The Golden Ratio: Phi, 1.618

Population Growth and the Fibonacci Sequence

Gary Meisner

The Fibonacci sequence was discovered by studying population growth.

Population growth is also related to the Fibonacci series.  In 1202, Leonardo Fibonacci investigated the question of how fast rabbits could breed under ideal circumstances.  Here is the question that he posed:

Suppose a newborn pair of rabbits, one male and one female, is put in the wild. The rabbits mate at the age of one month. At the end of its second month, a female can produce another pair of rabbits. Suppose that the rabbits never die and that each female always produces one new pair, with one male and one female, every month from the second month on.  How many pairs will there be in one year?

The answer is found in series of numbers now known as the Fibonacci series.  Pair A of rabbits gives birth to pairs B, C, D and E.  Each of these new pairs in turn gives birth to other pairs B1, B2, B3, C1, and C2, who in turn give birth to B11, etc.  At the end of each month, the total population of rabbits will be a number in the Fibonacci series:

Month Rabbits from A: from B: from C: D: B1: Total
0 A 1
1 A 1
2 A B 2
3 A B C 3
4 A B C D B1 5
5 A B C D E B1 B2 C1 8
6 A B C D E F B1 B2 B3 C1 C2 D1 B11 13
etc. 1 2 3 4 5 6 7 8 9 10 11 12 13 etc.

 

The Fibonacci series can be used to predict urban populations

It appears that the Fibonacci series can even be used to predict populations of major cities, as shown by the relationships of various U.S. urban areas in 1970:

Area Census
Rank		 Actual
Population		 Predicted Population
Method 1 Method 2
New York, NE NJ 1 16,206,841
LA Long Beach CA 2 8,351,266 10,016,379 10,016,379
Chicago NW IN 3 6,714,578 6,190,462 5,161,366
Detroit, MI 5 3,970,584 3,825,916 4,149,837
Washington DC 8 2,481,459 2,364,546 2,453,956
Houston, TX 13 1,677,863 1,461,370 1,533,626
Cincinnati, OH 21 1,110,514 903,176 1,036,976
Dayton, OH 34 685,942 558,194 686,335
Richmond, VA 55 416,563 344,983 423,935
Las Vegas, NV 89 236,681 213,211 257,450
New London, CT 144 139,121 131,772 146,277
Great Falls, MT 233 70,905 81,439 85,982

 

Method 1 takes the population of the largest city and divides it again and again by phi.  Method 2 takes the population of each successive city and divides it by phi.

Source: http://www-personal.umich.edu/~sarhaus/

Multicellular organisms

In biology, once an egg is fertilized, it divides and multiplies in count until it reaches a point at which the ratio of the succeeding number of cells to the previous number of cells is phi (1.618 …).

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