Fisher's reproductive value

Fisher's reproductive value was defined by R. A. Fisher (1930) 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


 * $$v_x = \mbox{either }\frac{\sum_{y=x}^\infty \ell_y m_y}{R}\mbox{ or }\frac{\int_{y=x}^\infty \ell_y m_y\,dy}{R}$$

where


 * $$R = \mbox{ either }\sum_{y=0}^\infty \ell_y m_y\mbox{ or } \int_0^\infty \ell_x m_x\,dx,$$

the net reproductive rate of the population.

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


 * $$T = \mbox{either }\sum_{y=0}^\infty \ell_y v_y\mbox{ or } \int_0^\infty \ell_x v_x\,dx.$$