Bregman divergence

Fisher's reproductive value was defined by R. A. Fisher in his 1930 book The Genetical Theory of Natural Selection as the expected reproduction of an individual from their current age onward, given that they have survived to their current age. It is used in describing populations with age structure.

Definition

Consider a species with a life history table with survival and reproductive parameters given by ${\ell }_x$ and $m_x$, where

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

and

$m_x$ = average number of offspring produced by an individual of age $x.$

Depending on whether the breeding is discrete or continuous, Fisher's reproductive value is calculated as

$v_x=\text{either }\frac{\sum _{y=x}^{\mathrm{\infty }}{\ell }_y{m}_y}{R}\text{ or }\frac{\underset{y=x}{\overset{\mathrm{\infty }}{\int }}{\ell }_y{m}_{y}dy}{R}$

where

$R=\text{ either }\sum _{y=0}^{\mathrm{\infty }}{\ell }_y{m}_y\text{ or }\underset{0}{\overset{\mathrm{\infty }}{\int }}{\ell }_x{m}_{x}dx,$

the net reproductive rate of the population.

The average age of a reproducing adult is the generation time and is

$T=\text{either }\sum _{y=0}^{\mathrm{\infty }}{\ell }_y{v}_y\text{ or }\underset{0}{\overset{\mathrm{\infty }}{\int }}{\ell }_x{v}_{x}dx.$

See also

• Effective population size
• Senescence

References

Fisher, R. A. (1930) The Genetical Theory of Natural Selection. Oxford University Press, Oxford.