The Fibonacci sequence results from an age-structured population model.
The sequence of numbers
\[ \begin{array}{ccccccc} x_{0} & x_{1} & x_{2} & x_{3} & \cdots & x_{j} & \cdots \\ 0 & 1 & 1 & 2 & \cdots & x_{j-1}+x_{j-2} & \cdots \end{array} \] is often called the Fibonacci numbers and is generated by the recurrence relation
\[ x_{n+2} = x_{n+1} + x_{n}, \ \ n=0,1,2,\ldots \] In the book A Short History of Mathematical Population Dynamics, the author Bacaër explains how Leonardo of Pisa aka Fibonacci arrived at this sequence starting from a problem of population dynamics stated in his book Liber abaci. A translation of Fibonacci’s problem is, quoting from Bacaër:
A certain man had one pair of rabbits together in a certain enclosed place. One wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair and in the second month those born to bear also.
An analysis of this problems leads to the difference equation \(x_{n+2} = x_{n+1} + x_{n}\) which generates the Fibonacci numbers whenever the starting values \(x_{0}\) and \(x_{1}\) are 0 and 1 respectively. From the statement of Fibonacci’s problem, it is apparent that an age-structured model is appropriate. We elaborate on this in this post.
An important assumption made in the statement of Fibonacci’s rabbit problem is that rabbits are not fertile until they are at least two months old. So, if we have a new-born pair of rabbits, they will not give birth until the third month after they are born. Thus, if we let \(x_{n}\) be the number of rabbits at the beginning of month \(n\), and we assume as stated that each pair of fertile rabbits produces a pair of new-born rabbits and no rabbits die, then the number \(x_{n+1}\) of pairs of rabbits at the beginning of the next month is
\(x_{n+1} = x_{n} + x_{n-1}\)
which is equivalent to the recurrence stated above. There are several ways, one of which is given as a homework assignment in Topics in Biomathematics, to show that
\(x_{n} = \frac{1}{\sqrt{5}}\left( \frac{1+\sqrt{5}}{2}\right)^{n} - \frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2} \right)^n\)
satisfies the Fibonacci recurrence. It is well known that
\[ \frac{x_{n+1}}{x_{n}} \rightarrow \frac{1+\sqrt{5}}{2} \]
as \(n\rightarrow \infty\), the number \(\frac{1+\sqrt{5}}{2}\) is often called the golden ratio and it appears in many mathematical and natural patterns.