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?