## Introduction

So I got pointed to the http://www.icr.org, which if you haven't poked around on the site, you really should. It is quite interesting (not really in a good way). However, I want to look at a particular article on the young earth, arguing from population growth rates. The relevant sentences are:

Using census records from the last 400 years and a bit of algebra, and assuming a natural logarithmic growth, eight Flood survivors 4,500 years ago produce 7 billion people almost exactly. This is powerful evidence that biblical history is accurate, and man-made evolutionary history is not.

Then they footnote the following:

The formula for logarithmic human world population growth is $P = Po e^{rt}$, where P = the current population, Po = the initial population, e = the base of natural logarithms (2.718), r = the average annual population growth rate (0.456% or 0.00456 in the calculator), and t = the time interval from $P_o$ to $P$.

Finally, in the middle of the article, they are incredulous of the low net birth rate in pre-agricultural human populations:

Thus, the first human couple that supposedly evolved from ape-like ancestors would have had only 6 million descendants after 2.4 million years. This requires a population growth rate of about 0.000000009-essentially zero. Virtually no growth for 2.4 million years?

So, how do we respond to this sort of analysis? One could try to simply write it off as the rantings of a person blinded, by their religious bias, to a true understanding of science. However I prefer an alternative approach. It may be tilting at windmills, but I prefer to take a claim at face value, and see why it fails, and to ask followup questions. Since the math here is so simple, it is easy to demonstrate any errors, assumptions, etc.. and get to the bottom of it. So, someone has made a claim, I don't agree, how do I proceed?

## Did they do their arithmetic right? Yes!

The first thing I like to do is, assuming they are right in all of their numbers, do they do the calculation correctly. This eliminates a pretty obvious error that could have caused the problem. I suspected they got it right, but I wanted to make sure. The math is

$$
P=P_o e^{rt}

$$

The code is:

r=0.00456 t=4500.0 Po=8.0 print Po*exp(r*t)/1e9,"billion"

Yielding

6.52849018681 billion

The other direction is, how long ago were there only 8 people?

\begin{eqnarray}

P(t)&=&P_o \times e^{r\cdot t} \\

\log P(t) &=& \log(P_o) + r\cdot t\\

t&=& \frac{\log(P(t))-\log(P_o)}{r}

\end{eqnarray}

or

Po=6.5e9 # current population r=0.00456 P=8 # target population t=(log(P)-log(Po))/r print t, "years"

resulting in

-4499.04089302 years.

## Is their growth rate correct?

Here I am using data from census.gov. It is possible (in fact likely) that the original post used a slightly different data set, but they didn't cite it so I can't confirm that. The conclusions will not be qualitatively different, but that would be one of the questions I'd have for the author (see below).

It is instructive to look at the data set both on a normal scale and a
log scale. The key thing here is that **a constant growth rate
translates to a straight line on the log scale**. Let's see if this is
the case (see the lower of the two plots, especially):

Linear? Not so much.

I then chose a time-frame over which to calculate the average which gave (for my data set) as close to their growth rate above - this is essentially the growth rate consistent with the Biblical chronology. In my data set that average is from 1300 to 2010, or 700 years (theirs, they claim, it was 400 years so there is clearly a difference in data sets here - however, they don't reference their actual data, and regardless, this same analysis would work in their case too).

You can see from here that this average yields the Biblical number of
4500 years ago, but that it is clearly not a good assumption to think
that the growth rate, even over this time-frame, is constant. Because of
this, depending on how far back you do the averaging, you get a
*different* time of the Flood. For example, the following show cases for
1600-2010, and 1950-2010. The last one is the closest to a constant
growth rate, but gives a ridiculous -1200 years back to 8 people!

(As an aside, Richard Dawkins does the same calculation with rats, and comes up with an astonishing 1900 or so for the beginning of the Earth.)

## Bottom Line

The biggest problem with this calculation is assuming a constant growth
rate. Once it isn't constant, then the time-frame over which you
calculate the average makes a big difference. Only *one* possible
time-frame will result in something consistent with the Biblical
account, so one would have to justify:

- Why you can possibly assume constant growth rate in the distant past knowing it wasn't constant in the time-frame we know about.
- Even given a solid justification (which I don't believe actually
exists...but I'm open to suggestions), you have to further justify
why the
*particular*time average you take, that is consistent with the Biblical account, is the one you*should*take.

You can't use as a reason "because it's consistent with the Biblical
account", because the entire point is to *justify* the Biblical account,
or to demonstrate that the data is consistent with it. It's not, if you
have to postulate that the Biblical account is correct, and choose the
specifics of the calculations to meet that.

## Questions for Brian Thomas, M.S. at the ICR.

Here are the questions I'd like answered. I'm not expecting any, but I'd be thrilled to get one!

- What data set did you use? I'm sure it won't make any difference, but it is good to make sure that we all agree on the numbers. I'd be happy to run my analysis through any data set that you provide.
- Using your data set, and your analysis, do you confirm my observation that your result is sensitive to the time-frame (i.e. the 400 years in your case, 700 years in mine) over which you are obtaining your "average" growth rate?
- If it is, do you have a justification for using that,
**and only that**, time-frame? - Do you have a justification for using an average growth rate model
in the face of data that clearly shows that the growth rate is
*not*constant?

## Comment about Growth Rates before Agriculture

The original article was astounded that the growth rate, prior to
agriculture, could be so low. Essentially, they seem to believe that
exponential growth should have started once you had people, and gone at
a constant rate the entire time. However, before agriculture, there
would have been a fairly low carrying capacity, and human population
would follow *logistic growth*, not exponential. Near the carrying
capacity, the growth rate is essentially zero, exactly as observed. A
little bit of reading on how populations *actually change* might be
helpful to these people.