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# Question on Effective Population Size

Fred Bacon bacon at arlut.utexas.edu
Tue Nov 21 18:48:12 EST 1995

```Recently, in a different newsgroup, the textbook _Fundamentals of
Molecular Evolution_ (by Wen-Hsiung Li and Dan Graur) was recommended as a
good introduction to molecular evolution.  Since I am a physicist and not
a biologist, my background in the subject is weak, but I have a personal
interest in the topic.

To come to my point, I am having trouble understanding a formula presented
in the second chapter of FoME (page 30 in my copy).  The authors are
discussing the concept of effective population size, denoted N_e.  They
offer a formula to calculate N_e when the number of males and females in a
populations is different.

4 N_m N_f
N_e = -------------
N_m + N_f

My problem is that they offer no argument for the validity of this
equation, and they give no reference for where it was first derived.  As a
consequence, I can't really understand the formula and what it
represents.  Furthermore, it would seem to me that there is not enough
freedom in this equation to accurately describe different species with
different mating behaviors.  For instance, they present the case in which
one breeding male is present in a population while many females are
available.  In the limit that N_f -> infinity (or at least really big),
the effective population goes to 4 (or so they say).  But this ignores the
fact that different species may behave differently.  Shouldn't we expect a
different result for swan's, which I am told mate for life, and mountain
gorillas which live in "harems" by nature?

I admit that it gives the proper answer in the limit that N_m = N_f, but
then so will an equation of the form

4 N_m N_f
N_e = -------------  + (N_f - N_m) P(N_f - N_m)
N_m + N_f

where P(N_f - N_m) is any polynomial in (N_f - N_m).

Would some kind biologist either fill in the missing arguments from the
textbook or verify that the formula is nonsense?

Fred Bacon

===================================================================================
Most people believe they are thinking
when they really rearranging their prejudices.
---Edward R. Murrow
===================================================================================

```