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If I knew that by the year 2000 that there were 1 trillion humans, and that they started reproducing 2000 years before that, how would I calculate their birth rate?

Assume an initial population of 1 million, and a reproduction age of 15.

Also take into account a standard death rate, where death occurs after 75 years, and reproduction age ends at 50.

Update : Assume a uniform age distribution.

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    Yes, I did gather there were human beings before the year 0. What different models do I have to choose from?2011-02-01

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The simplest model would just say the population increases by a fixed fraction every year: $P(t)=P_0g^t$ where $P_0$ is the initial population, $g$ the growth rate, and $t$ is time. Given two points, you can solve this with logarithms. Using the values you supply:

$10^{12}=10^6g^{2000}$

$12=6+2000 \log_{10} g$

$\log_{10} g = .003$

$g\approx 1.007$

If you want to take into account the age distribution, you need to specify a starting one. Then you will have a reproduction rate for those in a certain age bracket and you have to specify also what age people die at. But if you define the inputs completely, you can write a set of equations to cover it-you might have one for every age, for example.

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    You can make models as complex as you want. Imagine a spreadsheet with 150 columns and lots of rows. The rows are the years. Each column is the number of people alive at that age. Each cell except the left row is just the cell above and one to the left less the death rate of people of that age. The left column is people of each age times the birth rate summed over the reproductive ages (where the birth rate can vary with age if you like).2011-02-02