# Reproductive value (population genetics)

Reproductive value (not to be confused with breeding value) is a concept in demography and population genetics that represents the discounted number of future female children that will be born to a woman of a specific age. Ronald Fisher first defined reproductive value in his 1930 book The Genetical Theory of Natural Selection where he proposed that future offspring be discounted at the rate of growth of the population; this implies that sexually reproductive value measures the contribution of an individual of a given age to the future growth of the population.[1][2]

## Definition

Consider a species with a life history table with survival and reproductive parameters given by ${\displaystyle \ell _{x}}$ and ${\displaystyle m_{x}}$, where

${\displaystyle \ell _{x}}$ = probability of surviving from age 0 to age ${\displaystyle x}$

and

${\displaystyle m_{x}}$ = average number of offspring produced by an individual of age ${\displaystyle x.}$

In a population with a discrete set of age classes, Fisher's reproductive value is calculated as

${\displaystyle v_{x}=\sum _{y=x}^{\infty }\lambda ^{-(y-x+1)}{\frac {\ell _{y}}{\ell _{x}}}m_{y}}$

where ${\displaystyle \lambda }$ is the long-term population growth rate given by the dominant eigenvalue of the Leslie matrix. When age classes are continuous,

${\displaystyle v(x)=\int _{x}^{\infty }e^{-r(y-x)}{\frac {\ell (y)}{\ell (x)}}m(y)dy}$

where ${\displaystyle r}$ is the intrinsic rate of increase or Malthusian growth rate.

## References

1. ^ Grafen, A (2006). "A theory of Fisher's reproductive value". J Math Biol. 53 (1): 15–60. doi:10.1007/s00285-006-0376-4. PMID 16791649.
2. ^ The Relation Between Reproductive Value and Genetic Contribution Published by the Genetics journal