Appendix 2: Mathematical Modelling | A2.1

Question 1

Consider the following situation.

A problem dating back to the early 13th century, posed by Leonardo Fibonacci asks how many rabbits you would have if you started with just two and let them reproduce. Assume that a pair of rabbits produces a pair of offspring each month and that each pair of rabbits produces their first offspring at the age of 2 months. 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 the 1st months.

After just 16 months, you have nearly 1600 pairs of rabbits! Clearly state the problem and the different stages of mathematical modelling in this situation.

Solution

We will model rabbit population growth using a recursive sequence.

Step 1 — Formulate the Problem

We need to count rabbit pairs. We start with one pair of rabbits. Each pair produces a new pair monthly. Rabbits begin reproducing at 2 months old. We assume no rabbits die. The goal is to find pairs after 16 months.

Diagram 1

Step 2 — Formulate the Mathematical Model

Let $F_n$ be the number of rabbit pairs. This is at month $n$ . At month 0, we have 1 pair. So, $F_0 = 1$ . At month 1, we still have 1 pair. So, $F_1 = 1$ . From month 2, new pairs are added. New pairs come from pairs 2 months old. The number of pairs is $F_n = F_{n-1} + F_{n-2}$ . This is the Fibonacci sequence.

Step 3 — Solve the Model

We use the formula $F_n = F_{n-1} + F_{n-2}$ . Let's calculate the first few values. $F_0 = 1$ $F_1 = 1$ $F_2 = F_1 + F_0 = 1 + 1$ $= 2$ $F_3 = F_2 + F_1 = 2 + 1$ $= 3$ $F_4 = F_3 + F_2 = 3 + 2$ $= 5$ We continue this pattern. The table shows all values. At month 16, we find the value. $F_{16} = 1597$

$\boxed{1597 \text{ pairs}}$

Step 4 — Interpret and Validate

The model shows rapid growth. After 16 months, we have 1597 pairs. This model is a simplification. It ignores real-world factors. For example, rabbits can die. But it helps us understand growth.

Answer

(i) The problem is to determine the number of rabbit pairs over time, starting with one pair, given specific reproduction rules. (ii) The different stages of mathematical modelling in this situation are: Problem Formulation, Mathematical Formulation, Solving the Model, and Interpretation and Validation. (iii) After 16 months, there are 1597 pairs of rabbits.