random and it will be driven by two antagonistic
forces: mutation and replication.
Mutation. We assume that the mutation mechanism is the same for all the loci,
and that mutations occur independently. Moreover we choose the most symmetric
mutation scheme. We denote by q ∈]0, 1 1/κ[ the probability of the occurrence of
a mutation at one particular locus. If a mutation occurs, then the letter is replaced
randomly by another letter, chosen uniformly over the κ 1 remaining letters. We
encode this mechanism in a mutation matrix
M(u, v) , u, v A
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