On the stochastic evolution of finite populations
Fabio A. C. C. Chalub, Max O. Souza
This work is concerned with Markov chain models used in population genetics. Namely, we start from two celebrated models, the Moran and the Wright-Fisher processes, and study them from a very general viewpoint. Our aim is three fold: to identify the algebraic structures associated to time-homogeneous processes; to study the monotonicity properties of the fixation probability, with respect to the initial condition— in particular we show that there are situations, that are by no means exceptional, where an increase in the initial presence of a type can lead to a decrease in the fixation probability of this type; to understand time-inhomogeneous processes in a more systematic way. In addition, we also discuss the traditional identification of frequency dependent fitnesses and pay-offs, extensively used in evolutionary game theory, the role of weak selection when the population is finite, and the relations between jumps in evolutionary processes and frequency dependent fitnesses.