Hello and welcome to unit four of the course.
This is the first of two units
on bifucations.
Sudden qualitative changes in the behavior
of dynamical system has a parameter
is varied continuously.
In this unit we will look
at bifurcations in differential equations.
The next unit will be on bifurcation in iterated functions.
like the logistical equation we studied in unit three.
I will begin this unit
by reviewing qualitative solution techniques
for differential equations.
I will then introduce the differential equation version
of the logistical equation from the last unit
and I will use this example to introduce the idea of bifurcations.
So, let's get started.
So here is the logistic differential equation.
dp/dt = rP(1-p/k)
It looks similar to the logisitcal equations
from the previous unit.
However this is a differential equation.
This is a dp/dt on the left.
So it means it will have somewhat different properties.
I'll compare and contrast the differential equation
and the iterated function in the next video.
For now I want to focus on properties
of this equation
So this equation has two parameters .
R as before is a measure of the growth rate
The larger the r, the larger the population will be.
P here again, I'm picturing
as population as some animal or something.
And K is the parameter known
as the carrying capacity.
For the iterated function,
the iterated logistical function
this was a and it was known as the
annihilation parameter.
K and A are they mathematical appear in the same place.
They have different meanings or do different things.
For this differential equation.
So, this is the differential equation that we studed
in unit 2.
dp/dt the rate of change of P.
is a function of P.
How fast the population grows
depends on the current population and these
two parameter values.
Here is a plot of the right hand side of this.
And for concreteness,
What did I choose?
I choose a k of 100 and of a r of I think 3.
So, let me just make a note of that.
Here k is a 100 and r is 3.
so from a graph like this
we saw in unit two
that we can get alot of qualitative information.
about the behavior of this dynamical system.
When this function is positive.
That means the growth rate is positive.
And the population is increasing.
So any population that is a little larger than zero
or less than 100.
will increase up to zero.
If the population is larger than 100.
The growth rate is negative
so the population increases.
If the population is less than zero.
That doesn't really make any sense
but mathematically if the population were negative.
The growth rate would be negative.
So it would push that number
to the left.
So we can draw a phase line
which were this.
So let me do this
Here is the phase line
And, we have two fixed points.
One of the fixed points is at zero
And the other fixed point is at 100.
And this fixed point is stable
It is an attractor
Anything between zero and 100
get pulled toward it
anything larger than 100 get pulled toward it as well..
Population larger than 100 decrease
until they reach 100
Population between 0 and 100 increase
until they reach 100 as well.
And then zero would be an unstable fixed point.
A repeller.
If you are near zero and you move a bit to each side
you get pushed away and in this case towards
the attractor at 100.
And in this case you would go towards negative infinity
So this is the phase line
for this differential equation.
We can also sketch solutions
to the differential equations.
And the solution in this context
is population as a function of time.
So, let's see.
We know 100 is a fixed point.
And sorry let me move that up.
There we go.
We know that 100 is a fixed point
Let's see. I'll draw some solution curves
in blue.
We started around 20, we would increase
We would increase rapidly until we get 100.
If we started above 100.
We would decrease and approach 100.
So here are three different solutions.
P as a function of time.
as this blue curve is showing.
This is P, this is T.
And in all cases they approach this fixed point
at 100.
Draw that through with a dashed line
So again without doing any caclulus
or using Eulers method
We can get a reliable shape of these curves.
So lastly, let me say a little bit
about this quanitity of K.
Why it is known about carrying capacity.
We this says--this equation says
that any population positive population
Real population.
you start of with, is going to go to 100.
So a 100 is in a sense the equilibrium population.
It's the number of creatures this system
whatever it is
can support.
Ok, so this is a qualitiative approach
to solutions of a logistical equation
And, after this is a quiz
to refresh your memory on how all this works
Then, I'll discuss the logistical iterated function
and I'll compare and contrast
the differential equation and the iterated function.