Problem Statement:

The problem, dating back to the early 13th century and posed by Leonardo Fibonacci, asks how many rabbits you would have if you started with just two rabbits and let them reproduce. The assumptions are that a pair of rabbits produces a pair of offspring each month, and each pair produces their first offspring at the age of 2 months.

Stages of Mathematical Modelling:

Step 1 (Understanding the Problem): We need to determine the number of rabbit pairs over time given the reproduction rules. The key observation is that month by month, the number of pairs of rabbits is given by the sum of the rabbits in the two preceding months, except for the 0th and 1st months.

Step 2 (Mathematical Description): The problem translates to finding a sequence where each term equals the sum of the two preceding terms. This gives us the recurrence relation:

$$F(n) = F(n-1) + F(n-2)$$

with $F(0) = 1$ and $F(1) = 1$.

Step 3 (Solving the Mathematical Problem): Using the recurrence relation, we calculate the number of pairs month by month. The sequence obtained is:

$$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597$$

for months 0 to 16 respectively.

Step 4 (Interpreting the Solution): The model predicts nearly 1600 pairs after just 16 months. Modellers look for a balance between simplification and accuracy, hoping to approximate reality closely enough to make predictions with reasonable accuracy.

Step 5 (Validating the Model): We examine whether the assumptions used for simplifying the problem are reasonable. Different assumptions can lead to different models. There are no perfect models, only good ones and yet better ones.