2 1. INTRODUCTION

and Schuster [32] constructed a birth and death model to approximate Eigen’s

model. This birth and death model plays a key role in our analysis, as we shall

see later. Alves and Fontanari [1] study how the error threshold depends on the

population in a simplified model. More recently, Musso [29] and Dixit, Srivastava,

Vishnoi [10] considered finite population models which approximate Eigen’s model

when the population size goes to ∞. These models are variants of the classical

Wright–Fisher model of population genetics. Although this is an interesting ap-

proach, it is already a delicate matter to prove the convergence of these models

towards Eigen’s model. We adopt here a different strategy. Instead of trying to

prove that some finite population model converges in some sense to Eigen’s model,

we try to prove directly in the finite model an error threshold phenomenon. To this

end, we look for the simplest possible model, and we end up with a Moran model

[28]. The model we choose here is not particularly original, the contribution of this

work is rather to show a way to analyze this kind of finite population model.

Let us describe informally the model (see chapter 2 for the formal definition).

We consider a population of size m of chromosomes of length over the alphabet

{ A, T, G, C }. We work only with the sharp peak landscape: there is one specific

sequence, called the master sequence, whose fitness is σ 1, and all the other

sequences have fitness equal to 1. The replication rate of a chromosome is propor-

tional to its fitness. When a chromosome replicates, it produces a copy of himself,

which is subject to mutations. Mutations occur randomly and independently at

each locus with probability q. The offspring of a chromosome replaces a chromo-

some chosen at random in the population.

The mutations drive the population towards a totally random state, while the

replication favors the master sequence. These two antagonistic effects interact in a

complicated way in the dynamics and it is extremely diﬃcult to analyze precisely

the time evolution of such a model. Let us focus on the equilibrium distribution

of the process. A fundamental problem is to determine the law of the number

of copies of the master sequence present in the population at equilibrium. If we

keep the parameters m, , q fixed, there is little hope to get useful results. In order

to simplify the picture, we consider an adequate asymptotic regime. In Eigen’s

model, the population size is infinite from the start. The error threshold appears

when goes to ∞ and q goes to 0 in a regime where q = a is kept constant. We

wish to understand the influence of the population size m, thus we use a different

approach and we consider the following regime. We send simultaneously m, to ∞

and q to 0 and we try to understand the respective influence of each parameter on

the equilibrium law of the master sequence. By the ergodic theorem, the average

number of copies of the master sequence at equilibrium is equal to the limit, as the

time goes to ∞, of the time average of the number of copies of the master sequence

present through the whole evolution of the process. In the finite population model,

the number of copies of the master sequence fluctuates with time. Our analysis of

these fluctuations relies on the following heuristics. Suppose that the process starts

with a population of size m containing exactly one master sequence. The master

sequence is likely to invade the whole population and become dominant. Then the

master sequence will be present in the population for a very long time without

interruption. We call this time the persistence time of the master sequence.

The destruction of all the master sequences of the population is quite unlikely,

nevertheless it will happen and the process will eventually land in the neutral