Price equation

In the theory of evolution and natural selection, the Price equation (also known as Price's equation or Price's theorem) describes how a trait or allele changes in frequency over time. 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. The Price equation was derived by George R. Price, working in London to re-derive W.D. Hamilton's work on kin selection. Examples of the Price equation have been constructed for various evolutionary cases. The Price equation also has applications in economics.

It is important to note that the Price equation is not a physical or biological law. It is not a concise or general expression of experimentally validated results. It is rather a purely mathematical relationship between various statistical descriptors of population dynamics. It is mathematically valid, and therefore not subject to experimental verification. In simple terms, it is a mathematical restatement of the expression "survival of the fittest" which is actually self-evident, given the mathematical definitions of "survival" and "fittest".

Statement

The Price equation shows that a change in the average amount $z$ of a trait in a population from one generation to the next ($\Delta z$) is determined by the covariance between the amounts $z_i$ of the trait for subpopulation $i$ and the fitnesses $w_i$ of the subpopulations, together with the expected change in the amount of the trait value due to fitness, namely $\mathrm(w_i \Delta z_i)$:
:$\Delta = \frac\operatorname(w_i, z_i) + \frac\operatorname(w_i\,\Delta z_i).$
Here $w$ is the average fitness over the population, and $\operatorname$ and $\operatorname$ represent the population mean and covariance respectively. 'Fitness' $w$ is the ratio of the average number of offspring for the whole population per the number of adult individuals in the population, and $w_i$ is that same ratio only for subpopulation $i$.

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

The second term, $\mathrm(w_i \Delta z_i)$, represents factors other than direct selection that can affect trait evolution. This term can encompass genetic drift, mutation bias, or meiotic drive. Additionally, this term can encompass the effects of multi-level selection or group 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:

Proof

Assume there is an n-person population in which the amount of a certain feature changes. Those n people may be divided into groups based on how much of each feature they have. There can be just one group of all n people (consisting of a single common characteristic value) or as many as n groups of one individual each (consisting of n distinct values of the characteristic). Index each group with $i$ so that the number of members in the group is $n_i$ and the value of the characteristic shared among all members of the group is $z_i$. Now assume that having $z_i$ of the characteristic is associated with having a fitness $w_i$ where the product $w_i n_i$ represents the number of offspring in the next generation. Denote this number of offspring from group $i$ by $n_i'$ so that $w_i=n_i'/n_i$. Let $z_i'$ be the average amount of the characteristic displayed by the offspring from group $i$. Denote the amount of change in characteristic in group $i$ by $\Delta z_i$ defined by

:$\Delta \;\stackrel\; z_i' - z_i$

Now take $z$ to be the average characteristic value in this population and $z'$ to be the average characteristic value in the next generation. Define the change in average characteristic by $\Delta$. That is,

:$\Delta \;\stackrel\; z' - z$

Note that this is ''not'' the average value of $\Delta$ (as it is possible that $n_i \neq n'_i$). Also take $w$ to be the average fitness of this population. The Price equation states:

:$w\,\Delta = \operatorname(w_i, z_i) + \operatorname(w_i\,\Delta z_i)$

where the functions $\operatorname$ and $\operatorname$ are respectively defined in Equations (1) and (2) below and are equivalent to the traditional definitions of sample mean and covariance; however, they are not meant to be statistical estimates of characteristics of a population. In particular, the Price equation is a deterministic difference equation that models the trajectory of the actual mean value of a characteristic along the flow of an actual population of individuals. Assuming that the mean fitness $w$ is not zero, it is often useful to write it as

:$\Delta = \frac\operatorname(w_i, z_i) + \frac\operatorname(w_i\,\Delta z_i)$

In the specific case that characteristic $z_i = w_i$ (i.e., fitness itself is the characteristic of interest), then Price's equation reformulates Fisher's fundamental theorem of natural selection.

To prove the Price equation, the following definitions are needed. If $n_i$ is the number of occurrences of a pair of real numbers $x_i$ and $y_i$, then:

* The mean of the $x_i$ values is:
:

* The covariance between the $x_i$ and $y_i$ values is:
:

The notation $\langle x_i \rangle = \operatorname(x_i)$ will also be used when
convenient.

Suppose there is a population of organisms all of which have a genetic characteristic described by some real number. For example, high values of the number represent an increased visual acuity over some other organism with a lower value of the characteristic. Groups can be defined in the population which are characterized by having the same value of the characteristic. Let subscript $i$ identify the group with characteristic $z_i$ and let $n_i$ be the number of organisms in that group. The total number of organisms is then $n$ where:

:$n = \sum_i n_i$

The average value of the characteristic $z$ is defined as:

:

Assume that the population grows and the number of persons in each group grows. $i$ in the next generation is represented by $n'_i$. The so-called fitness $w_i$ of group $i$ is defined to be the ratio of the number of its individuals in the next generation to the number of its individuals in the previous generation. That is,

:

As a result, a group's "greater fitness" equates to its members producing more offspring per individual in the following generation. Similarly, n' stands for the total number of people in all groups, which may be written as:

:$n' = \sum_i n'_i$

Furthermore, the average fitness of the population can be shown to be the growth rate of the population as a whole, as in:

:

Although the overall population may increase, the proportion of people in each category may alter. If one group is more physically fit than another, the higher-fitness group will have a greater rise in representation in the following generation than the lower-fitness group. The average fitness indicates how the population increases, and individuals with below-average fitness will see their proportions drop, while those with above-average fitness will see their proportions increase.

In addition to the number of people in each group shifting over time, characteristic values within a single group may fluctuate slightly from generation to generation (e.g., due to mutation). Together, these two forces will lead the characteristic's average value to rise across the board.

:

where $z'_i$ are the (possibly new) values of the characteristic in group $i$. Equation (2) shows that:

:

Call the change in characteristic value from parent to child populations $\Delta z_i$ so that $\Delta z_i = z'_i - z_i$. As seen in Equation (1), the expected value operator $\operatorname$ is linear, so

:

Combining Equations (7) and (8) leads to

:

Now, let's compute the first term in the equality above. From Equation (1), we know that:

:$\operatorname(w_iz'_i) = \frac\sum_i w_i z'_i n_i$

Substituting the definition of fitness, $w_i=\frac$ (Equation (4)), we get:

:

Next, substituting the definitions of average fitness ($w=\frac$) from Equation (5), and average child characteristics ($z'$) from Equation (6) gives the Price equation:

:$\operatorname(w_i, z_i) + \operatorname(w_i\,\Delta z_i) = wz' - wz = w\,\Delta z\,$

Derivation of the continuous-time Price equation

Consider a set of groups with $i = 1,...,n$ that are characterized by a particular trait, denoted by $x_$. The number $n_$ of individuals belonging to group $i$ experiences exponential growth:$= f_n_$where $f_$ corresponds to the fitness of the group. We want to derive an equation describing the time-evolution of the expected value of the trait:$\mathbb(x) = \sum_p_x_ \equiv \mu, \quad p_ =$Based on the chain rule, we may derive an ordinary differential equation:$\begin &= \sum_ + \sum_ \\ &= \sum_ x_ + \sum_ p_ \\ &= \sum_ x_ + \mathbb\left( \right) \end$A further application of the chain rule for $dp_/dt$ gives us:$= \sum_, \quad = \begin -p_/N, \quad &i\neq j \\ (1-p_)/N, \quad &i=j \end$Summing up the components gives us that:\begin
&= p_\left(f_ - \sum_p_f_\right) \\
&= p_\left _ - \mathbb(f)\right\end

which is also known as the replicator equation. Now, note that:
\begin
\sum_ x_ &= \sum_ p_x_\left _ - \mathbb(f)\right\\
&= \mathbb\left\ \\
&= \text(x,f)
\endTherefore, putting all of these components together, we arrive at the continuous-time Price equation:$\mathbb(x) = \underbrace_ + \underbrace_$

Simple Price equation

When the characteristic values $z_i$ 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:

:$w\,\Delta z = \operatorname\left(w_i, z_i\right)$

which can be restated as:

:$\Delta z = \operatorname\left(v_i, z_i\right)$

where $v_i$ is the fractional fitness: $v_i=w_i/w$.

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."

Applications

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 a heterozygote 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

Dynamical sufficiency and the simple Price equation

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 $z$ can be written:
:$w(z' - z) = \langle w_i z_i \rangle - wz$

For the second generation:
:$w'(z'' - z') = \langle w'_i z'_i \rangle - w'z'$

The simple Price equation for $z$ only gives us the value of $z'$ for the first generation, but does not give us the value of $w'$ and $\langle w_iz_i\rangle$, which are needed to calculate $z''$ for the second generation. The variables $w_i$ and $\langle w_iz_i\rangle$ can both be thought of as characteristics of the first generation, so the Price equation can be used to calculate them as well:

:$\begin w(w' - w) &= \langle w_i^2\rangle - w^2 \\ w\left(\langle w'_i z'_i\rangle - \langle w_i z_i\rangle\right) &= \langle w_i ^2 z_i\rangle - w\langle w_i z_i\rangle \end$

The five 0-generation variables $w$, $z$, $\langle w_iz_i\rangle$, $\langle w_i^2\rangle$, and $\langle w_i^2z_i$ must be known before proceeding to calculate the three first generation variables $w'$, $z'$, and $\langle w'_iz'_i\rangle$, which are needed to calculate $z''$ 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 $\langle w_i^n\rangle$ and $\langle w_i^nz_i\rangle$ 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 the dynamics of the model forward in time.

Full Price equation

The simple Price equation was based on the assumption that the characters $z_i$ do not change over one generation. If it is assumed that they do change, with $z_i$ 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.

Genotype fitness

We focus on the idea of the fitness of the genotype. The index $i$ indicates the genotype and the number of type $i$ genotypes in the child population is:
:$n'_i = \sum_j w_n_j\,$

which gives fitness:
:$w_i = \frac$

Since the individual mutability $z_i$ does not change, the average mutabilities will be:

:$\begin z &= \frac\sum_i z_i n_i \\ z' &= \frac\sum_i z_i n'_i \end$

with these definitions, the simple Price equation now applies.

Lineage fitness

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 $i$-type organism has is:

:$n'_i = n_i\sum_j w_\,$

which gives fitness:

:$w_i = \frac = \sum_j w_$

We now have characters in the child population which are the average character of the $i$-th parent.
:$z'_j = \frac$

with global characters:

:$\begin z &= \frac\sum_i z_i n_i \\ z' &= \frac\sum_i z_i n'_i \end$

with these definitions, the full Price equation now applies.

Criticism

The use of the change in average characteristic ($z'-z$) 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.

A critical discussion of the use of the Price equation can be found in van Veelen (2005), van Veelen ''et al''. (2012), and van Veelen (2020). Frank (2012) discusses the criticism in van Veelen ''et al''. (2012).

Cultural references

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. The game is set in the 1950s, substantially before Price's work.

See also

* Price equation examples

References

Further reading

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{{Population genetics

Equations
Evolutionary dynamics
Evolutionary biology
Population genetics