Do you like mathematics over biology? Or maybe you’re a fan of biology, but hate mathematics. We all know that both intersect each other frequently, and the intersections are often very interesting. You might remember studying in your Biology class how rabbits reproduce or how evolve according to the environment. We may also want to know to how many rabbits a mother rabbit can give birth to. Leonardo of Pisa, an Italian mathematician, discovered a solution to this called the “Fibonacci Sequence”.
The Fibonacci Sequence
A fibonacci sequence is a series in which each number is a sum of previous two consecutive numbers. The sequence is 0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 and so on. It is written as F(n) where n is the term number.
The golden ratio (also called phi) is a number that is obtained after a number in the fibonacci sequence is divided by previous consecutive number and has a value of 1.61803 approximately. The two terms taken for golden ratio must be large, for example 377 divided by 233.
The golden ratio can be observed all around us in nature. The number of petals in a flower is a classical example. The lily, has three petals, and each petal is placed at 0.618034 per turn (out of a 360° circle), allowing for ideal packing and better exposure to sunlight. Another example is our fingers. Our finger’s length, from tip of the base to the wrist is larger than the preceding one by roughly the ratio of phi.
Assume there are two consecutive terms in the Fibonacci sequence: x and y. The general formula for finding the next term is F(n) = F(n-2) + F(n-1). This tells that the value of F(n) is equal to the value of F(n-2) added to the value of F(n-1). Because the ratio of y/x is the Golden Ratio, we can also multiply x with 1.61803 to get y. y can then further be multiplied with 1.61803 to obtain the next number in the sequence, and so on. Hence, we can also find a term that is far away from the given term.
Suppose we know that F(14)= 377 and we want to find F(20). The difference between the terms is 6. Then we raise 1.61803 to that power and multiply with the given term. 377(1.61803)⁶ which is approximately equal to 6765; this is the value of F(20)
How rabbits grow according to Fibonacci Sequence
If a pair of rabbits mature in a month, live for a long time and give birth to a pair every month, then it follows that in the beginning of Month 1, we have 1 pair. Then,
• Ending of 1st month, we have 1 pair but mature.
• Month 2 , we have 2 pairs.
• Month 3, we have 3 pairs.
• Month 4, we 5 pairs.
• Month 5, we have 8 pairs.
• Month 6, we have 13 pairs.
• Month 7, we have 21 pairs.
• Month 8, we have 34 pairs.
• Month 9, we have 55 pairs .
• Month 10, we have 89 pairs.
• Month 11, we have 144 pairs.
• Month 12 (end of 12th month), we have 233 pairs.
According to fibonacci sequence, we have 233 pairs or 466 rabbits after 1 year, where half are male and half are female.
How this plays out in reality
I bought two rabbits in January 2021 and they didn’t reproduce until April 2021. They gave birth to a pair of rabbits which were identical to their mother. Both of them died after a few days. They original pair did not reproduce again until June 2021, when 5 rabbits were born. One of them was a male rabbit, the rest were females. 2 of these female rabbits were killed by a predator. Later, another of the female rabbits died. They original pair reproduced again in August 2021 and 8 babies were born, from which two were deformed and died. The father from the original pair died. The remaining six from the 8 died one by one, where the last one died in December 2021. The two rabbits other than the original pair that were still alive matured but never reproduced, and the male died due to illness in March 2022. Later, the mother from original pair also died due to illness. One female rabbit is still alive as of August 2022.
It is plausible to say that:
• A rabbit does not mature in a month.
• It doesn’t only give birth to a pair- more than 2 rabbits can be born at a time as well.
• Rabbits may die in less than year.
• A rabbit may not be able to reproduce.
• A deformed baby may be born.
Despite its popularity, The Fibonacci Sequence is not widely applicable. It is still a matter of controversy whether such a phenomenon exists at all. A mathematician might insist that it doesn’t, but an opinion from a biologist would be required too. Regardless, it does illustrate to us the fascinating intersection between mathematics and biology.