I wrote to a mathematician friend FR, about this...
I wrote:
I am bothered not knowing how to work out exponent growth patterns. To use a simple population pattern, no twins, no deaths, let’s consider one birth a year by one person for 13 years when the oldest child also starts having children. All children start having children at 13 years old and keep having children every year. How many people are there in 125 years?
Yeehaw. Branching processes.
Let’s do the simplest case we can first, which we can arrive at because in the case you have here, everything is deterministic and happens in 13 year increments, so you can take one generation to be 13 years and work only in terms of generations. In generation 0, there is 1 person. Each person in the previous generation produced 13 people in the next generation, plus themselves, so, notating the number of people in generation i as N_i,
N_i = 14 * N_{i-1}
with N_0 = 1, the solution to this is N_i = 14^i.
125 YEARS IS BETWEEN 9 AND 10 GENERATIONS, SO THERE ARE BETWEEN 20 BILLION AND 289 BILLION PEOPLE IN THIS MODEL.
The general program is to set up a recurrence for N_i in terms of earlier values of N, and set an initial condition (N_0). Then you are into the theory of difference equations, which is a well studied area of mathematics that just happens to be rarely taught. You also don’t have to get exact solutions via difference equations. You can just plug in numbers and work your way through:
i | N_i
————
0 | 1
1 | 14
2 | 196
3 | 2744
4 | 38416
5 | 537824
6 | 7529536
7 | 105413504
This is only possible if your expression for N_i only depends on the past, not the future. If it depends on the future as well, then you actually need the difference equations. It doesn’t usually depend on the future, but that problem does show up when calculating with boundary values in physics problems (temperature is X at this end, Y at that end, calculate what it is in the middle).
Let’s take a little more complicated case and work in terms of years. The next ingredient in such problems is using a structured population, that is, dividing it into groups so that the relations for each group are simple. In this case we are going to divide into juveniles and adults, the numbers of which we will write as J_i and A_i. The total population N_i = J_i + A_i.
The number of juveniles in a given year is given by the number that existed the previous year minus those that reach child bearing age and become adults (which is the number that were born 14 years ago, which is the same as the number of adults 14 years ago), plus those born in this year, or
J_i = J_{i-1} + A_{i-1} - A_{i-14}
and the number of adults is the number that existed last year plus the number of children born 14 years ago who have now become adults, which is again equal to the number of adults 14 years ago.
A_i = A_{i-1} + A_{i-14}
When we actually calculate this, we have to specify A_i and J_i for negative values of i. We just set that to be 0 since we assume we’re starting with a single adult that popped into existence at A_0, so A_0 = 1 and J_0 = 0. Then we can work forward the same way as before
A_0 = 1, J_0 = 0
A_1 = 1, J_1 = 1
A_2 = 1, J_2 = 2
…
A_13 = 1, J_13 = 13
A_14 = 2, J_14 = 13
A_15 = 3, J_15 = 14
A_16 = 4, J_16 = 15
...
So this is your general approach. Break your population into subpopulations that each behave uniformly, then write down recurrence relations for each subpopulation, simulate forward with the recurrence relations, and add up the subpopulations. You can do these calculations in Excel really easily and not have to worry about solving difference equations.
FR
