# Calculating population size at any future time

If current population size is $P$ and average life-time of any member is $L$. Any pair of member of the population is allowed to have $C$ children on an average. They can have children between the age of $A$ to $B$.Then how can I tell population size at any future time?

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You are missing some data. You need some information on when in life they have children. – Ross Millikan May 17 '11 at 22:14
And how are the lifetimes distributed? Is it possible to have all three children immediately at age 18? What is the starting age distribution? – Ross Millikan May 17 '11 at 22:17
All the distributions are uniform. And you can have at max 3 children at a time. – Pratik Deoghare May 17 '11 at 22:20

Ross has given you a formula for the first $A$ years. Beyond that you could model this, or as an approximation assume that the shape of the population distribution does not change, in which case you will get exponential population change.

For simplification, let's assume everybody lives exactly $L$ years and each person gives birth to $C/2$ children (or that the half of them that are female each give birth to $C$) after exactly $A$ years. We want to find $k$ where the population after $N$ years is $Pk^N$. The shape of the population distribution will be a truncated exponential or geometric distribution with each year group having $k$ times as many people as the one a year older, so the number dying each year will be $P\dfrac{k^N-k^{N-1}}{k^L-1}$ and the number being born $Pk^{L}\dfrac{k^N-k^{N-1}}{k^L-1}$. But the number being born is also $\dfrac{C}{2}Pk^{L-A}\dfrac{k^N-k^{N-1}}{k^L-1}$ so $k=\left(\dfrac{C}{2}\right)^{1/A}$ and that means that the population after $N$ years is

$$P\left(\frac{C}{2}\right)^{N/A}.$$

This seems fairly intuitive: note that it suggests that earlier parental age at childbirth means faster population growth (or decline) even if lifetime fertility stays the same, and that the population will grow iff $C>2$.

To make this more sophisticated, you can introduce $B$, so childbirth takes place over an interval (this will affect the denominator of the exponent, putting it between $A$ and $B$), and you can also widen the distribution of death with substantial effects if it is possible to die before childbearing age $A$ and the average number of children per couple does not take this into account. You could also start with a different population shape, or allow random effects, or allow fertility to vary over time, in a not particularly complicated model.

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Though this is still not a well-specified problem, here are some thoughts. You start with $\frac{P}{L}$ people of each age, which will be strange as the distribution will not stay uniform. If the childbearing is random over the span, you will have $\frac{3PL}{2(B-A)}$ children born in the first year and $\frac{P}{L}$ die. These numbers will hold for $A$ years so you will have $P+\left(\frac{CPL}{2(B-A)}-\frac{P}{L}\right)N$ in year $N$

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Something linear in N seems unlikely; I would expect something broadly exponential in time, as with most population modelling with a constant fertility rate. – Henry May 17 '11 at 22:41
@Henry: It is an artifact of the uniform age distribution at the start and the fact that I stopped at $A$ years when the new babies start having children. It would then become exponential. – Ross Millikan May 17 '11 at 22:45
OK(ish). You also seem to have missed $C$ from the calculation (I think the 3 mentioned earlier was a prohibition on quadruplets not total children). – Henry May 17 '11 at 23:00
@Henry: you are right. It was 3 originally, but had been changed before I posted. I'll change the 3. – Ross Millikan May 17 '11 at 23:06