The Fundamental Theorem of Natural Selection

TL;DR

This paper presents a mathematical proof that the rate of information change in natural selection equals the variance in fitness, offering a modified perspective on Fisher's fundamental theorem.

Contribution

It provides a rigorous proof linking the speed of information update in natural selection to fitness variance, extending Fisher's theorem.

Findings

Speed of natural selection equals fitness variance

Provides a modified version of Fisher's fundamental theorem

Connects information theory with evolutionary dynamics

Abstract

Suppose we have $n$ different types of self-replicating entity, with the population $P_i$ of the $i$th type changing at a rate equal to $P_i$ times the fitness $f_i$ of that type. Suppose the fitness $f_i$ is any continuous function of all the populations $P_1, \dots, P_n$. Let $p_i$ be the fraction of replicators that are of the $i$th type. Then $p = (p_1, \dots, p_n)$ is a time-dependent probability distribution, and we prove that its speed as measured by the Fisher information metric equals the variance in fitness. In rough terms, this says that the speed at which information is updated through natural selection equals the variance in fitness. This result can be seen as a modified version of Fisher's fundamental theorem of natural selection. We compare it to Fisher's original result as interpreted by Price, Ewens and Edwards.