In the last section, I introduced the logistic equation.
A simple model of population growth, where there's some limit on how large the population can get.
The details of the derivation of the logistic equation aren't essential,
but we'll be using the logistic equation so much in the next several units
that I thought it was worth talking a little bit about its origins, and hopefully giving you a little bit of intuition about the equation.
In this unit, we'll look at properties of the logistic equation.
When you iterate the logistic equation, what happens? What sorts of behavior do you see?
Iterating the logistic equation--and we'll want to iterate it a lot--is tedious and not that interesting to do by hand,
so we'll use a computer program to do it for us.
The program that we'll use is on the web, and there's a link to it right here, underneath me.
If you downloaded the video to watch seperately,
you'll need to go back to the complexity explorer site and click on this link, so the program appears.
So if you click on this link, a new window should appear. You might need to disable a pop-up blocker,
If you click on it and nothing happens, try disabling the pop-up blocker on your browser.
You should get a window that says, "Logistic Equation Time Series Plots."
Let's look at that program and you'll see how to use it.
Here's the website that will calculate orbits and time series plots of the logistic equation.
There are three things that you'll have to enter.
First, you need to choose the number of iterates you want plotted.
I'll choose 20.
Next you need to choose the initial condition.
I'll stick with 0.2; that's the default.
And then you need to choose the growth parameter, r.
I'll use 1.5, because this was the example that I did at the end of the previous section.
In that section, we found that if the seed is .2, the first iterate is .24, and then the next iterate is .2736.
Let's see...I'm going to scroll down. Beneath the plot, the orbits are listed.
We see it starts at .2, .24, .2736; just like we calculated with the calculator previously.
We can see that the orbits are growing and then they reach a fix point at .3333... or a third,
So we suspect this is an attracting fix point.
Then the time series plot shows the orbit starting at .2 and growing to a third.
Let's see if this really is a fixed point. We could check with Algebra, but we could also do an experiment using the computer.
I put in .333, and look at the time series plot, and we can see that the point is indeed fixed; it doesn't go up or down.
You can see that also in the table of numbers.
Let's try another value; why don't we try 0.6?
Remember, this would be sixty percent of the way to the annihilation population.
And we see that the population decreases and approaches the same fix point at .333.
This is the time series plot, and here are the raw numbers that the plot was made from.
Let's try a few more things, just to get a feel for the logistic equation.
I'll try an initial condition of 0.99.
This means we're 99 percent of the way to the annihilation population.
So, perhaps not surprisingly,
we initially see a very large population drop, because we are very close to the annihilation parameter;
we're very close to having so many rabbits that there is no food left for anybody.
So the population dips very dramatically,
and then it grows up to the same stable fix point we're seeing at one third.
Here are the raw numbers for that.
Lastly, I'm going to try an initial condition of zero, in this case there are no rabbits to start off with,
And sure enough, there remain no rabbits.
So if there are no rabbits on the island today, there will be no rabbits on the island next year or next generation.
It's not surprising that zero is a fixed point, but perhaps comforting to see it on this numerical example.
We've just been investigating the logistic equation for r=1.5.
Let's summarize its behavior.
We saw an attracting fixed point, a stable fixed point, at 0.33.
I'm going to summarize this with something that's very similar to a phase line but not identical,
it's something I call a final state diagram.
In a final state diagram, one just draws a dot for the final state or final states.
So here, this is just supposed to go from zero to one, there is just a single dot at .33
because if I let it iterate in the system for a long time and watched it, I would just see it fixed at this point.
Let's go back to the program and investigate orbits of the logistic equation for another value of r.