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.
If the covariance between fitness (formula_5) and trait value (formula_3) is positive, the trait value is expected to rise on average across population formula_4. If the covariance is negative, the characteristic is harmful, and its frequency is expected to drop.
The second term, formula_6, represents the portion of formula_2 due to all factors other than direct selection which 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 we are given four equal-length lists of real numbers formula_19, formula_3, formula_21, formula_22 from which we may define formula_23. formula_19 and formula_3 will be called the parent population numbers and characteristics associated with each index "i". Likewise formula_21 and formula_22 will be called the child population numbers and characteristics, and formula_28 will be called the fitness associated with index "i". (Equivalently, we could have been given formula_19, formula_3, formula_5, formula_22 with formula_33.) Define the parent and child population totals:
and the probabilities (or frequencies):
Note that these are of the form of probability mass functions in that formula_34 and are in fact the probabilities that a random individual drawn from the parent or child population has a characteristic formula_3. Define the fitnesses:
The average of any list formula_37 is given by: 
so the average characteristics are defined as:
and the average fitness is:
A simple theorem can be proved:
formula_40
so that:
and
The covariance of formula_5 and formula_3 is defined by:
Defining formula_46, the expectation value of formula_47 is
The sum of the two terms is:
Using the above mentioned simple theorem, the sum becomes 
where
formula_51.
Derivation of the continuous-time Price equation.
Consider a set of groups with formula_52 that are characterized by a particular trait, denoted by formula_53. The number formula_54 of individuals belonging to group formula_4 experiences exponential growth:formula_56where formula_57 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_58Based on the chain rule, we may derive an ordinary differential equation:formula_59A further application of the chain rule for formula_60 gives us:formula_61Summing up the components gives us that:formula_62
which is also known as the replicator equation. Now, note that:
formula_63Therefore, putting all of these components together, we arrive at the continuous-time Price equation:formula_64
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_68 is the fractional fitness: formula_69.
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_74 for the first generation, but does not give us the value of formula_75 and formula_76, which are needed to calculate formula_77 for the second generation. The variables formula_5 and formula_76 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_76, formula_84, and formula_85 must be known before proceeding to calculate the three first generation variables formula_75, formula_74, and formula_88, which are needed to calculate formula_77 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_90 and formula_91 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_106) 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 formula_107, formula_108, and formula_109, then for the child population, formula_110 showing that the peak fitness at formula_111 is in fact fractionally increasing the population of individuals with formula_112. However, the average characteristics are "z=2" and "z'=2" so that formula_113. The covariance formula_114 is also zero. The simple Price equation is required here, and it yields "0=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), 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.