In the theory of evolution and natural selection, the Price equation (also known as Price's equation or Price's theorem) describes how a "characteristic" of a population changes in frequency over time as the result of reproduction and natural selection. A characteristic may be a physical or behavioral trait (phenotype) or a particular genetic makeup (allele). The equation uses a covariance between a characteristic 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 and/or phenotypes 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. For example Collins and Gardner use the Price equation to partition the total change in toxin resistance in microbial communities into evolutionary change, ecological change and physiological change. Ellner et al. use the Price equation to disentangle "ecological impacts of evolution vs. non-heritable trait change", using examples from data on birds, fish and zooplankton. The Price equation also has applications in economics.
Statement
The Price equation shows that a change in the average amount z {\displaystyle z} of a characteristic in a population from one generation to the next ( Δ z {\displaystyle \Delta z} ) is determined by the covariance between the amounts z i {\displaystyle z_{i}} of the characteristic for subpopulation i {\displaystyle i} and the fitnesses w i {\displaystyle w_{i}} of the subpopulations, together with the expected change in the characteristic due to fitness, namely E ( w i Δ z i ) {\displaystyle \mathrm {E} (w_{i}\Delta z_{i})} :
Δ z = 1 w cov ( w i , z i ) + 1 w E ( w i Δ z i ) . {\displaystyle \Delta {z}={\frac {1}{w}}\operatorname {cov} (w_{i},z_{i})+{\frac {1}{w}}\operatorname {E} (w_{i}\,\Delta z_{i}).}
Here w {\displaystyle w} is the average fitness over the population, and E {\displaystyle \operatorname {E} } and cov {\displaystyle \operatorname {cov} } represent the population mean and covariance respectively. 'Fitness' w {\displaystyle 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 {\displaystyle w_{i}} is that same ratio only for subpopulation i {\displaystyle i} . If the covariance between fitness ( w i {\displaystyle w_{i}} ) and the characteristic ( z i {\displaystyle z_{i}} ) is positive, the characteristic is expected to rise on average across population i {\displaystyle i} . If the covariance is negative, the characteristic is harmful, and its frequency is expected to drop. By noting that the covariance is the standard deviation multiplied by the correlation ( cov ( X , Y ) = r X , Y σ X σ Y {\displaystyle \operatorname {cov} (X,Y)=r_{X,Y}\sigma _{X}\sigma _{Y}} ) we can rewrite this as
Δ z σ z = r w , z w / σ w + 1 w / σ w E ( w i σ w Δ z i σ z ) {\displaystyle {\frac {\Delta {z}}{\sigma _{z}}}={\frac {r_{w,z}}{w/\sigma _{w}}}+{\frac {1}{w/\sigma _{w}}}\operatorname {E} \left({\frac {w_{i}}{\sigma _{w}}}{\frac {\Delta z_{i}}{\sigma _{z}}}\right)}
Δ z σ z = C V w r w , z + C V w E ( w i σ w Δ z i σ z ) {\displaystyle {\frac {\Delta {z}}{\sigma _{z}}}=CV_{w}r_{w,z}+CV_{w}\operatorname {E} \left({\frac {w_{i}}{\sigma _{w}}}{\frac {\Delta z_{i}}{\sigma _{z}}}\right)}
where C V w = σ w w {\displaystyle CV_{w}={\frac {\sigma _{w}}{w}}} is the coefficient of variation of the fitness. That is, we can rewrite the quantities in standardized form. We see that if the correlation between the characteristic and fitness is held constant, more variance increases the magnitude of selection. The second term, E ( w i Δ z i ) {\displaystyle \mathrm {E} (w_{i}\Delta z_{i})} , represents the portion of Δ z {\displaystyle \Delta z} due to all factors other than direct selection which can affect the evolution of the characteristic. 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.
Proof As with any concept, the more general its application, the more difficult it usually is to understand. The Price equation is no exception, and its proof is rather extended and detailed. There are a number of equations in evolutionary biology which are special cases of the Price equation and are more easily understood, such as the Robertson-Price identity (Simple Price equation), the breeder's equation, and Fisher's fundamental theorem of natural selection. The Price equation relies on a simplified description of the properties of evolutionary systems. These can be summarized as:
Individuals in a population have characteristic that can be quantified. The average characteristic of the population changes over time (such as between a past observation and a present observation). Individuals in the past (i.e. ancestors) are associated with some individuals in the present (descendants). All these properties can be summarized in a diagram representing two generations, in which the ancestors are parents and the children are the descendants.
This simplified description of evolution can be formalized by studying four equal-length lists of real numbers, the abundance of individuals of type i {\displaystyle i} in the past, n i {\displaystyle n_{i}} , the characteristic of individuals of type i {\displaystyle i} in the past z i {\displaystyle z_{i}} , the abundance of individuals of type i {\displaystyle i} in the present n i ′ {\displaystyle n_{i}'} , and the characteristic of individuals of type i {\displaystyle i} in the present z i ′ {\displaystyle z_{i}'} . From this from we may define w i = n i ′ / n i {\displaystyle w_{i}=n_{i}'/n_{i}} . n i {\displaystyle n_{i}} and z i {\displaystyle z_{i}} will be called the parent population numbers and characteristics associated with each index i. Likewise n i ′ {\displaystyle n_{i}'} and z i ′ {\displaystyle z_{i}'} will be called the child population numbers and characteristics, and w i ′ {\displaystyle w_{i}'} will be called the fitness associated with index i. (Equivalently, we could have been given n i {\displaystyle n_{i}} , z i {\displaystyle z_{i}} , w i {\displaystyle w_{i}} , z i ′ {\displaystyle z_{i}'} with n i ′ = w i n i {\displaystyle n_{i}'=w_{i}n_{i}} .) 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 ∑ i q i = ∑ i q i ′ = 1 {\displaystyle \sum _{i}q_{i}=\sum _{i}q_{i}'=1} and are in fact the probabilities that a random individual drawn from the parent or child population has a characteristic z i {\displaystyle z_{i}} . Define the fitnesses:
w i = d e f n i ′ / n i {\displaystyle w_{i}\;{\stackrel {\mathrm {def} }{=}}\;n_{i}'/n_{i}}
The average of any list x i {\displaystyle x_{i}} is given by:
E ( x i ) = ∑ i q i x i {\displaystyle E(x_{i})=\sum _{i}q_{i}x_{i}}
so the average characteristics are defined as:
and the average fitness is:
w = d e f ∑ i q i w i {\displaystyle w\;{\stackrel {\mathrm {def} }{=}}\;\sum _{i}q_{i}w_{i}}
A simple theorem can be proved:
q i w i = ( n i n ) ( n i ′ n i ) = ( n i ′ n ′ ) ( n ′ n ) = q i ′ ( n ′ n ) {\displaystyle q_{i}w_{i}=\left({\frac {n_{i}}{n}}\right)\left({\frac {n_{i}'}{n_{i}}}\right)=\left({\frac {n_{i}'}{n'}}\right)\left({\frac {n'}{n}}\right)=q_{i}'\left({\frac {n'}{n}}\right)}
so that:
w = n ′ n ∑ i q i ′ = n ′ n {\displaystyle w={\frac {n'}{n}}\sum _{i}q_{i}'={\frac {n'}{n}}}
and
q i w i = w q i ′ {\displaystyle q_{i}w_{i}=w\,q_{i}'}
The covariance of w i {\displaystyle w_{i}} and z i {\displaystyle z_{i}} is defined by:
cov ( w i , z i ) = d e f E ( w i z i ) − E ( w i ) E ( z i ) = ∑ i q i w i z i − w z {\displaystyle \operatorname {cov} (w_{i},z_{i})\;{\stackrel {\mathrm {def} }{=}}\;E(w_{i}z_{i})-E(w_{i})E(z_{i})=\sum _{i}q_{i}w_{i}z_{i}-wz}
Defining Δ z i = d e f z i ′ − z i {\displaystyle \Delta z_{i}\;{\stackrel {\mathrm {def} }{=}}\;z_{i}'-z_{i}} , the expectation value of w i Δ z i {\displaystyle w_{i}\Delta z_{i}} is
E ( w i Δ z i ) = ∑ i q i w i ( z i ′ − z i ) = ∑ i q i w i z i ′ − ∑ i q i w i z i {\displaystyle E(w_{i}\Delta z_{i})=\sum _{i}q_{i}w_{i}(z_{i}'-z_{i})=\sum _{i}q_{i}w_{i}z_{i}'-\sum _{i}q_{i}w_{i}z_{i}}
The sum of the two terms is:
cov ( w i , z i ) + E ( w i Δ z i ) = ∑ i q i w i z i − w z + ∑ i q i w i z i ′ − ∑ i q i w i z i = ∑ i q i w i z i ′ − w z {\displaystyle \operatorname {cov} (w_{i},z_{i})+E(w_{i}\Delta z_{i})=\sum _{i}q_{i}w_{i}z_{i}-wz+\sum _{i}q_{i}w_{i}z_{i}'-\sum _{i}q_{i}w_{i}z_{i}=\sum _{i}q_{i}w_{i}z_{i}'-wz}
Using the above mentioned simple theorem, the sum becomes
cov ( w i , z i ) + E ( w i Δ z i ) = w ∑ i q i ′ z i ′ − w z = w z ′ − w z = w Δ z {\displaystyle \operatorname {cov} (w_{i},z_{i})+E(w_{i}\Delta z_{i})=w\sum _{i}q_{i}'z_{i}'-wz=wz'-wz=w\Delta z}
where
Δ z = d e f z ′ − z {\displaystyle \Delta z\;{\stackrel {\mathrm {def} }{=}}\;z'-z} .
Derivation of the continuous-time Price equation Consider a set of subgroups with populations of size x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\dots ,x_{n}} . Suppose they each grow exponentially over time with their own rates, r 1 , r 2 , … , r n {\displaystyle r_{1},r_{2},\dots ,r_{n}} : x ˙ i = d e f r i x i . {\displaystyle {\dot {x}}_{i}\;{\overset {\underset {\mathrm {def} }{}}{=}}\;r_{i}x_{i}.} Furthermore, suppose there is some characteristic z {\displaystyle z} that each individual presents, where z {\displaystyle z} is represented as a real number. We want to describe the evolution of the expected value of z {\displaystyle z} over the entire population, E [ z ] {\displaystyle \mathbb {E} [z]} . We can expand this using the definition of an expected value: E [ z ] = x i z i ∑ i = 1 n x i = ∑ i = 1 n f i z i , {\displaystyle \mathbb {E} [z]={\frac {x_{i}z_{i}}{\sum _{i=1}^{n}x_{i}}}=\sum _{i=1}^{n}f_{i}z_{i},} where we are defining z i {\displaystyle z_{i}} to be the expected value of z {\displaystyle z} over subgroup i {\displaystyle i} and f i = x i / ∑ j = 1 n x j {\textstyle f_{i}=x_{i}/\sum _{j=1}^{n}x_{j}} to be the relative size of subgroup i {\displaystyle i} . Notice that both f i {\displaystyle f_{i}} and z i {\displaystyle z_{i}} may vary over time. Using the product rule, this means that the derivative of E [ z ] {\displaystyle \mathbb {E} [z]} has two parts to it: d d t E [ z ] = ∑ i = 1 n ( f ˙ i z i + f i z ˙ i ) , = ( ∑ i = 1 n f ˙ i z i ) + ( ∑ i = 1 n f i z ˙ i ) . {\displaystyle {\begin{aligned}{\frac {d}{dt}}\mathbb {E} [z]&=\sum _{i=1}^{n}({\dot {f}}_{i}z_{i}+f_{i}{\dot {z}}_{i}),\\&=\left(\sum _{i=1}^{n}{\dot {f}}_{i}z_{i}\right)+\left(\sum _{i=1}^{n}f_{i}{\dot {z}}_{i}\right).\end{aligned}}} The second term is just E [ z ˙ ] {\displaystyle \mathbb {E} [{\dot {z}}]} , the expected change in the characteristic z i {\displaystyle z_{i}} per group i {\displaystyle i} , averaged across groups by their current sizes. It accounts for z {\displaystyle z} varying within subgroups but not the variation in z {\displaystyle z} due to some subgroups growing faster than others. The first term accounts for this selection pressure. To calculate the first term, we can use the chain rule: f ˙ i = ∑ j = 1 n d x j d t ∂ f i ∂ x j = ∑ j = 1 n ( r i x i ) ∂ f i ∂ x j . {\displaystyle {\dot {f}}_{i}=\sum _{j=1}^{n}{\frac {dx_{j}}{dt}}{\frac {\partial f_{i}}{\partial x_{j}}}=\sum _{j=1}^{n}(r_{i}x_{i}){\frac {\partial f_{i}}{\partial x_{j}}}.} We can derive that ∂ f i ∂ x j = { ( 1 − f i ) / ∑ k = 1 n x k if i = j , − f i / ∑ k = 1 n x k otherwise , {\displaystyle {\frac {\partial f_{i}}{\partial x_{j}}}={\begin{cases}(1-f_{i})/\sum _{k=1}^{n}x_{k}&{\text{if }}i=j,\\-f_{i}/\sum _{k=1}^{n}x_{k}&{\text{otherwise}},\end{cases}}} so that f ˙ i = ( r i x i ) − f i ∑ j = 1 n (