The first concept to understand in
dynamics and chaos is

iteration, that is doing something again
and again.

For example, let's think 
about population growth.

Population growth is an iterative process

since reproduction happens over and over
again. We are going to be looking at a

extremely simple model 
of population growth

called SimplePopulationGrowth.nlogo

As usual this is available on the course
materials page for website.

Let's open it up. In this model,

when we click setup, it starts off

with a single individual,
making up our population.

The birth rate is 2 which means that

at every time step, this little bunny will

produce two offspring and die.

So if we click reproduce you can see that
happening. There are the two offspring.

Now the next time step, each one of these
bunnies produces two offspring

and dies and so on and so on

and very quickly the world 
starts to fill up

with more and more of these bunnies. 
The model shows us two plots

the top one here gives us 
the population overtime

so at every time step 
the population is doubling

you can see that goes up quickly. 
It also gives us another plot, another way

of looking at this in which
we plot last year's population

with this year's population,
if we think of a year as

each time step represents a year. 
If last year's population is 100

this year's population will be 200

you could see that here and if it's 300

last year, it will be 600 this year and so
you can see we get a straight line

this couldn't be simpler.
Because it is so simple

I'm gonna show you how to 
turn this into a mathematical model

so let's close netlogo here,
let's put this aside

and let's write down some equations.

First some terminology.

Let's call a population n

little n - this is the population.

Now let's label the initial population n_0

In our model

n_0 was equal to 1 there
was only one individual

so we can similarly label

n_1 equals the population at year 1

and so on. For us that was 2

because our initial one bunny
had two bunnies

and then died.

More generally, we can label

the population at year t as n_t

We also had a value

for the birth rate which was

the number of offspring produced each year

We noticed that

n_1 was equal to the birthrate

times n_0, right? So we had

1 at the beginning and

we multiply that by 2

to get two offspring.

Similarly n_2 equals 
the birthrate times n_1

We can generalize that

to say that n_t+1

that is the population at year t+1

is the birth rate times

the population at the previous year n_t.

so this is our model.

The iterative part comes from

the fact that we always take 
the last year's population

and use it to calculate 
the next year's population

and then in the year after that 
we would take

this new population and use it to
calculate the following year's population

and so on and that's called an iteration.

This is called exponential population
growth

and we can see that as follows:

Let's write down a table (up here)
of the population growth

We will write down here the year

and here we write down n_t

So it is the year t

and n_t

so when we have year 0 
we remember this was 1

and this is for birth rate equals 2

At year 1 we had 2, at year 2 we had 4

because we doubling these each year

at year 3 we had 8 and so on

I hope you are seing a pattern here

because at year t we see that our

population is to 2 raise to the t power

so 2 to the 1, 2 to the 2, 2 to the 3

2 to the t

This is called an exponential function

because we have this exponent t

and that causes the population
to grow very very fast.

Of course, this is completely unrealistic

it is uncontrolled population growth 
and of course, in the real world,

there are limits to growth.

A population will run out of resources,

it will run out of food or 
space in which to grow.

But we just for now will assume 
there is no limit to growth.

And now let's look again at the 
netlogo version of this model.

So remember here we have our world
filling up with bunnies,

we have our plot of population versus time

but now we can plot

this year's population versus 
last year's population.

Now this one here population versus time
is an exponential function

that was that function that had
the 2 raised to the t power in it.

That is what an exponential function looks like

but if we plot this year's population 
versus last year's population,

we get a linear function.

We could put a little note on it

to remind ourselves what the function was

and note the function n_t+1

equals birthrate times n_t.

For those of you who vaguely remember 
algebra, algebra 1 even,

this is the equation of a line.

We have Y equals slope times X.

So here is our y-axis,

our slope is the birthrate that's 2.

This just shows that every time we
look at X we double it to get Y.

Well this is a linear equation and the
reason it is linear is because

this is in essence a linear system.

If we look at this year's population
versus last year's population.

We have talked in Unit 1 about the
notion of nonlinear versus linear.

Linear comes about because

there is no interactions 
among these bunnies:

you just have reproduction

and the bunnies are going along all
independent of each other,

and independence in that sense, 
in a system,

yields linearity, linear growth.

OK now it is time for a quiz.

Now suppose the birthrate goes up to 3.

We will let the initial population
n_0 be 1 bunny again.

The question is:

what is the population at time 4?