Worked example: Logistic model word problem | Differential equations | AP Calculus BC | Khan Academy

The population ( p ) of ( t ) of bacteria in a petri dish satisfies the logistic differential equation. The rate of change of population with respect to time is equal to ( 2 ) times the population times the difference between ( 6 ) and the population divided by ( 8000 ), where ( t ) is measured in hours and the initial population is ( 700 ) bacteria. What is the carrying capacity of the population, and what is the population's size when it's growing the fastest?

Alright, so in order to even attempt to answer these questions, and at any point, if you're inspired, definitely pause the video and try to answer on your set by yourself. Let's just remind ourselves what we're talking about, or what they're talking about with the logistic differential equation and the carrying capacity.

So in general, a logistic differential equation is one where we're seeing the rate of change of—and it's often referring to population—so let's just stick with population. The rate of change of our population with respect to time is proportional to the product of the population and the difference between what's known as the carrying capacity and the population.

Now, why is this a model that you will see a lot, and especially why is it useful for studying things like populations? Well, when a population is small, the environment really isn't limiting it. So, assuming it starts from some non-zero value, this thing grows—this thing is not going to get much smaller. Our population's rate of change is going to increase.

Let me just draw a little graph here to show the typical solution to a logistic differential equation. So this is our population; this is time. When our population is low, let's say it's going to start from some non-zero value. If it was zero, what would happen? Well, then our rate of change would just be zero, and our population would never grow. That makes sense: if you have no bunnies on your island, then there never will be any bunnies on your island.

But if you have a few bunnies, well, initially, their rate of change—the rate of population—is going to keep accelerating. As this thing grows, it's going to keep accelerating, but then at some point, your environment is going to limit how many bunnies or, for example, bacteria can grow in your environment. Because once the population gets close to a, this thing over here is going to approach zero and it's going to make our rate of change smaller and smaller.

You can imagine in the limiting case, as ( p ) gets very, very, very, very close to ( a ), as ( p ) gets very, very, very close to ( a ), you can imagine as ( t ) approaches infinity, our rate of change is going to approach zero. So one way to think about it is our population would asymptote towards the carrying capacity. That is right over here; that is our carrying capacity.

So there's a couple of ways of answering this first question. One way is we can actually put our logistic differential equation in this form, and then we can recognize what the carrying capacity is. The other way is to think about well, what happens as ( t ) approaches infinity? As ( t ) approaches infinity, this thing approaches zero.

You can see the rate of change approaches zero, so when that approaches zero, what does ( p ) approach? We can just solve for ( 6 - \frac{p}{8000} ). Well, there are two situations—two ( p )'s for which our derivative is equal to zero. There's one case when this is equal to zero, in which case our population is zero, or the other case is when this is equal to zero.

So let's do that: ( 6 - \frac{p}{8000} = 0 ), or we could say ( \frac{p}{8000} = 6 ), and so we can say multiply both sides by ( 8000 ) ( p = 48000 ), which is exactly what we had right over there.

So now let's answer the second part: what is the population's size when it's growing the fastest? So intuitively, you can see when that is right over here. The population, the rate of change is going to grow, grow, grow, grow, grow, but then as we approach the carrying capacity the rate of change is going to start slowing down. So your maximum rate of change—it's growing the fastest—is right about, right about there.

But how do we figure it out exactly? Well, you could go back to the logistic differential equation. You can see that it's really our rate of change as a function—you could view it as a function of our population right over here, and this is actually a quadratic expression. This is a concave downward quadratic expression.

It would look like this: if you were graphing, if you were graphing rate of change ( \frac{dp}{dt} ) as a function of population. Well, when the population is small—so when the population is say ( 700 ), or that's where we're starting—well, let's just speak in generalities. When your population is small, ( \frac{dp}{dt} )—your rate—is small, but then it increases, and at some point, around there, it will start—the rate of change starts decreasing and it approaches zero as our population approaches the carrying capacity.

This right over here, for example, would actually be our carrying capacity. So one way to think about, well, what is this maximum point right over here? There are a couple of ways that you can approach it, with calculus and even algebra. We have many tools for identifying this maximum point, which is really just the vertex of this downward-opening, this concave downward parabola.

So let's just do that, and let's find this vertex. The vertex is just halfway between the zeros of this quadratic. So let's find the ( p )-values that make this equal to zero, which we actually just figured out. The maximum point—the vertex—is just going to be halfway between that.

So when the population is zero, our rate of change is zero. When the population is ( a ), which we know is ( 48000 ), our rate of change is zero. So our carrying—sorry, our maximum rate of change is going to happen halfway between those two points, which is ( 24000 ).

So it is a population of ( 24000 ). We got that by really just saying, well, at what point does this quadratic as a function of ( p ) hit a maximum point. Well, that's going to be halfway between the zeros—zeros happen when ( p ) equals zero and ( p ) is equal to ( 48000 ).

So that's going to happen when the population is ( 24000 ). This type of problem seems very intimidating at first: logistic differential equation—how do I actually solve this and then analyze it? But the key is to one, recognize the logistic differential equation, see what it's talking about, and then maybe think of this in terms of our rate of change as a function of our population.

Remember that the carrying capacity is what happens as ( t ) approaches infinity. As of ( t ) approaches infinity, our rate of change approaches zero. If that's approaching zero, what must our population be approaching?