Why Does the Flap of a Butterfly Cause A Tornado?

To understand that, we have to start with rabbits on an island …

This is an adaptation from my academic essay on Chaos, Systems and Mathematical Models.

A Simple Model

Let us start by having an x number of rabbits on the island in the first year.

We multiply the number x by r and get rx to denote the growth rate of the number of rabbits in the next year.

If r = 2, the population doubles every year. The number of rabbits goes from 2 to 4, 4 to 8, 8 to 16, so on and so forth.

This is, however, unrealistic as the number of rabbits would grow exponentially forever. Can we possibly get infinitely many rabbits? No!

the logistic equation

To represent the constraint of the environment, we can add the term (1 − x) and get rx(1 − x).

The term x here represents the theoretical maximum of the population as a percentage point, and it goes from 0 to 1. By extension x_(n + 1) and x_n represent the respective percentage point in the (n+1)th and nth generations.

And as x approaches the maximum, the term (1 − x) goes to zero, constraining the population.

If we were to plot a graph of x_(n + 1) against x_n, we should expect to see a concave down parabola since we can rewrite the equation as x_(n + 1) = -r(x_n)² + rx_n. The leading coefficient is negative r.