Talk:Reproductive value (population genetics)

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Removed from page

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

== 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?[reply]

Verbal definitions:

A rough definition: expected contribution of an individual in class x to future population size.
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)[reply]