Population Growth

Biological experiments on the computer

Manil SuriProfessor, Department of Mathematics and StatisticsUniversity of Maryland Baltimore CountyBaltimore, MD 21250www.manilsuri.com

Copyright © Manil Suri 2009

“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

Say the population at hour 0 is 0.1 million, i.e.X0=0.1

Then, at hour 1, hour 2, hour 3, etc, we have:X1=0.2X2=0.4X3=0.8X4=1.6X5=3.2X6=6.4

The rule is: X 2X The population grows very fast.

Doubling of Bacteria

IS THIS REALISTIC?

Limited space, limited food. We need a better model.

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.

Let’s try it! x 2x x 2x(1-x)

X0=0.1X1=0.2X2=0.4X3=0.8X4=1.6X5=3.2X6=6.4

X0=0.1000X1=0.1800X2=0.2952X3=0.4161X4=0.4859X5=0.4996X6=0.4999

What is happening to the population now?

Comparison of Growth

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

CONTROLS

Slide Cursor tochange to 2.0 in formula for F(x)(TRY IT ON WEB)

http://math.bu.edu/DYSYS/applets/nonlinear-web.html

CONTROLS

Move your arrow aroundYou will see this lowerright box gives the x-value.

Now place the arrow so that the box reads x=0.1Click.

Q: What are the two curves?

Q: What is the intersection of the two curves?

http://math.bu.edu/DYSYS/applets/nonlinear-web.html

http://math.bu.edu/DYSYS/applets/nonlinear-web.html

Now click “Iterate” boxThis gives the line segments shown

Q: What is the point marked on the straight line?

Notice that thisleft box gives the value of x0,i.e. x0=0.1 (initial population)

The right boxalways gives thex-value of the location of your arrow.

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 finallyends up as being. The answer is 0.5.Now press the “Del Trans” button. Only the final answer, the red dot, should remain.

HOW PROGAM WORKSEach time you click, the program calculates

the population up to x200, 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.

What if you start with a different initial population?

http://math.bu.edu/DYSYS/applets/nonlinear-web.html

What if you start with x0 bigger than 0.5, for instance x0=0.9?Q: What is the final value of the population you get?

Try x0=0.2, 0.4,etc – different values.

Now, the population actually decreases, then goes again to 0.5

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 tochange 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 x0. Q: What is the final population?

Next, repeat with F(x)=2.5x(1-x).What is the final population now?

Final Population=0.33333Final Population=0.5Final 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

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

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

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 redpart – what happens after a long time.)

Now try c=3.02. What do you see?

http://math.bu.edu/DYSYS/applets/nonlinear-web.html

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

“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.

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 withperiod=4.

CHAOSTaking 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)

CHAOS

Starting with twoinitial values very close to each other,the populations become quickly farapart.

This is why it is impossible to predict weather in the long term.

Other ExperimentsYou 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