### Estimating Population Growth

(Originally created Jan 2012)
(Finally revised and uploaded Oct 2017)

### Algebraic Solution for Population Growth (the hard way)

P = P0 * e^[r*t]

Where

P = Population at the end of the interval
P0 = Population at the beginning of the interval.
t = Elapsed Time between PStart and PEnd
e = Base of natural logarithms. (2.71828183)
r = Rate of natural increase in decimal format

### Example:

The global human population was around 5 million in 8000 BC. What would the human population be in 2000 BC if we used the 4.1793% (in decimal format, remember!) per century rate of natural increase from the April 1986 Bulletin chart below?

First, we define 8,000 BC and 2,000 BC as -8,000 and -2,000 respectively; then subtract them to get a range of 6,000 years, or 60 centuries.

5,000,000 * 2.71828183^[0.041793*60] = 61,375,941

Thus the human population was around 61.3 million in 2000 BC according to these crude calculations.

### 2012 Wikipedia Calculations

These rates of increase are based off Wikipedia’s World Population Estimates (LINK) page, with all the various estimates averaged together.

Time Frame

Growth Per Century
(Percentage)

Growth Per Century
(Decimal)

Centuries
Req’d For
Doubling

10,000 BC to 4,000 BC

0.9371%

0.009371

74.31

4,000 BC to 1,000 BC

6.7732%

0.067732

10.58

16.4563%

0.164563

4.55

1.8411%

0.018411

37.99

10.3299%

0.103299

7.05

23.4171%

0.234171

3.29

65.7685%

0.657685

1.37

136.1627%

1.361627

0.81

480.9947%

4.809947

0.39

### 1986 Bulletin Calculations

These rates of increase appeared in the April 1986 Bulletin of the Atomic Scientists.

(Doubling times apparently were calculated via the rule of 72, e.g 72 divided by the growth in percentage points, for example: [72 / 5.6964].)

Time Frame

Growth Per Century
(Percentage)

Growth Per Century
(Decimal)

Centuries
Req’d For
Doubling

2,000,000 BC to 50,000 BC

0.0079%

0.000079

8,774

50,000 BC to 10,000 BC

0.4015%

0.004015

140

4.1793%

0.041793

17

5.6964%

0.056964

12

116.1868%

1.161868

0.6

### Computing a Population’s Growth Rate (the Easy Way)

Load up Excel; and use the following formula:

=RATE(Interval, , PopStart,PopEnd)

where

Interval = Interval in years or days or centuries, or whatever.
PopStart: Starting population (MUST BE IN NEGATIVE FORMAT!)
PopEnd: Ending Population

So if a population grew from 250,000 to 500,000 during a 30 year period, you’d enter:

=RATE(30,,(0-250000),500000)

to get the growth rate.

### Computing a Population’s Doubling Time through the “Rule of 72” (The Easy Way)

Doubling Time = (72 / Growth)

Example: A population grows at 0.5% per year. How long until the population roughly doubles?

72 / 0.5 = 144.0 years

### Computing a Population’s Doubling (or tripling or quadrupling) Time

T = LN(IntervalFactor) / LN(1+r)

Where:

T = Doubling (or tripling, etc) time in the amount used to originally calculate the Growth Rate
IntervalFactor = Place a 2 here if you want to know the doubling time, a 3 if you want to know the quadrupling time, etc.
r = The Growth Rate

Example:

A population grows at 0.4015% (0.004015 in decimal) per year. What is it’s doubling time?

LN(2) / LN(1+0.004015) = 173 years