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In this article we present a new p-adic generalization of the Eigen-Schustermodel where the genomes (sequences) are represented by words written in thealphabet 0,1,,p-1, where p is a prime number, with a time variable length. Thetime evolution of the concentration of a sequence is controlled by a p-adicevolution equation. The long term behavior of the concentration of a sequencedepends on a fitness function f, a mutation measure Q, and an initialconcentration distribution. The new model provides essentially two types ofasymptotic scenarios for evolution. If the complexity of sequences grows at theright pace, then in the long term the survival is assured. This agrees with thefact that larger genome size improves the replication fidelity. In other case,the sequences cannot copy themselves with sufficiently fidelity, and in thelong term they will not survive. Eigen's paradox is one, among the infinitelymany, possible scenarios of the evolution in the long term. The mathematicalformulation of this fact requires solving the Cauchy problem for the p-adicEigen-Schuster model in a rigorous mathematical way. This requires imposingrestrictions on the fitness function and on the mutation measure, among otherconditions. The study of the mentioned initial value problem requirestechniques of p-adic wavelets and heat kernels developed in the last 35 years.

Eigen's Paradox and the Quasispecies Model in a Non-Archimedean Framework