**EDIT- ***I can't help but be drawn back to this question... with this method not only can you get actual total population, you can know the number of kids, teens, adults, whatever you want. Additionally, it takes into account BOTH survival rates AND birth rates for each group, AND it takes into account that children do not have babies, and old people have much fewer babies than young people. Logicpolice, I do believe THIS is what you are looking for.*

Another thing you could do is use a Leslie matrix. It is much more accurate, but much more tedious for very long range population growth. Also, you have to have some basic linear algebra skills to use it, or you at least have to be familiar with the TI-83 calculator and it's matrix operations.

Leslie devised a method for calculating the population growth of the females of a species that also includes age groups, survival rates, and (implicitly) reproduction rates. All the computation you need to do is (iterative) matrix multiplication. At the end of the computation, just multiply everything by two, since after all usually there are about as many females as males in a given population (although that is a function of time, but for the purposes of this I will neglect it)

EDIT 2- I would suggest though that instead of multiplying the female population by two to get the correct results, multiply each age group by the correct amount of times the male group is larger or smaller than the female group. You can find these statistics pretty easily.

As a first example, divide the population into the following ten year age groups:

Children (1-10), teens (11-20), youths (21-30) and adults (31-40) - since the average lifespan wasn't much higher than 40 for a long time anyway.

Then, let's make up some rates that each group will make it into the next age class:

Children: 40% Teens: 50%; Youths: 60%; Adults: 0% (there is no class after adults. All adults are guaranteed to die before making it into the next class, which naturally is residency in the grave yard).

Now we need a starting population.

Let's start with 5 children, 2 teens, 2 young adults and 2 old adults.

Now we need to know how many offspring each produce in on iteration (10 years).

Children: 0; Teens: 2; Young Adults: 5; Old Adults; 2

So, here are the two matrices we must multiply in general

Code:

f1 f2 f3 ...
s1 0 0 ...
0 s2 0 ...

times

where f1, f2, f3... are the number of females born each year per capita per group,

s1, s2, s3... are fraction of individuals who survive into the next age class

n1, n2, n3... are the number of individuals in each age class at the beginning of a particular iteration.

So, the starting n's are:

Code:

n1 = 5
n2 = 2
n3 = 2
n4 = 2

The fractional survival rates are (as we defined)

Code:

s1 = 0.40
s2 = 0.50
s3 = 0.60
s4 = 0

and the number born per capita will be:

born by children: 0

born by teens, young adult, and old adults: 2,5,2 respectfully.

So, now plug these in and do some matrix multiplication. I'll use paint instead of code so it is easier to see. Here is the first matrix multiplication.

*(You can see on the left most matrix, the first row is the number of offspring each group contributes per iteration. The second, third and forth rows represent each group's chances of making it into the next age group (the zero's you see must be exactly where they are in order for this to work. Don't worry about them). The left most matrix WILL NOT EVER change, unless you want to adjust survival rates or birth rates (for example, if the survival rate of children is greater the tenth iteration than it was the first) The middle matrix (just to the left of the equal sign) is obviously the number of each group we started with (5 kids, 2 teens, 2 young adults, 2 old adults). The matrix on the right side of the equal sign is the number of each group AFTER ten years.)*

Now, the product of that multiplication (shown after the equal sign) will be the number of people in each age group after 10 years (one iteration). So, based upon our initial parameters, after ten years, you will have:

18 Children

2 Teenagers

1 Young Adult

~1 Old Adult

Keep doing this iteration, but each time replacing the matrix in the middle (or column vector, whatever you want to call it) with the matrix that was on the one on the far right (after the equal sign) the previous iteration.

After 6 iterations, here is what the matrix will look like:

Basically, if we assume that there are roughly the same amount of males as females in a human population, this means that after FIFTY years you will have:

32 Children * 2 = 64

11 Teenagers*2 = 22

4 Young Adults*2 = 8

~1 Old Adult * 2 = 2

For a total population of 96 people.

and after SIXTY years you will have:

46 Children *2 = 92

13 Teenagers* 2 = 26

6 Young Adults * 2 = 12

~ 2 Old Adults * 2 = 4

For a total population of 134 people, most of whom will be young children.

**For Adam and Eve**

Now, if we wanted to start this off with only ONE young adult female (ala Adam and Eve), instead of two teenage females, two young adult females and two old adult females, we would have, after sixty years:

Which amounts to

11.4 children * 2 = 23

4.6 teenagers * 2 = 9

1.12 young adults * 2 = 2

.715 Old adults * 2 = 1

For a total population of 35 people after SIXTY years, with Grandfather Able recently deceased at the hands of Cain, and Cain and his wife dead from old age, Able's long suffering widow barely clinging to life. (of course, this is all based upon the realistic life spans we assigned rather than the ridiculous ones in the Bible. But all you have to do is adjust the initial parameters.)

Now, if you want to calculate the amount after 8 thousand years, this WILL work, but it will take a lot of work on your part.

To make it easier, find a program that will do iterative matrix multiplication, with the value multiplying changing every iteration.

To make it more accurate, even after you find this program (I don't know if TI-83 will do this, btw), then you need to estimate how birthrates and survival rates changed over time. Basically, run with a matrix for as many years as you believe a given birthrate and survival rates are accurate. Then take your total value of the population and create a NEW leftmost matrix that represents each new value for those parameters and multiply it by the total population. Continue as before until the parameters change.

**As far as actual lifespans, and this is the huge part, **__that doesn't really matter__! All you have to worry about is the birth rate WITHIN some given time frame. So, if instead of 40 years your lifespan is 80, all you have to do is adjust your groups.

Instead of children, teens, young adults and old adults, you might have:

Teens and children (1-20 years)

Young adults (21-40 years)

Middle age (41-60)

Old adults (60-80)

And then just adjust the birth rate for each group, *now using a span of 20 years instead of 10*, and PRESTO! Continue as you were. For example you might have the following birth and survival rates:

Teens and children: Each contributes .7 births every 20 years: 65% chance of making it to the next level.

Young Adults: Each contributes 6 births every 20 years: 50% chance of making it to the next level.

Middle age: Each contributes .7 births every 20 years 25% chance of making it to the next level.

Old adults: Each contributes 0 births every 20 years: 0% chance of surviving death.

I mean, this is so freakin' easy mathematically that it makes me tingle.

Let's say after sixty years, Adam and Eve's family suddenly jumps to 80 year lifespans with the parameters listed. So what happens 80 years later?

Start with the population of **females** we had after sixty years:

11.4 children

4.6 teenagers

1.12 young adults

.715 Old adults

Now multiply the new left most matrix (made by the new parameters) by the population matrix after sixty years, and then iterate the new multiplication. So, after 1 NEW iteration:

Which means after 80 more years we have:

73 Children and teenagers, 15 young adults, 5 middle age adults, and 1 old folk barely clinging to life.

Now, do this 3 more times, so...

So, during that 320 year span, the population grew to:

433 teens and children * 2 = 866

128.7 young adults * 2 = 257

23.55 middle age * 2 = 47

2.98 old folks * 2 = 6

Which means that the TOTAL population after 320 + 60 = 380 years is:

866 + 257 + 47 + 6 = 1176 people.

Just remember, the parameters you chose will have an affect on the final results. If you REALLY want to do this, do some careful research to determine birth rates and survival rates for each age group. I recommend studying modern tribal peoples as well as historical estimates on these parameters.

However, if you put in the right numbers on this model and actually take the time to do the matrix iterations, I see no reason at all for you to not get a very reasonably accurate estimate.

## Bookmarks