Some Exact Results from a Coarse Grained Formulation of Genetic Dynamics

TL;DR

This paper extends an exact schema-based formulation of genetic dynamics to arbitrary selection and crossover, providing new analytical results for population distributions under specific conditions.

Contribution

It generalizes the coarse grained genetic dynamics framework and derives exact solutions for population distributions with 1-point crossover and weak selection.

Findings

Hierarchical solutions for genetic distribution equations.

Exact population distribution over time for flat landscapes.

Analytic form of distribution for 1-point crossover and weak selection.

Abstract

We extend a recently developed exact schema based, or coarse grained, formulation of genetic dynamics \cite{stewael,stewael1,stewael2} and its associated exact Schema theorem to an arbitrary selection scheme and a general crossover operator. We show that the intuitive``building block'' interpretation of the former is preserved leading to hierarchical formal solutions of the equations that upon iteration lead to new results for the limiting distribution of a population in the case of 1-point crossover and ``weak'' selection, where we define quantitatively ``weak''. We also derive an exact, analytic form for the population distribution as a function of time for a flat landscape and 1-point crossover.