Fibonacci numbers are so appealing to our human propensity to see patterns that they have their own quarterly published continuously for more than 50 years.

http://www.fq.math.ca/

Fibonacci illustrated the sequence using the growth of a rabbit population, though one where rabbit immortality was assumed. As Knuth says:

Fibonacci numbers have often been observed in nature, probably for reasons similar to the original assumptions of the rabbit problem. [TAoCP: Volume 1, 2nd edition.p 79]