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We consider a population evolving under mutation and selection. The genotype of an individual is a word of length $\ell$ over a finite alphabet. Mutations occur during reproduction, independently on each locus; the fitness depends on the Hamming class (the distance to a reference sequence $w^*$). Evolution is driven according to the classical Wright--Fisher process. We focus on the proportion of the different classes under the invariant measure of the process. We consider the regime where the length of the genotypes $\ell$ goes to infinity, and both the population size and the inverse of the mutation rate are of order $\ell$. We prove the existence of a critical curve, which depends both on the population size and the mutation rate. Below the critical curve, the proportion of any fixed class converges to $0$, whereas above the curve, it converges to a positive quantity, for which we give an explicit formula.

We study Eigen's quasispecies model in the asymptotic regime where the length of the genotypes goes to infinity and the mutation probability goes to 0. A limiting infinite system of differential equations is obtained. We prove the convergence of the trajectories, as well as the convergence of the equilibrium solutions. We give the analogous results for a discrete-time version of Eigen's model, which coincides with a model proposed by Moran.

The quasispecies model was introduced in 1971 by Manfred Eigen to discuss the first stages of life on Earth. It provides an appealing mathematical framework to study the evolution of populations in biology, for instance viruses. We present briefly the model and we focus on its stationary solutions. These formulae have a surprisingly rich combinatorial structure, involving for instance the Eulerian and Stirling numbers, as well as the up--down coefficients of permutations.

The quasispecies model was introduced in 1971 by Manfred Eigen to discuss the first stages of life on Earth. It provides an appealing mathematical framework to study the evolution of populations in biology, for instance viruses. We present briefly the model and we focus on its stationary solutions. These formulae have a surprisingly rich combinatorial structure, involving for instance the Eulerian and Stirling numbers, as well as the up--down coefficients of permutations.

We study a classical multitype Galton--Watson process with mutation and selection. The individuals are sequences of fixed length over a finite alphabet. On the sharp peak fitness landscape together with independent mutations per locus, we show that, as the length of the sequences goes to $\infty$ and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on $k$ digits from the master sequence approaches $$(\sigma e^{-a}-1)\frac{a^k}{k!}\sum_{i\geq1}\frac{i^k}{\sigma^i}\,,$$ where $\sigma$ is the selective advantage of the master sequence and $a$ is the product of the length of the chains with the mutation probability. The probability distribution $\mathcal{Q}(\sigma,a)$ on the non negative integers given by the above formula is the quasispecies distribution with parameters $\sigma$ and $a$., Comment: Some formulas contained errors that were corrected. References were updated

We consider the classical Wright-Fisher model with mutation and selection. Mutations occur independently in each locus, and selection is performed according to the sharp peak landscape. In the asymptotic regime studied in [3], a quasispecies is formed. We find explicitly the distribution of this quasispecies, which turns out to be the same distribution as for the Moran model.

We prove that a Moran model converges in probability to Eigen's quasispecies model in the infinite population limit.

We consider the Moran model on the sharp peak landscape, in the asymptotic regime studied in [3], where a quasispecies is formed. We find explicitly the distribution of this quasispecies.