Dynamics of two-element populations in the space of population states

An analysis of population dynamics in the space of population states is presented. The simplest case of the phenotypic evolution-a population consisting of two individuals with one real-valued trait, evolving under proportional selection and mutation with an underlying normal distribution-is considered. The focus is on the trajectories of the expected population state values generating a discrete dynamical system. The system models the expected asymptotic behavior of the evolutionary process. The analysis and the simulation results shed light on the dynamics of approaching evolutionary equilibria. The effect of two-speed convergence is observed: 1) initially fast convergence toward an approximately homogenous population and then 2) a slow drift of the population toward optima. The system´s fixed points and their stability are determined. Periodic and chaotic behaviors are observed for some fitness functions.