The Mathematics of Evolution: The Price Equation, Natural Selection, and Environmental Change

TL;DR

This paper extends Price's equation and Fisher's theorem to quantum and measurable cases, introduces entropy functionals, and formulates four Laws of Natural Selection that relate evolutionary dynamics to thermodynamic principles.

Contribution

It develops a unified framework for evolutionary processes incorporating entropy and extends classical theorems to quantum and environmental contexts.

Findings

Decomposition of evolutionary processes into selective and environmental components.

Introduction of selective and environmental entropy functionals.

Formulation of four Laws of Natural Selection analogous to thermodynamic laws.

Abstract

George Price introduced his famous equation to study selective and environmental effects in discrete populations. We extend Price's framework to the measurable and quantum cases, decomposing all evolutionary processes into selective and environmental components. We also extend Fisher's fundamental theorem, showing that selective change of relative fitness equals variance of relative fitness. We introduce novel selective and environmental entropy functionals. Selective entropy is non-positive, representing biological negentropy, and environmental entropy is non-negative, representing physical entropy. Environmental entropy further decomposes into dispersion and mixing entropies. We prove four novel Laws of Natural Selection, showing that selection consistently acts to increase selection, but can be disrupted by environmental change. We apply convex analysis to variance and entropy functionals and their selective changes, and equilibrium processes arise to optimize these inequalities. These laws are inspired by but distinct from the classical Thermodynamic Laws. Our Zeroth Law is a refinement of Fisher's theorem, showing that variance of relative fitness is bounded below by a quantity depending on the child-bearing population. Our First Law shows that selective acceleration of relative fitness is also bounded below, depending on the variance. This is a non-conservative, selective version of the Thermodynamic First Law. Our Second Law shows that the selective change of selective entropy and its selective acceleration are similarly bounded by non-positive constants. This is a formal, rigorous version of the Thermodynamic Second Law. Our Third Law shows that for a class of equilibrium processes, selective change of environmental entropy vanishes, and otherwise may vary in an open window around zero. This is a selective version of the Third Law of Thermodynamics.