In the theory ofevolution andnatural selection, thePrice equation (also known asPrice's equation orPrice's theorem) describes how a trait orallele changes in frequency over time. The equation uses acovariance 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. The Price equation was derived byGeorge R. Price, working in London to re-deriveW.D. Hamilton's work onkin selection. Examples of the Price equation have been constructed for various evolutionary cases. The Price equation also has applications ineconomics.^[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".

The Price equation shows that a change in the average amount${\displaystyle z}$ of a trait in a population from one generation to the next (${\displaystyle \mathrm{\Delta }z}$) is determined by thecovariance between the amounts${\displaystyle z_i}$ of the trait for subpopulation${\displaystyle i}$ and the fitnesses${\displaystyle w_i}$ of the subpopulations, together with the expected change in the amount of the trait value due to fitness, namely${\displaystyle \mathrm{E}(w_i\mathrm{\Delta }{z}_i)}$:

${\displaystyle \mathrm{\Delta }z=\frac{1}{w}\mathrm{cov}(w_i,z_i)+\frac{1}{w}\mathrm{E}(w_i\mathrm{\Delta }{z}_i).}$

Here${\displaystyle w}$ is the average fitness over the population, and${\displaystyle \mathrm{E}}$ and${\displaystyle \mathrm{cov}}$ represent the population mean and covariance respectively. 'Fitness'${\displaystyle w}$ is the ratio of the average number of offspring for the whole population per the number of adult individuals in the population, and${\displaystyle w_i}$ is that same ratio only for subpopulation${\displaystyle i}$.

If the covariance between fitness (${\displaystyle w_i}$) and trait value (${\displaystyle z_i}$) is positive, the trait value is expected to rise on average across population${\displaystyle i}$. If the covariance is negative, the characteristic is harmful, and its frequency is expected to drop.

The second term,${\displaystyle \mathrm{E}(w_i\mathrm{\Delta }{z}_i)}$, represents the portion of${\displaystyle \mathrm{\Delta }z}$ due to all factors other than direct selection which can affect trait evolution. This term can encompassgenetic drift,mutation bias, ormeiotic drive. Additionally, this term can encompass the effects of multi-level selection orgroup selection. Price (1972) referred to this as the "environment change" term, and denoted both terms using partial derivative notation (∂_NS and ∂_EC). This concept of environment includes interspecies and ecological effects. 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 onM 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.

— G.R. Price (1972),Fisher's fundamental theorem made clear^[2]

Suppose we are given four equal-length lists of real numbers^[3]${\displaystyle n_i}$,${\displaystyle z_i}$,${\displaystyle n_i^{\prime }}$,${\displaystyle z_i^{\prime }}$ from which we may define${\displaystyle w_i=n_i^{\prime }/n_i}$.${\displaystyle n_i}$ and${\displaystyle z_i}$ will be called the parent population numbers and characteristics associated with each indexi. Likewise${\displaystyle n_i^{\prime }}$ and${\displaystyle z_i^{\prime }}$ will be called the child population numbers and characteristics, and${\displaystyle w_i^{\prime }}$ will be called the fitness associated with indexi. (Equivalently, we could have been given${\displaystyle n_i}$,${\displaystyle z_i}$,${\displaystyle w_i}$,${\displaystyle z_i^{\prime }}$ with${\displaystyle n_i^{\prime }=w_i{n}_i}$.) Define the parent and child population totals:

${\displaystyle n\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sum _i{n}_i}$

${\displaystyle n^{\prime }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sum _i{n}_i^{\prime }}$

and the probabilities (or frequencies):^[4]

${\displaystyle q_i\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{n}_i/n}$

${\displaystyle q_i^{\prime }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{n}_i^{\prime }/n^{\prime }}$

Note that these are of the form of probability mass functions in that${\displaystyle \sum _i{q}_i=\sum _i{q}_i^{\prime }=1}$ and are in fact the probabilities that a random individual drawn from the parent or child population has a characteristic${\displaystyle z_i}$. Define the fitnesses:

${\displaystyle w_i\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{n}_i^{\prime }/n_i}$

The average of any list${\displaystyle x_i}$ is given by:

${\displaystyle E(x_i)=\sum _i{q}_i{x}_i}$

so the average characteristics are defined as:

${\displaystyle z\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sum _i{q}_i{z}_i}$

${\displaystyle z^{\prime }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sum _i{q}_i^{\prime }{z}_i^{\prime }}$

and the average fitness is:

${\displaystyle w\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sum _i{q}_i{w}_i}$

A simple theorem can be proved:${\displaystyle q_i{w}_i=\left(\frac{n_i}{n}\right)\left(\frac{n_i^{\prime }}{n_i}\right)=\left(\frac{n_i^{\prime }}{n^{\prime }}\right)\left(\frac{n^{\prime }}{n}\right)=q_i^{\prime }\left(\frac{n^{\prime }}{n}\right)}$so that:

${\displaystyle w=\frac{n^{\prime }}{n}\sum _i{q}_i^{\prime }=\frac{n^{\prime }}{n}}$

and

${\displaystyle q_i{w}_i=w{q}_i^{\prime }}$

The covariance of${\displaystyle w_i}$ and${\displaystyle z_i}$ is defined by:

${\displaystyle \mathrm{cov}(w_i,z_i)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}E(w_i{z}_i)-E(w_i)E(z_i)=\sum _i{q}_i{w}_i{z}_i-wz}$

Defining${\displaystyle \mathrm{\Delta }{z}_i\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{z}_i^{\prime }-z_i}$, the expectation value of${\displaystyle w_i\mathrm{\Delta }{z}_i}$ is

${\displaystyle E(w_i\mathrm{\Delta }{z}_i)=\sum q_i{w}_i(z_i^{\prime }-z_i)=\sum _i{q}_i{w}_i{z}_i^{\prime }-\sum _i{q}_i{w}_i{z}_i}$

The sum of the two terms is:

${\displaystyle \mathrm{cov}(w_i,z_i)+E(w_i\mathrm{\Delta }{z}_i)=\sum _i{q}_i{w}_i{z}_i-wz+\sum _i{q}_i{w}_i{z}_i^{\prime }-\sum _i{q}_i{w}_i{z}_i=\sum _i{q}_i{w}_i{z}_i^{\prime }-wz}$

Using the above mentioned simple theorem, the sum becomes

${\displaystyle \mathrm{cov}(w_i,z_i)+E(w_i\mathrm{\Delta }{z}_i)=w\sum _i{q}_i^{\prime }{z}_i^{\prime }-wz=w{z}^{\prime }-wz=w\mathrm{\Delta }z}$

where${\displaystyle \mathrm{\Delta }z\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{z}^{\prime }-z}$.

Consider a set of groups with${\displaystyle i=1,...,n}$ that are characterized by a particular trait, denoted by${\displaystyle x_i}$. The number${\displaystyle n_i}$ of individuals belonging to group${\displaystyle i}$ experiences exponential growth:$${\displaystyle \frac{d{n}_i}{dt}=f_i{n}_i}$$where${\displaystyle f_i}$ corresponds to the fitness of the group. We want to derive an equation describing the time-evolution of the expected value of the trait:$${\displaystyle \mathbb{E}(x)=\sum _i{p}_i{x}_i\equiv \mu ,p_i=\frac{n_i}{\sum _i{n}_i}}$$Based on thechain rule, we may derive anordinary differential equation:A further application of the chain rule for${\displaystyle d{p}_i/dt}$ gives us:Summing up the components gives us that:

which is also known as thereplicator equation. Now, note that:Therefore, putting all of these components together, we arrive at the continuous-time Price equation:$${\displaystyle \frac{d}{dt}\mathbb{E}(x)=\underset{\text{Selection effect}}{\underbrace{\text{Cov}(x,f)}}+\underset{\text{Dynamic effect}}{\underbrace{\mathbb{E}(\dot{x})}}}$$

When the characteristic values${\displaystyle z_i}$ do not change from the parent to the child generation, the second term in the Price equation becomes zero resulting in a simplified version of the Price equation:

${\displaystyle w\mathrm{\Delta }z=\mathrm{cov}\left(w_i,z_i\right)}$

which can be restated as:

${\displaystyle \mathrm{\Delta }z=\mathrm{cov}\left(v_i,z_i\right)}$

where${\displaystyle v_i}$ is the fractional fitness:${\displaystyle v_i=w_i/w}$.

This simple Price equation can be proven using the definition in Equation (2) above. 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 evolution of sight provides an example of simple directional selection. The evolution of sickle cell anemia shows how aheterozygote advantage can affect trait evolution. 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. This property is referred to as dynamical sufficiency. For simplicity, the following looks at dynamical sufficiency for the simple Price equation, but is also valid for the full Price equation.

Referring to the definition in Equation (2), the simple Price equation for the character${\displaystyle z}$ can be written:

${\displaystyle w(z^{\prime }-z)=⟨w_i{z}_i⟩-wz}$

For the second generation:

${\displaystyle w^{\prime }(z^{″}-z^{\prime })=⟨w_i^{\prime }{z}_i^{\prime }⟩-w^{\prime }{z}^{\prime }}$

The simple Price equation for${\displaystyle z}$ only gives us the value of${\displaystyle z^{\prime }}$ for the first generation, but does not give us the value of${\displaystyle w^{\prime }}$ and${\displaystyle ⟨w_i{z}_i⟩}$, which are needed to calculate${\displaystyle z^{″}}$ for the second generation. The variables${\displaystyle w_i}$ and${\displaystyle ⟨w_i{z}_i⟩}$ 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${\displaystyle w}$,${\displaystyle z}$,${\displaystyle ⟨w_i{z}_i⟩}$,${\displaystyle ⟨w_i^2⟩}$, and${\displaystyle ⟨w_i^2{z}_i}$ must be known before proceeding to calculate the three first generation variables${\displaystyle w^{\prime }}$,${\displaystyle z^{\prime }}$, and${\displaystyle ⟨w_i^{\prime }{z}_i^{\prime }⟩}$, which are needed to calculate${\displaystyle z^{″}}$ for the second generation. 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${\displaystyle ⟨w_i^n⟩}$ and${\displaystyle ⟨w_i^n{z}_i⟩}$ from the lower moments in a way that is independent of the generation. Dynamical sufficiency means that such equations can be found in the genetic model, allowing the Price equation to be used alone as a propagator of thedynamics of the model forward in time.

The simple Price equation was based on the assumption that the characters${\displaystyle z_i}$ do not change over one generation. If it is assumed that they do change, with${\displaystyle z_i}$ being the value of the character in the child population, then the full Price equation must be used. A change in character can come about in a number of ways. The following two examples illustrate two such possibilities, each of which introduces new insight into the Price equation.

We focus on the idea of the fitness of the genotype. The index${\displaystyle i}$ indicates the genotype and the number of type${\displaystyle i}$ genotypes in the child population is:

${\displaystyle n_i^{\prime }=\sum _j{w}_{ji}{n}_j}$

which gives fitness:

${\displaystyle w_i=\frac{n_i^{\prime }}{n_i}}$

Since the individual mutability${\displaystyle z_i}$ 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. Note that we now have two methods of grouping, by lineage, and by genotype. It is this complication that will introduce the need for the full Price equation. The number of children an${\displaystyle i}$-type organism has is:

${\displaystyle n_i^{\prime }=n_i\sum _j{w}_{ij}}$

which gives fitness:

${\displaystyle w_i=\frac{n_i^{\prime }}{n_i}=\sum _j{w}_{ij}}$

We now have characters in the child population which are the average character of the${\displaystyle i}$-th parent.

${\displaystyle z_j^{\prime }=\frac{\sum _i{n}_i{z}_i{w}_{ij}}{\sum _i{n}_i{w}_{ij}}}$

with global characters:

with these definitions, the full Price equation now applies.

The use of the change in average characteristic (${\displaystyle z^{\prime }-z}$) per generation as a measure of evolutionary progress is not always appropriate. There may be cases where the average remains unchanged (and the covariance between fitness and characteristic is zero) while evolution is nevertheless in progress. For example, if we have${\displaystyle z_i=(1,2,3)}$,${\displaystyle n_i=(1,1,1)}$, and${\displaystyle w_i=(1,4,1)}$, then for the child population,${\displaystyle n_i^{\prime }=(1,4,1)}$ showing that the peak fitness at${\displaystyle w_2=4}$ is in fact fractionally increasing the population of individuals with${\displaystyle z_i=2}$. However, the average characteristics arez=2 andz'=2 so that${\displaystyle \mathrm{\Delta }z=0}$. The covariance${\displaystyle \mathrm{c}\mathrm{o}\mathrm{v}(z_i,w_i)}$ is also zero. The simple Price equation is required here, and it yields0=0. In other words, it yields no information regarding the progress of evolution in this system.

A critical discussion of the use of the Price equation can be found in van Veelen (2005),^[5] van Veelenet al. (2012),^[6] and van Veelen (2020).^[7] Frank (2012) discusses the criticism in van Veelenet al. (2012).^[8]

Price's equation features in the plot and title of the 2008 thriller filmWΔZ.

The Price equation also features in posters in the computer gameBioShock 2, in which a consumer of a "Brain Boost" tonic is seen deriving the Price equation while simultaneously reading a book. The game is set in the 1950s, substantially before Price's work.

• Thebreeder's equation, which is a special case of the Price equation.

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• Evolutionary dynamics
• Evolutionary biology
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