Systematic Approximations For Genetic Dynamics

Although much progress has been made in recent years in describing the dynamics of genetic systems, both in population genetics and evolutionary computation, there is still a conspicuous lack of tools with which to derive systematic, approximate solutions to their dynamics. In this article, we propose and study perturbation theory and the renormalization group as potential tools to fill this gap. We concentrate mainly on selection–mutation systems, showing different implementations of the perturbative framework, developing, for example, perturbative expansions for the eigenvalues and eigenvectors of the transition matrix. The main focus, however, is on diagrammatic methods, taken from physics, where we show how approximations can be built up using a pictorial representation generated by a simple set of rules, and how the renormalization group can be used to systematically improve the perturbation theory.