Exponential Growth and the Fixation of Rare Beneficial Alleles

Introduction

Exponential growth is a fundamental concept in population genetics, particularly in understanding how rare beneficial alleles spread within a growing population. When a population expands exponentially, the probability of fixation – the likelihood that a rare beneficial allele will eventually become the only allele in the population – differs significantly from that in a stable or declining population.

Exponential Growth and Allele Fixation

In a population growing at a rate r, the average number of offspring per parent is not static. Instead, it is influenced by (1 + r) for wild-type individuals and (1 + s + r) for individuals carrying a rare beneficial allele, where:

• r represents the population growth rate.

• s represents the selective advantage of the beneficial allele.

This distinction means that when r > 0 (positive growth), beneficial mutations fixate at a faster rate because:

• Each individual has more offspring, increasing the chances of passing on the beneficial allele.

• The growing population provides a “buffer” against genetic drift, reducing the likelihood of allele loss due to random fluctuations.

Conversely, when r < 0 (negative growth), beneficial mutations fixate at a slower rate or may be lost entirely because: