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## Price equation
The Suppose there is a population of Now take Note that this is where the functions and are respectively defined in Equations (1) and (2) below but are In the specific case that characteristic Price's equation is, importantly, a tautology. It is a statement of mathematical fact between certain variables, and its value lies in the insight gained by assigning certain values encountered in evolutionary genetics to the variables. For example, the statement "if every pair of birds has two offspring, then among ten pairs of birds there will be twenty offspring" is a tautology. It doesn't really impart any new information about birds so much as it organizes our concepts about birds and their offspring. The Price equation is much more sophisticated than the above statement, but at its core, it too is a mathematically provable tautology. The Price equation also has applications in economics. ## Additional recommended knowledge
## Proof of the Price equationTo prove the Price equation, the following definitions are needed. If - The average or expected value of the
*x*_{i}values is:
- The covariance between the
*x*_{i}and*y*_{i}values is:
The notation 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 The average value of the characteristic is
Now suppose that the population reproduces, all parents are eliminated, and then there is a selection process on the children, by which less fit children are removed from the reproducing population. After reproduction and selection, the population numbers for the child groups will change to The total number of children is The fitness of group
with average fitness of the population being
The average value of the child characteristic will be
where
Call the change in characteristic value from parent to child populations Δ
Combining Equations (7) and (8) leads to
but from Equation (1) gives: and from Equation (4) gives:
Applying Equations (5) and (6) to Equation (10) and then applying the result to Equation (9) gives the Price Equation: ## Simple Price equationWhen the characters which can be restated as: where ## Example: Evolution of sightAs an example of the simple Price equation, consider a model for the evolution of sight. Suppose
**i****0****1****2***n*_{i}10 20 30 *z*_{i}3 2 1
Using Equation (4), the
child population (assuming the character **i****0****1****2***n*_{i}30 40 30 *z*_{i}3 2 1
We would like to know how much average visual acuity has increased or decreased in the population. From Equation (3), the average sightedness of the parent population is which indicates that the trait of sightedness is
increasing in the population. Applying the Price equation we
have (since ## Dynamical sufficiency and the simple Price equationSometimes 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 For the second generation: The simple Price equation for The five 0-generation variables ## Example: Evolution of sickle cell anemia
As an example of dynamical sufficiency, consider the case of sickle cell anemia. Each person has two sets of genes, one set inherited from the father, one from the mother. Sickle cell anemia is a blood disorder which occurs when a particular pair of genes both carry the 'sickle-cell trait'. The reason that the sickle-cell gene has not been eliminated from the human population by selection is because when there is only one of the pair of genes carrying the sickle-cell trait, that individual (a "carrier") is highly resistant to malaria, while a person who has neither gene carrying the sickle-cell trait will be susceptible to malaria. Let's see what the Price equation has to say about this. Let - type 0 children (unaffected)
- type 1 children (carriers)
- type 2 children (affected)
Suppose a fraction To find the concentration where In other words the ratio of carriers to non-carriers will be equal to the above constant non-zero value. In the absence of malaria, The situation has been effectively determined for the infinite (equilibrium) generation. This means that there is dynamical sufficiency with respect to the Price equation, and that there is an equation relating higher moments to lower moments. For example, for the second moments: ## Example: sex ratiosIn a 2-sex species or deme with sexes 1 and 2 where ## Full Price equationThe simple Price equation was based on the assumption that the characters
## Examples## Evolution of altruismTo study the evolution of a genetic predisposition to altruism, altruism will be defined as the genetic predisposition to behavior which decreases individual fitness while increasing the average fitness of the group to which the individual belongs. First specifying a simple model, which will only require the simple Price equation. Specify a fitness where where var( It can be seen that, by this model, in order for altruism to persist it must be uniform throughout the group. If there are two altruist types the average altruism of the group will decrease, the more altruistic will lose out to the less altruistic. Now assuming a hierarchy of groups which will require the full Price equation. The population will be divided into groups, labelled with index The and global averages: It can be seen that since the In this case, the first term isolates the advantage to each group conferred by having altruistic members. The second term isolates the loss of altruistic members from their group due to their altruistic behavior. The second term will be negative. In other words there will be an average loss of altruism due to the in-group loss of altruists, assuming that the altruism is not uniform across the group. The first term is: In other words, for ## Evolution of mutabilitySuppose there is an environment containing two kinds of food. Let α be the amount of the first kind of food and β be the amount of the second kind. Suppose an organism has a single allele which allows it to utilize a particular food. The allele has four gene forms: Let α 0 0 0 mα (1−3 *m*)α*m*β*m*β0 0 β 0 *m*α*m*α*m*β(1−3 *m*)β
Mutators are at a disadvantage when the food supplies α and β are constant. They lose every generation compared to the non-mutating genes. But when the food supply varies, even though the mutators lose relative to an With the introduction of mutability, the question of identity versus lineage arises. Is fitness measured by the number of children an individual has, regardless of the children's genetic makeup, or is fitness the child/parent ratio of a particular genotype?. Fitness is itself a characteristic, and as a result, the Price equation will handle both. Suppose we want to examine the evolution of mutator genes. Define the in other words, 0 for non-mutator genes, 1 for mutator genes. There are two cases: ## Genotype fitnessLets focus on the idea of the fitness of the genotype. The index which gives fitness: Since the individual mutability with these definitions, the simple Price equation now applies. ## Lineage fitnessIn 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 which gives fitness: We now have characters in the child population which are the average character of the with global characters: with these definitions, the full Price equation now applies. ## References- Frank, S.A. (1995). "George Price's contributions to Evolutionary Genetics".
*Journal of Theoretical Biology***175**: 373-388.
- Frank, S.A. (1997). "The Price Equation, Fisher's Fundamental Theorem, Kin Selection, and Causal Analysis".
*Evolution***51**(6): 1712-1729.
- Grafen, M. (2000). "Developments of the Price equation and natural selection under uncertainty".
*Proc. R. Soc. London B***267**: 1223-1227.
- Langdon, W. B. (1998). "Evolution of GP Populations: Price's Selection and Covariance Theorem".
*Genetic Programming and Data Structures*: 167-208.
- Price, G.R. (1970). "Selection and covariance".
*Nature***227**: 520-521.
- Price, G.R. (1972). "Fisher's "fundamental theorem" made clear".
*Annals of Human Genetics***36**: 129-140.
- Van Veelen, M. (2005). "On the use of the Price equation".
*Journal of Theoretical Biology***237**: 412-426.
Categories: Evolutionary dynamics | Evolutionary biology | Population genetics |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Price_equation". A list of authors is available in Wikipedia. |