Let’s write down our simple model for population growth one more time.
And let’s suppose that the birth rate is equal to two.
In the previous subunit I said that this simple model is linear
when you plot the population at time t plus 1
versus the population at time t.
This is because individual bunnies didn’t interact with one another
and there was no limit to growth.
Another way to describe linearity is to say that
the whole is the sum of the parts.
In the simple population growth model we can see this.
For example, suppose we start off our initial population with 1 bunny
and a birth rate of 2
and we reproduce for 3 years.
We end up with 8 bunnies.
Now, let’s set our initial population to 10.
That is 10 times the original.
Setup
Reproduce 1, 2, 3
We end up with 80 bunnies.
10 times the original.
So if we had taken 10 different runs of initial population at 1
and run it for 3 years and summed up the results
We get the same thing as taking an initial population of 10 for 3 years.
That is 80 bunnies in either case.
So here the whole is equal to the sum of the parts.
But this is not the case if the system is nonlinear.
We’ll make our system nonlinear by putting in
nonlinear interactions between the individuals.
Here’s our model of nonlinear interaction.
Here we have n sub t plus 1 is equal to the birth rate times n sub t
But now we’re gonna subtract the number of offspring who die due to overcrowding.
We’re gonna assume that overcrowding occurs due to limits on food or space
and that there is some maximum population that can live in this particular habitat.
So we’re gonna define the number who die due to overcrowding is equal to n sub t squared.
The square of the number in the current population divided by this maximum possible population.
This model is a nonlinear interaction.
The n sub t squared term comes from the fact that there are on the order of n squared
possible pairwise interactions between the individuals.
This is a simplification of course from reality.
But this model is highly simplified model of real world population growth.
It’s meant to capture the effects of nonlinear interactions
and it’s only very approximate to what happens in reality.
Let’s extend the model and make even more general by allowing
some of offspring bunnies to die of causes not due to overcrowding.
We’ll call this the death rate.
Our extended model is the same except
now we’re gonna have birth rate minus death rate.
This is called the logistic model
and was developed by the mathematician Pierre Verhulst
in the early eighteen hundreds.
Now let’s see this run in Netlogo
The model was called the Logistic model dot nlogo.
And as usual you can download it from the link below
or from the course material page.
Here’s what the model looks like.
We have an initial population slider,
and sliders for here birth rate and death rate.
Notice that these are now real numbers not integers
and if the model comes up with a real numbered value
for the population size at the next generation,
it just rounds off that real number to produce an integer.
There is also a slider for carrying capacity
which is the scientific name for the maximum population
that the habitat can support.
Ok so let’s set the initial population here to 1
The birth rate to 2 point 0
Let’s leave the death rate at 0
And let’s set the carrying capacity to 50
Do set up and now let’s just reproduce several times in a row
So the population is growing, growing, growing
and notice here that it has fixed at 25
that is if I do another run of reproduce
it just stays at 25
now note that this population versus time plot
is no longer exponential in its shape
it doesn’t go up really fast
instead, starts out by going up quickly but then
starts slowing down in its growth until it flattens out.
This is called the logistic function instead of an exponential function
and the plot of this year’s population versus last year’s population
is no longer a straight line.
Because we no longer have a linear system.
Now let’s go back to the netlogo model.
An obvious question is why we didn’t the number of bunnies go to 50
which is the carrying capacity.
Well, this is just the way the model works
if there are 25 individuals then 25 squared divided by 50
equals 12 point 5 of those individuals will die off
and the remainder have a birth rate of 2 to create offspring
for the next generation so we get 25.
It’s a mathematical model not necessarily a realistic model of population growth
especially with these small numbers.
It’s meant to capture the nonlinear nature of how population
actually grows in the real world but not necessarily to be precise.
So what we’re interested in here is not how realistic the model is
but what effect the nonlinear aspect of it has on its behavior.
Let’s go back to the model.
So our question is
Is the whole the sum of the parts?
as we saw it was in the linear model.
Let’s do what we did before.
Run, reproduce for 3 time steps
Here we get 7 bunnies, because of rounding off.
But now let’s set the initial population to 5
That is 5 times the original.
Do set up and again run for 3 time steps.
And you see here that the number of bunnies is not 5 times 7.
Which would have been sum of the parts of the first version of this.
That is one individual produced 7 individuals after 3 time steps.
So you think that 5 individuals would reproduce 5 times that
or 35 individuals after 3 times steps.
But instead that, we get 21
So in this nonlinear case
the whole is different from the sum of the parts.
So now you’ve seen the difference between linear and nonlinear systems.
That is what happens when the parts interact in a nonlinear way.
This is a key concept in complex systems that will come back to again and again
Now it’s time for you to do a few short exercises.