A stochastic field theory for the evolution of quantitative traits in finite populations

TL;DR

This paper develops a stochastic field theory to model the eco-evolutionary dynamics of finite populations with quantitative traits, capturing the effects of demographic noise and density dependence.

Contribution

It introduces stochastic field equations for finite populations that unify classical models and account for demographic fluctuations and eco-evolutionary feedbacks.

Findings

Derives equations recovering classical models like replicator-mutator and Price equation.

Describes how noise influences evolutionary trajectories in finite populations.

Provides a physics-inspired approach to population genetics modeling.

Abstract

Infinitely many distinct trait values may arise in populations bearing quantitative traits, and modeling their population dynamics is thus a formidable task. While classical models assume fixed or infinite population size, models in which the total population size fluctuates due to demographic noise in births and deaths can behave qualitatively differently from constant or infinite population models due to density-dependent dynamics. In this paper, I present a stochastic field theory for the eco-evolutionary dynamics of finite populations bearing one-dimensional quantitative traits. I derive stochastic field equations that describe the evolution of population densities, trait frequencies, and the mean value of any trait in the population. These equations recover well-known results such as the replicator-mutator equation, Price equation, and gradient dynamics in the infinite population limit. For finite populations, the equations describe the intricate interplay between natural selection, noise-induced selection, eco-evolutionary feedback, and neutral genetic drift in determining evolutionary trajectories. My work uses ideas from statistical physics, calculus of variations, and SPDEs, providing alternative methods that complement the measure-theoretic martingale approach that is more common in the literature.