Replicator–mutator equation, universality property and population dynamics of learning

Replicator–mutator equation is used to describe the dynamics of complex adaptive systems in population genetics, biochemistry and models of language learning. We study “localized”, or “coherent”, solutions, which are especially relevant in the context of learning and correspond to the existence of a predominant language in the population. There is a coherence threshold for learning fidelity, above which coherent communication can be maintained. We prove the following surprising universality property of coherence threshold: for typical realizations of random coefficients in the fitness matrix, the value of the coherence threshold does not depend on the size of the system.