Fitness (biology)

The absolute fitness (${\displaystyle W}$) of a genotype is defined as the proportional change in the abundance of that genotype over one generation attributable to selection. For example, if ${\displaystyle n(t)}$ is the abundance of a genotype in generation ${\displaystyle t}$ in an infinitely large population (so that there is no genetic drift), and neglecting the change in genotype abundances due to mutations, then[4]

${\displaystyle n(t+1)=Wn(t)}$.

An absolute fitness larger than 1 indicates growth in that genotype's abundance; an absolute fitness smaller than 1 indicates decline.

Whereas absolute fitness determines changes in genotype abundance, relative fitness (${\displaystyle w}$) determines changes in genotype frequency. If ${\displaystyle N(t)}$ is the total population size in generation ${\displaystyle t}$, and the relevant genotype's frequency is ${\displaystyle p(t)=n(t)/N(t)}$, then

${\displaystyle p(t+1)={\frac {w}{\overline {w}}}p(t)}$,

where ${\displaystyle {\overline {w}}}$ is the mean relative fitness in the population (again setting aside changes in frequency due to drift and mutation). Relative fitnesses only indicate the change in prevalence of different genotypes relative to each other, and so only their values relative to each other are important; relative fitnesses can be any nonnegative number, including 0. It is often convenient to choose one genotype as a reference and set its relative fitness to 1. Relative fitness is used in the standard Wright–Fisher and Moran models of population genetics.

Absolute fitnesses can be used to calculate relative fitness, since ${\displaystyle p(t+1)=n(t+1)/N(t+1)=(W/{\overline {W}})p(t)}$ (we have used the fact that ${\displaystyle N(t+1)={\overline {W}}N(t)}$, where ${\displaystyle {\overline {W}}}$ is the mean absolute fitness in the population). This implies that ${\displaystyle w/{\overline {w}}=W/{\overline {W}}}$, or in other words, relative fitness is proportional to ${\displaystyle W/{\overline {W}}}$. It is not possible to calculate absolute fitnesses from relative fitnesses alone, since relative fitnesses contain no information about changes in overall population abundance ${\displaystyle N(t)}$.

Assigning relative fitness values to genotypes is mathematically appropriate when two conditions are met: first, the population is at demographic equilibrium, and second, individuals vary in their birth rate, contest ability, or death rate, but not a combination of these traits.[5]

Increase in frequency over time of genotype ${\displaystyle A}$, which has a 1% greater relative fitness than the other genotype present, ${\displaystyle B}$.

The change in genotype frequencies due to selection follows immediately from the definition of relative fitness,

${\displaystyle \Delta p=p(t+1)-p(t)={\frac {w-{\overline {w}}}{\overline {w}}}p(t)}$.

Thus, a genotype's frequency will decline or increase depending on whether its fitness is lower or greater than the mean fitness, respectively.

In the particular case that there are only two genotypes of interest (e.g. representing the invasion of a new mutant allele), the change in genotype frequencies is often written in a different form. Suppose that two genotypes ${\displaystyle A}$ and ${\displaystyle B}$ have fitnesses ${\displaystyle w_{A}}$ and ${\displaystyle w_{B}}$, and frequencies ${\displaystyle p}$ and ${\displaystyle 1-p}$, respectively. Then ${\displaystyle {\overline {w}}=w_{A}p+w_{B}(1-p)}$, and so

${\displaystyle \Delta p={\frac {w-{\overline {w}}}{\overline {w}}}p={\frac {w_{A}-w_{B}}{\overline {w}}}p(1-p)}$.

Thus, the change in genotype ${\displaystyle A}$'s frequency depends crucially on the difference between its fitness and the fitness of genotype ${\displaystyle B}$. Supposing that ${\displaystyle A}$ is more fit than ${\displaystyle B}$, and defining the selection coefficient ${\displaystyle s}$ by ${\displaystyle w_{A}=(1+s)w_{B}}$, we obtain

${\displaystyle \Delta p={\frac {w-{\overline {w}}}{\overline {w}}}p={\frac {s}{1+sp}}p(1-p)\approx sp(1-p)}$,

where the last approximation holds for ${\displaystyle s\ll 1}$. In other words, the fitter genotype's frequency grows approximately logistically.