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Population Growth Biological experiments on the computer Manil Suri Professor, Department of Mathematics and Statistics University of Maryland Baltimore County Baltimore, MD 21250 www.manilsuri.com Copyright © Manil Suri 2009

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“Doubling” (Exponential Growth) Consider bacteria growing in a dish. Growth depends on factors like amount of food, temperature, size of dish, etc. We want to find out how the population growth proceeds. Suppose first the number of bacteria doubles every hour

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Say the population at hour 0 is 0.1 million, i.e. X 0 =0.1 Then, at hour 1, hour 2, hour 3, etc, we have: X 1 =0.2 X 2 =0.4 X 3 =0.8 X 4 =1.6 X 5 =3.2 X 6 =6.4 The rule is: X 2X The population grows very fast.

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Doubling of Bacteria

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IS THIS REALISTIC? Limited space, limited food. We need a better model.

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More Realistic Growth Model Instead of X 2X, we now modify this to X 2X (1-X). What does this extra factor (1-X) do for us? As x becomes larger, this factor becomes smaller. It puts the brakes on growth.

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Let’s try it! x 2x x 2x(1-x) X0=0.1 X1=0.2 X2=0.4 X3=0.8 X4=1.6 X5=3.2 X6=6.4 X0=0.1000 X1=0.1800 X2=0.2952 X3=0.4161 X4=0.4859 X5=0.4996 X6=0.4999 What is happening to the population now?

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Comparison of Growth

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Let’s continue on the computer We will use link below for “Nonlinear Web” applet (can also find this by typing “nonlinear web” on Google – first link that appears in list) (instructions follow) http://math.bu.edu/DYSYS/applets/nonlinear-web.html

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CONTROLS Slide Cursor to change to 2.0 in formula for F(x) (TRY IT ON WEB) http://math.bu.edu/DYSYS/applets/nonlinear-web.html

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CONTROLS Move your arrow around You will see this lower right box gives the x-value. Now place the arrow so that the box reads x=0.1 Click. Q: What are the two curves? Q: What is the intersection of the two curves? http://math.bu.edu/DYSYS/applets/nonlinear-web.html

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Now click “Iterate” box This gives the line segments shown Q: What is the point marked on the straight line? Notice that this left box gives the value of x 0, i.e. x 0 =0.1 (initial population) The right box always gives the x-value of the location of your arrow.

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Click “Iterate” several times. Q: What points are marked out on the straight line? http://math.bu.edu/DYSYS/applets/nonlinear-web.html Next, click “Iterate All.” Q: Can you see a tiny red dot? What the program is doing is calculating the iterations for you. The red dot shows what the population finally ends up as being. The answer is 0.5. Now press the “Del Trans” button. Only the final answer, the red dot, should remain.

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HOW PROGAM WORKS Each time you click, the program calculates the population up to x 200, i.e. it calculates 200 steps using the rule! It shows the first 25 steps in black, and the next 175 in red. In the previous case, all 175 steps landed in just one spot, at 0.5! The red tells us where the population will finally end up.

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What if you start with a different initial population? http://math.bu.edu/DYSYS/applets/nonlinear-web.html What if you start with x 0 bigger than 0.5, for instance x 0 =0.9? Q: What is the final value of the population you get? Try x 0 =0.2, 0.4, etc – different values. Now, the population actually decreases, then goes again to 0.5

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More Experiments We have seen that the population always ends up at 0.5, no matter what it starts with. Let us now see what happens if we change the number 2 in the rule F(x)=2.0 x(1-x). This is the growth constant. Slide Cursor to change 2.0 to 1.5 in formula for F(x) Growth Constant=1.5 http://math.bu.edu/DYSYS/applets/nonlinear-web.html Try this with different initial x 0. Q: What is the final population?

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Next, repeat with F(x)=2.5x(1-x). What is the final population now? Final Population=0.33333 Final Population=0.5 Final Population=0.6 To summarize (Different Growth Constants): F(x)=1.5x(1-x) F(x)=2x(1-x) F(x)=2.5x(1-x) http://math.bu.edu/DYSYS/applets/nonlinear-web.html

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Experiments: Growth Rate < 1 Try growth rate=0.5, 0.7, 0.9. Q What is the final population? Q Why does this happen? http://math.bu.edu/DYSYS/applets/nonlinear-web.html

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SUMMARY SO FAR Let c be the growth constant. Our experiments so far have shown that: 1.If c<=1, then the population dies out. 2.If 1<c<=3, then population ends up at a final value given by the intersection of the two graphs For c=3, the population may go to this value very slowly (try it!) http://math.bu.edu/DYSYS/applets/nonlinear-web.html

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The case c=3 Try different initial populations. Q: How does the size of the red box change? Click on the “Del Trans” button. What happens? (From now on, we will only be interested in the red part – what happens after a long time.) Now try c=3.02. What do you see? http://math.bu.edu/DYSYS/applets/nonlinear-web.html

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Experiments: Growth Rate c>3 Q: What is the population doing at c=3.32? Try it with c= 3.02, 3.12, 3.32. http://math.bu.edu/DYSYS/applets/nonlinear-web.html

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“Boom and Bust” Phenomena When the growth constant becomes high enough, the population no longer settles at one value, but varies between two values. (A “periodic orbit” with period=2) Seen in crab populations, insect/bird populations, etc.

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Growth Constant above 3.45 Try taking c=3.45, then 3.47, 3.50, 3.52. What do you observe? Now take c=3.54. What happens? http://math.bu.edu/DYSYS/applets/nonlinear-web.html A periodic orbit with period=4.

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CHAOS Taking the growth constant even higher, we get increasingly unpredictable behavior, until at c=4, there is complete chaos. For what value of c between 3.5 and 4 does the picture suddenly clear up? http://math.bu.edu/DYSYS/applets/nonlinear-web.html (You should see a value for which you suddenly get an “L” shape)

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CHAOS Starting with two initial values very close to each other, the populations become quickly far apart. This is why it is impossible to predict weather in the long term.

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Other Experiments You can try other population functions by changing “logistic” (upper left box) to “quadratic” or “tent” or “doubling.” These can be thought of as different growth rules. http://math.bu.edu/DYSYS/applets/nonlinear-web.html As shown in the video, the quadratic map gives rise to fractals and the Coverly Set. (Use link below to experiment) http://www.ibiblio.org/e-notes/MSet/Anim/ManJuOrb.htm

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