Popular Populations & Power (or a fistful of Fibonacci!)
What does you fist have to with exponential functions? Well aside from wishing you could hit your teacher for making you suffer through Exponential (&Transcendental) functions, it shows you an important sequence of numbers – the Fibonacci (Fibonacci was the first person to bring decimal numerals and place value notation to the west!) sequence. The fibonacci sequence is:
1, 1, 2, 3, 5, 8, 13 ,21 ,34,…
If you look at the side of your fist, you can see this pattern in the knuckles (not knuckle joints). The first knuckle is 1. The second knuckle is 1. When you fold your finger over, it makes a curl of three knuckles. When you close your fist (without your thumb) it makes a sequence of five sections. When you include your thumb, it goes to eight – in other words, a fistful of fibonacci (1 knuckle, 1 knuckle, 3 sections in a curl, 5 sections in a fist, 8 sections in a fist and thumb). The Fibonacci sequnce shows up everywhere in nature and art as a ratio called the golden mean. It shows up in places as different as the petals of a rose, sections of a pine cone, the shape of a shell, the Athenean Acropolis, Leonardo Da Vinci’s Mona Lisa and Vetruvian Man, the stock market, the spiral galaxy, DNA, even popular music!
You’d have to be a tool not to appreciate them! Fibonacci numbers are everywhere and they are particularly useful for investigating (unconstrained) population growth. Fibonacci numbers are also found in Pascal’s Triangle (you didn’t think that we would let that one go, did you? Lots of cool stuff here)!The so called “Rabbit problem” is a classic example of using the fibonacci sequence to model population. If we have a pair of newborn rabbits, which will breed after two months, producing another pair of newborn rabbits, who will also breed after two months, How many rabbits will we have in twelve months? You should make a prediction before you look below this point…
|Month||Mature Rabbits||Immature Rabbits||Total Rabbits|
|The simulation starts with one pair of immature rabbits|
|The immature rabbits mature and will be breeding next month|
|The first offspring are produced; now one pair of mature and immature each|
|The first pair breed again, and the first offspring become mature|
|The first offspring and the original pair breed; 2 immature rabbit pairs born|
|The pattern continues; the previous month’s breeding rabbits each produce new offspring, the previous month’s offspring become mature.|
In twelve months, you will have 144 rabbit pairs!
This is actually not a very good simulation, because it relies on many assumptions (including incestuous rabbits… ick!). We need a better modelling tool – one that shows population growth rate as smoother function, in order to reflect that population growth (when looked at across the whole population) is a gradual process, not a stepped function like Fibonacci simulates. To do this, we use exponential functions. Exponential functions are functions where the variable is the power, like this:
a must be positive, or the function would oscillate above and below the x-axis (below if int(x) is odd, above is int(x) is even). a cannot equal one, or f(x) would be 1 for all values of x, as one raised to any power is one.
Plotting this function is relatively easy, as regardless of the value of a, and x = 0, f(x) = 1 (+ any constants), as any number raised to the zero power is always 1. Also, at x=1, f(x) = a (+ any constants), so we have two critical points. Draw a smooth curve through these points, steeper than you would draw for a quadratic, and you are mostly finished.
There is a little more to it than that, as the videos show:
Also, if you are interested in population modelling that isn’t so artificial (exponential modelling at the level we are using doesn’t take into account any interactions between the measured population and the environment), There is a fun game to play.
Last of all, there are some fantastic resources available throught Itunes:
I strongly recommend this podcast, Pam Watkins Math 103 Blended, particularly episodes 5.1 to 5.4 to support your learning on this issue – use your mp3 players for good, not evil!
Math According to Mike also has some fantastic resources; maybe if you could prove that your Ipod is loaded with math-y goodness, you might be able to use them in class…
See you in class and don’t forget your MathsOnline Homework!