Darwin's Elephants

In the first edition of On The Origin of Species, Darwin gives a example of the effect of even slow reproduction in producing large populations:

The elephant is reckoned to be the slowest breeder of all known animals, and I have taken some pains to estimate its probable minimum rate of natural increase: it will be under the mark to assume that it breeds when thirty years old, and goes on breeding until ninety years old, bringing forth three pair of young in this interval; if this be so, at the end of the fifth century there would be alive fifteen million elephants, descended from the first pair.
(Darwin. On the Origin of Species. Ist Ed. Ch 3.)

How did Darwin arrive at the figure of 15 million elephants? For, since elephants reproduce sexually, the question of how many elephants there will be has many possible answers. If, for example, the offspring of the first pair are all males, or all females, then there will be no further reproduction, and there will be zero elephants after 500 years. Equally, if each female elephant gives birth to five sons and one daughter, then there will be more or less as many elephants after 500 years as there were at the outset.

Assume that Darwin's elephants live to the age of 100 years, and that the adults are sexually fertile by the age of 30 years, and fertile for 60 years. Assume also that females give birth every 10 years from age 30 onwards to one son or daughter. Then if each female gives birth to 3 daughters, followed by 3 sons, there will be 1,864,333 elephants alive after 500 years - rather less than Darwin states.

Daughters per elephant	Population after 500 years	Number of Females per Male
0	0	-
1	22	0.22
2	64,771	0.89
3	1,864,333	2.32
4	7,315,931	5.53
5	13,368,523	15.81
6	0	-

If each female gives birth to some number of daughters and then some number of sons, so that the total is six in all, then the table on the left gives the total population after 500 years for 0 to 6 daughters. It is only when females give birth to 5 daughters and one son that Darwin's figure is approached. Oddly, using this calculation method, if each female gives birth to 4 daughters, a son, and then another daughter, the population after 500 years is 14,999,058 - almost exactly Darwin's figure, and the maximum value obtainable using this calculation method.

The procedure used to calculate these figures is very simple, and the QuickBasic source code is attached. Did Darwin use the same iterative process to arrive at the figure of 15 million elephants? It would have been laborious, but not impossible, to manually calculate the figure.

But the objection must be that, if he did, he biased the sex ratio as heavily as possible in favour of maximum reproduction. And someone must have objected, for in subsequent editions of Origin, the text has changed:

The elephant is reckoned to be the slowest breeder of all known animals, and I have taken some pains to estimate its probable minimum rate of natural increase: it will be safest to assume that it begins breeding when 30 years old, and goes on breeding till 90 years old, bringing forth six young in the interval, and surviving till one hundred years old; if this be so then, after a period of from 740 to 750 years there would be nearly nineteen million elephants alive descended from the the first pair.
(Darwin. On the Origin of Species. Ch 3.)

Clearly, to achieve this new result, Darwin had to use a more even sex ratio, and to give the more slowly reproducing elephants half as long again to reach a figure a little larger than he had first produced.

It is interesting that in neither case does Darwin state what ratio of male to female offspring he is using. Diligent readers must have assumed an even sex ratio, and found it impossible to achieve the figures Darwin gave. They would have come up with about 1.8 million elephants at the very most. They must have wondered whether Darwin's elephants were living for several centuries