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.