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 formula_1 of a trait in a population from one generation to the next (formula_2) is determined by the covariance between the amounts formula_3 of the trait for subpopulation formula_4 and the fitnesses formula_5 of the subpopulations, together with the expected change in the amount of the trait value due to fitness, namely formula_6: 
Here formula_8 is the average fitness over the population, and formula_9 and formula_10 represent the population mean and covariance respectively. 'Fitness' formula_8 is the ratio of the average number of offspring for the whole population per the number of adult individuals in the population, and formula_5 is that same ratio only for subpopulation formula_4.
The covariance term captures the effects of natural selection; if the covariance between fitness (formula_5) and the trait value (formula_3) is positive then the trait value is predicted to increase on average over population formula_4. If the covariance is negative then the trait is deleterious and is predicted to decrease in frequency.
The second term, formula_6, 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.
Suppose there is a population of formula_18 individuals over which the amount of a particular characteristic varies. Those formula_18 individuals can be grouped by the amount of the characteristic that each displays. There can be as few as just one group of all formula_18 individuals (consisting of a single shared value of the characteristic) and as many as formula_18 groups of one individual each (consisting of formula_18 distinct values of the characteristic). Index each group with formula_4 so that the number of members in the group is formula_24 and the value of the characteristic shared among all members of the group is formula_3. Now assume that having formula_3 of the characteristic is associated with having a fitness formula_5 where the product formula_28 represents the number of offspring in the next generation. Denote this number of offspring from group formula_4 by formula_30 so that formula_31. Let formula_32 be the average amount of the characteristic displayed by the offspring from group formula_4. Denote the amount of change in characteristic in group formula_4 by formula_35 defined by
Now take formula_1 to be the average characteristic value in this population and formula_38 to be the average characteristic value in the next generation. Define the change in average characteristic by formula_39. That is,
Note that this is "not" the average value of formula_41 (as it is possible that formula_42). Also take formula_8 to be the average fitness of this population. The Price equation states:
where the functions formula_9 and formula_10 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 formula_8 is not zero, it is often useful to write it as
In the specific case that characteristic formula_49 (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 formula_24 is the number of occurrences of a pair of real numbers formula_51 and formula_52, then:
The notation formula_56 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 formula_4 identify the group with characteristic formula_3 and let formula_24 be the number of organisms in that group. The total number of organisms is then formula_18 where:
The average value of the characteristic formula_1 is defined as:
Now suppose that the population reproduces and the number of individuals in group formula_4 in the next generation is represented by formula_64. The so-called fitness formula_5 of group formula_4 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,
Thus, saying that a group has "higher fitness" is equivalent to saying that its members produce more offspring per individual in the next generation. Similarly, formula_67 represents the total number of individuals across all groups, which can be expressed as:
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 total population may grow, the proportion of individuals from a certain group may change. In particular, if one group has a higher fitness than another group, then the higher-fitness group will have a larger increase in representation in the next generation than the lower-fitness group. The average fitness represents how the population grows, and those groups with below-average fitness will tend to decline in proportion, while those groups with above-average fitness will tend to increase in proportion.
Along with the proportion of individuals in each group changing over time, the trait values within a single group may vary slightly from one generation to another (e.g., due to mutation). These two pressures together will cause the average value of the characteristic over the whole population to change over time. Assuming the value of formula_69 has changed by exactly the same amount for all members of the original group, in the new generation of the group, the average value formula_38 of the characteristic is:
where formula_69 are the (possibly new) values of the characteristic in group formula_4. Equation (2) shows that:
Call the change in characteristic value from parent to child populations formula_35 so that formula_74. As seen in Equation (1), the expected value operator formula_9 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:
Substituting the definition of fitness, formula_77 (Equation (4)), we get:
Next, substituting the definitions of average fitness (formula_78) from Equation (5), and average child characteristics (formula_79) from Equation (6) gives the Price equation:
Derivation of the continuous-time Price equation.
Consider a set of groups with formula_81 that are characterized by a particular trait, denoted by formula_82. The number formula_83 of individuals belonging to group formula_4 experiences exponential growth:formula_85where formula_86 corresponds to the fitness of the group. We want to derive an equation describing the time-evolution of the expected value of the trait:formula_87Based on the chain rule, we may derive an ordinary differential equation:formula_88A further application of the chain rule for formula_89 gives us:formula_90Summing up the components gives us that:formula_91
which is also known as the replicator equation. Now, note that:
formula_92Therefore, putting all of these components together, we arrive at the continuous-time Price equation:formula_93
Simple Price equation.
When the characteristic values formula_3 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:
which can be restated as:
where formula_97 is the fractional fitness: formula_98.
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 provides an example of simple directional selection. The 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 . Additionally, the Price equation is flexible enough to model second order traits such as the . 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 formula_1 can be written:
For the second generation:
The simple Price equation for formula_1 only gives us the value of formula_38 for the first generation, but does not give us the value of formula_104 and formula_105, which are needed to calculate formula_106 for the second generation. The variables formula_5 and formula_105 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 formula_8, formula_1, formula_105, formula_113, and formula_114 must be known before proceeding to calculate the three first generation variables formula_104, formula_38, and formula_117, which are needed to calculate formula_106 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 formula_119 and formula_120 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 formula_3 do not change over one generation. If it is assumed that they do change, with formula_3 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 formula_4 indicates the genotype and the number of type formula_4 genotypes in the child population is:
which gives fitness:
Since the individual mutability formula_3 does not change, the average mutabilities will be:
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 formula_4-type organism has is:
which gives fitness:
We now have characters in the child population which are the average character of the formula_4-th parent.
with global characters:
with these definitions, the full Price equation now applies.
Criticism.
The use of the change in average characteristic (formula_135) 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.