 Welcome to the screencast on section 7.6, where we'll talk about setting up a logistic differential equation. Logistic DEs are a special type of differential equation that are especially good at modeling the growth of animals in a population. So let's take a look at our situation here. We have a population of bacteria growing in a petri dish, and the number of bacteria, called P of T, which is measured in thousands of bacteria, after T hours, is given in this table. You can see that we measured the number of bacteria every 15 minutes or quarter of an hour. And we want to create a model for the number of bacteria, presumably so that we could predict how they'll grow in the future. Well, as it turns out, this data is probably noisy, and so when we create a model it won't be perfect, but we want to create the best model that we can. And so the first thing that I would always do is I would plot the data, take a look at it, and see if that gives me any information about the shape of this information. So here's a plot that I created for you. Taking a look at this, you can see that this follows sort of an S-shaped curve. So it looks like the data goes about like this. This is different from the exponential model for population growth that we've already seen. Exponential models follow an exponential curve that looks something like this, and they just boom as the number of bacteria or whatever other animal you have grows. But in this case, it seems that we start out exponential, and then later on the rate of growth of this population seems to be decreasing. I'm drawing some tangent lines to demonstrate that. So we might suspect this is because as we get more and more bacteria, they're competing for space and food and other resources, and so they can't grow quite as quickly. We want to find a way to incorporate that into our model that we're creating. So the big idea that we're going to use to do that is called the per capita growth rate, which is p' over p, the rate of growth of the bacteria divided by the number of bacteria. This is very useful in creating population models. The reason we're thinking about it now is because we notice that as p got larger, p' seemed to get smaller, and so we want to compare the two of them, which we'll do by dividing them. Let's take a look at where this comes from also. In our previous exponential model, which looked like this, we had a constant rate of growth that we called k, and if we divide this by p to get p' over p, we'll have that as equal to k. So a way that we could read this new equation is that the per capita growth rate in an exponential model is a constant k. The bacteria are always growing at the same rate, and because we get more and more bacteria, that means they grow more and more and more, compounding as they go on. That doesn't seem to be happening in our situation here. So we're going to want to look at the per capita growth rate, p' over p, and try to figure out what's going on with it. To do that, we really need to calculate the p' over p using the data provided to us. But we don't have p' we only have p. And so in order to calculate p' we're going to use a central difference estimate, which is a way of estimating the derivative that we learned back in Calc 1. So you may want to remind yourself of that. As an example, if I wanted to calculate p' of 0.25 divided by p of 0.25, so this would be the per capita growth rate at time 0.25, I can do that by using the two p values on either side of 0.25 and basically calculating the average rate of change between them. So in order to do this, to calculate that derivative up top, so p of 0.5 minus p of 0 and I'll divide by the distance between them times p of 0.25. So this last bit that I calculated right here comes from the denominator of the per capita growth rate and the rest of it is what I get when I calculate the central difference estimate for the derivative. So we're using our given data to calculate a derivative and then calculate the per capita growth rate. So if you do the calculations, you can find that this comes out to approximately 8.88 over 7.04, which is approximately 1.26. And that's our per capita growth rate at time 0.25 and it's saying something about how the bacteria are growing when there are about 7,000 bacteria, which is how many there are at time 0.25. So you could do a lot more of these calculations by hand, but I don't recommend that. This is a perfect thing for technology to do. And so using Geogebra, I calculated the rest of the per capita growth rates for us and I put them together in a table right here. So you can see the 1.26 that we calculated in the previous screen, but then you can also see other numbers and you can also notice that as time goes on, the per capita growth rate is decreasing and this is just what we noticed in the graph. We saw that the rate of change of the bacteria was decreasing as there seemed to be more competition for space or food and so the bacteria couldn't grow as quickly. That's what we're noticing in this entire column right here. Now remember, we're interested in a relationship between the per capita growth rate and the number of bacteria. And so, again, we're going to plot this information, but we've transformed it instead of having time and population. Now we have population and per capita growth rate, a more useful set of data to compare. So once again we're going to plot this and again this is a perfect thing for technology to do. So here's a plot. So some things to notice about the plot. We now have population on the horizontal axis, measured in thousands, and we have the per capita growth rate on the vertical axis. And down here in the left corner, because we didn't have a population below about 7,000, we have a gap. So this doesn't begin at zero, but it ended at around 7 just for convenience of plotting. And of course, this graph looks like it's a nice straight line. That's a good simple relationship to have between p and the per capita growth rate. So if I drew this straight line in, it looks like I would get something with a negative slope, which makes sense, the per capita growth rate is decreasing. And I'd really like to calculate what that line is. So again, we can do this by hand by calculating the slope between some points but this is exactly what technology can do for us. So I'm going to switch over to GeoGebra, where I've put that information in already. You can see my list of points over here are the points that I had on that graph. And conveniently, GeoGebra has a command that lets you automatically create a straight line through any set of points. This is a linear regression, is what you would do on your calculator. And in GeoGebra, we call it fitLine. And all we have to do is type in a list of points and we'll automatically create and plot that line for us. And you can see the formula over here is even given to us, negative 0.1 plus 1.95. And rounding a little bit, I'd say this is approximately a line with a slope of negative 0.1 and a vertical intercept, or a y-intercept as we might call it, of about two. So going back to where we were here, that means that we've found a formula for this slope, or for this line, which is approximately given below. So a couple of important things to notice. I'm not using y and x. I'm using the correct names for the axes here. I have my vertical axis, p' over p, and I have my horizontal axis, p. And so I have a nice linear relationship between the per capita growth rate and the number of bacteria. Now I'd like to get rid of that fraction, so I just have it in the form p' and doing that, multiplying the p over, I get p times negative 0.1p plus 2. And also, by convention, if I pull out the 0.1 out front of everything, I'm going to get a 20 where the 2 was, and I'm going to get a minus p for the rest of it. And this form right here is what we call the logistic differential equation for this situation. It's the rate of growth, and it's a product between the population multiplied by some growth factor out front and this interesting number 20 on the inside. The number 20 is called the carrying capacity, and it refers to the maximum number of bacteria that this petri dish can handle, about 20,000 in this case. And that corresponds to what we saw, which was that the bacteria seemed to be leveling out as they got near 20,000. In the next video, we'll take a look at how to solve this differential equation and verify the qualitative statements we've made in this video.