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