Stochastic selection processes

Abstract

We propose a mathematical framework for natural selection in finite populations. Traditionally, many of the selection-based processes used to describe cultural and genetic evolution (such as imitation and birth-death models) have been studied on a case-by-case basis. Over time, these models have grown in sophistication to include population structure, differing phenotypes, and various forms of interaction asymmetry, among other features. Furthermore, many processes inspired by natural selection, such as evolutionary algorithms in computer science, possess characteristics that should fall within the realm of a "selection process," but so far there is no overarching theory encompassing these evolutionary processes. The framework of $\textit{stochastic selection processes}$ we present here provides such a theory and consists of three main components: a $\textit{population state space}$, an $\textit{aggregate payoff function}$, and an $\textit{update rule}$. A population state space is a generalization of the notion of population structure, and it can include non-spatial information such as strategy-mutation rates and phenotypes. An aggregate payoff function allows one to generically talk about the fitness of traits without explicitly specifying a method of payoff accounting or even the nature of the interactions that determine payoff/fitness. An update rule is a fitness-based function that updates a population based on its current state, and it includes as special cases the classical update mechanisms (Moran, Wright-Fisher, etc.) as well as more complicated mechanisms involving chromosomal crossover, mutation, and even complex cultural syntheses of strategies of neighboring individuals. Our framework covers models with variable population size as well as with arbitrary, measurable trait spaces.