Price equation

The equation uses a covariance between a trait and fitness, to give a mathematical description of evolution and natural selection.

It provides a way to understand the effects that gene transmission and natural selection have on the frequency of alleles within each new generation of a population.

[1] The Price equation is a mathematical relationship between various statistical descriptors of population dynamics, rather than a physical or biological law, and as such is not subject to experimental verification.

In simple terms, it is a mathematical statement of the expression "survival of the fittest".

of the subpopulations, together with the expected change in the amount of the trait value due to fitness, namely

If the covariance is negative, the characteristic is harmful, and its frequency is expected to drop.

due to all factors other than direct selection which can affect trait evolution.

This term can encompass genetic drift, mutation bias, or meiotic drive.

Price describes this as follows: Fisher adopted the somewhat unusual point of view of regarding dominance and epistasis as being environment effects.

For example, he writes (1941): ‘A change in the proportion of any pair of genes itself constitutes a change in the environment in which individuals of the species find themselves.’ Hence he regarded the natural selection effect on M as being limited to the additive or linear effects of changes in gene frequencies, while everything else – dominance, epistasis, population pressure, climate, and interactions with other species – he regarded as a matter of the environment.Suppose we are given four equal-length lists of real numbers[3]

and are in fact the probabilities that a random individual drawn from the parent or child population has a characteristic

We want to derive an equation describing the time-evolution of the expected value of the trait:

Based on the chain rule, we may derive an ordinary differential equation:

It makes this fundamental statement about evolution: "If a certain inheritable characteristic is correlated with an increase in fractional fitness, the average value of that characteristic in the child population will be increased over that in the parent population."

The Price equation can describe any system that changes over time, but is most often applied in evolutionary biology.

The Price equation can also be applied to population context dependent traits such as the evolution of sex ratios.

Additionally, the Price equation is flexible enough to model second order traits such as the evolution of mutability.

The Price equation also provides an extension to Founder effect which shows change in population traits in different settlements Sometimes the genetic model being used encodes enough information into the parameters used by the Price equation to allow the calculation of the parameters for all subsequent generations.

can both be thought of as characteristics of the first generation, so the Price equation can be used to calculate them as well: The five 0-generation variables

It can be seen that in general the Price equation cannot be used to propagate forward in time unless there is a way of calculating the higher moments

being the value of the character in the child population, then the full Price equation must be used.

The following two examples illustrate two such possibilities, each of which introduces new insight into the Price equation.

genotypes in the child population is: which gives fitness: Since the individual mutability

does not change, the average mutabilities will be: with these definitions, the simple Price equation now applies.

In this case we want to look at the idea that fitness is measured by the number of children an organism has, regardless of their genotype.

with global characters: with these definitions, the full Price equation now applies.

There may be cases where the average remains unchanged (and the covariance between fitness and characteristic is zero) while evolution is nevertheless in progress.

In other words, it yields no information regarding the progress of evolution in this system.

[8] Price's equation features in the plot and title of the 2008 thriller film WΔZ.

The Price equation also features in posters in the computer game BioShock 2, in which a consumer of a "Brain Boost" tonic is seen deriving the Price equation while simultaneously reading a book.