# The Fundamental Theorem of Natural Selection

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## Abstract

**:**

## 1. Introduction

The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.

The time derivative of the mean fitness of a population equals the variance of its fitness.

The precise statement, with all the hypotheses, is in Theorem 2. However, one lesson is this: variance in fitness may not cause ‘progress’ in the sense of increased mean fitness, but it does cause change.As a population changes with time, the rate at which information is updated equals the variance of fitness.

## 2. The Time Derivative of Mean Fitness

**replicators**. Let ${P}_{i}\left(t\right),$ or ${P}_{i}$ for short, be the population of the ith type of replicator at time t, which we idealize as taking positive real values. Then a very general form of the

**Lotka–Volterra equations**says that

**fitness function**of the ith type of replicator. One might also consider fitness functions with explicit time dependence, but we do not do so here.

**mean fitness**$\overline{f}$ by

**variance in fitness**by

**Theorem**

**1.**

**Proof.**

**replicator equation**:

## 3. The Fisher Speed

**Fisher metric**is the Riemannian metric g on the interior of the $(n-1)$-simplex such that given a point p in the interior of ${\Delta}^{n-1}$ and two tangent vectors $v,w$ we have

**Fisher speed**is defined by

**information of**q

**relative to**p is

**Theorem**

**2.**

**Proof.**

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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Baez, J.C. The Fundamental Theorem of Natural Selection. *Entropy* **2021**, *23*, 1436.
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Baez JC. The Fundamental Theorem of Natural Selection. *Entropy*. 2021; 23(11):1436.
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Baez, John C. 2021. "The Fundamental Theorem of Natural Selection" *Entropy* 23, no. 11: 1436.
https://doi.org/10.3390/e23111436