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 of a trait in a population from one generation to the next () is determined by the covariance between the amounts of the trait for subpopulation and the fitnesses of the subpopulations, together with the expected change in the amount of the trait value due to fitness, namely :
Here is the average fitness over the population, and and represent the population mean and covariance respectively. 'Fitness' is the ratio of the average number of offspring for the whole population per the number of adult individuals in the population, and is that same ratio only for subpopulation .
If the covariance between fitness () and trait value () is positive, the trait value is expected to rise on average across population . If the covariance is negative, the characteristic is harmful, and its frequency is expected to drop.
The second term, , represents the portion of 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 , , , from which we may define . and will be called the parent population numbers and characteristics associated with each index i. Likewise and will be called the child population numbers and characteristics, and will be called the fitness associated with index i. (Equivalently, we could have been given , , , with .) Define the parent and child population totals: { |cellspacing=20 |- | || |} and the probabilities (or frequencies): { |cellspacing=20 |- | || |} Note that these are of the form of probability mass functions in that and are in fact the probabilities that a random individual drawn from the parent or child population has a characteristic . Define the fitnesses:
The average of any list is given by:
so the average characteristics are defined as: { |cellspacing=20 |- | || |} and the average fitness is:
A simple theorem can be proved:
so that:
and
The covariance of and is defined by:
Defining , the expectation value of is
The sum of the two terms is:
Using the above mentioned simple theorem, the sum becomes
where
.
Derivation of the continuous-time Price equation
Consider a set of groups with that are characterized by a particular trait, denoted by . The number of individuals belonging to group experiences exponential growth:where corresponds to the fitness of the group. 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:A further application of the chain rule for gives us:Summing up the components gives us that:
which is also known as the replicator equation. Now, note that: Therefore, putting all of these components together, we arrive at the continuous-time Price equation: