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.