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