Stochastic modelling of evolutionary processes

Abstract

Evolutionary models are applied in various branches of science. The notions of births, deaths and mutations in a population of individuals can be interpreted in many different ways. We discuss some general modelling choices and universal properties of evolutionary systems. Most importantly, when certain individuals reproduce faster than others, evolution gives rise to a dynamical system that optimizes itself. After introducing evolutionary models in general, we focus on two specific models. Firstly, we investigate the relation of the population size to convergence properties in a population of solutions to the optimization problem 1-in-3-SAT. Secondly, we consider birth-death processes of diffusing particles. We derive an equation for the density of particles that describes the average behaviour and compare predictions to simulations of the micro-model. The average description does not account for the observed clustering of the population, so we consider methods to describe and measure deviations from the mean behaviour.