Talk:Reproductive value (population genetics)

Removed from page
I removed comments and questions from the article to here for further discussion. --TeaDrinker 00:39, 11 November 2007 (UTC)

The verbal and mathematical definitions are wrong!
Gafox1 (talk) 14:44, 2 June 2010 (UTC) And not what Fisher wrote, either. Not correct either for $$v_x$$ or for the more generally useful $$\frac{v_x}{v_0}$$. Are you making some extreme simplifying assumptions like \lambda =1 and population at stable age distribution?

Verbal definitions:
 * 1) A rough definition: expected contribution of an individual in class x to future population size.
 * 2) More precise definition: (proportion of future births in the population to individuals now in class x)/(proportion of the population now in class x)

This involves taking into account the rate at which the population is growing -- which is why Fisher's formula isn't the same as yours. There are a lot of ways to write this out, but they all end up with some terms involving (discrete time) λ or (continuous time) exp(r). Using my 2nd definition above, we have (I'll write it out in LaTeX notation)
 * Numerator: $$\sum_{y=x}^{\infty} \lambda^{-y} \ell_{y} m_{y}$$
 * Denominator: $$ \frac{\ell_{x}}{\lambda^{x-1}}$$
 * Putting them together and rearranging gives $$ \frac{v_{x}}{v_0} = \frac{\lambda^{x-1}}{\ell_{x}} \sum_{y=x}^{\infty} \lambda^{-y} \ell_{y} m_{y}$$

For continuous time, there's another expression involving exp(r) instead of \lambda and integrals rather than sums.

See Goodman's 1982 paper, which clarifies much: Goodman, D. 1982. Optimal life histories, optimal notation, and the value of reproductive value. Am. Nat. 119:803-823. There's also a clear discussion of RV in Caswell's book on matrix models, among other places.

Please get this fixed. Enough people get confused by RV as it is.

Fixed wrong definition
Scibike (talk) 05:40, 12 December 2014 (UTC)