Question Video: Using the Logistic Growth Model on a Real-World Problem to Estimate the Length of Time It Would Take a Population to Reach a Certain Amount Mathematics • Higher Education

Video Transcript

The population growth of wolves in a national park follows a logistic growth model with an initial population of 15 wolves, a 𝑘-value of 0.05 — using a year as the unit of time — and a carrying capacity of 80. In roughly how many years does the model predict a wolf population of 60?

To start, the question tells us the population growth of some wolves in a national park is following a logistics growth model. And we’re told some information about our logistics growth model. We’re told the initial population of our wolves is equal to 15. We’re told the 𝑘-value of our logistics growth model is equal to 0.05, where a year is used as our unit of time. Finally, we’re told the carrying capacity of our logistics growth model is equal to 80.

We need to find the amount of time in years it takes our wolves population to reach a value of 60. In other words, if we call a function 𝑃 the population of wolves after 𝑡 years as predicted by our logistics growth model, then we want to find the value of 𝑡 where 𝑃 of 𝑡 is equal to 60.

The first thing we need to do is recall exactly what the logistics growth model is. We recall the logistics growth model will tell us that d𝑃 by d𝑡 is equal to 𝑘 times 𝑃 multiplied by one minus 𝑃 divided by 𝐿, where 𝑘 is a measure of the growth rate of our population and 𝐿 is the carrying capacity. And we’re told the population growth of wolves in the national park follows a logistic growth model. So, we can approximate our population by using this formula.

In fact, we can see from the question we’re given information about our logistic growth model. We’re told our value of 𝑘 is equal to 0.05, where, of course, we’re measuring 𝑡 in years. And we’re also told the carrying capacity of our population is equal to 80. So, our value of 𝐿 will be equal to 80. So, at this point, we can substitute these values into our growth model and then solve the differential equation.

However, we already know the solution to the logistic growth model. First, we get the general solution to this differential equation 𝑃 of 𝑡 is equal to 𝐿 divided by one plus 𝐴 times 𝑒 to the power of negative 𝑘𝑡. And to find the specific solution for our population, we need to use the fact that 𝐴 is equal to 𝐿 minus 𝑃 evaluated at zero all divided by 𝑃 evaluated at zero.

And it’s worth pointing out you’ll often hear 𝑃 evaluated at zero called the initial population. This is because it’s the population when 𝑡 is equal to zero. In fact, the question actually tells us the initial population. We’re told the initial population of our wolves is equal to 15. So, we also know 𝑃 of zero is equal to 15. In this case, remember, the question wants us to find the value of 𝑡 where 𝑃 of 𝑡 is equal to 60. So, the first thing we’re going to need to do is find an expression for 𝑃 of 𝑡, our population of our wolves after 𝑡 years.

We see that 𝑃 of 𝑡 is equal to 𝐿 divided by one plus 𝐴 times 𝑒 to the power of negative 𝑘𝑡. And since we have the values of 𝑃 zero and 𝐿, we can calculate the value of 𝐴. And we know the value of 𝑘. So, we can find an expression for 𝑃 of 𝑡. So, let’s start by calculating the value of 𝐴. It’s equal to 𝐿 minus 𝑃 of zero divided by 𝑃 of zero. Substituting in 𝐿 is equal to 80 and 𝑃 of zero is equal to 15, we get 𝐴 is equal to 80 minus 15 divided by 15, which we can calculate is equal to 13 divided by three.

Now that we found the value of 𝐴, we’re ready to find an expression for 𝑃 of 𝑡. It’s equal to 𝐿 divided by one plus 𝐴 times 𝑒 to the power of negative 𝑘𝑡. Substituting in 𝐿 is equal to 80, 𝑘 is equal to 0.05, and 𝐴 is equal to 13 over three, we get 𝑃 of 𝑡 is equal to 80 divided by one plus 13 over three times 𝑒 to the power of negative 0.05𝑡.

Remember, we want to solve 𝑃 of 𝑡 is equal to 60. So, we’ll substitute this into our equation. So now, we want to solve 60 is equal to this expression for 𝑡. To start, we’ll multiply both sides of our equation through by the denominator. This gives us 60 times one plus 13 over three 𝑒 to the power of negative 0.05𝑡 is equal to 80. Next, we’ll divide both sides of our equation through by 60. And of course, we can simplify 80 divided by 60 to be equal to four over three.

So now, we have the equation one plus 13 over three 𝑒 to the power of negative 0.05𝑡 is equal to four over three. We can simplify this further. We’ll subtract one from both sides of the equation. And, of course, four over three minus one is equal to one over three. We can simplify this further by dividing both sides of our equation through by 13 over three. This is the same as multiplying both sides by three over 13.

And, of course, one-third times three over 13 is equal to one over 13. This gives us 𝑒 to the power of negative 0.05𝑡 is equal to one over 13. To solve this, we’re going to need to take the natural logarithm of both sides of the equation. Keeping in mind that both sides of this equation are positive, taking the natural logarithms of both sides, we get the natural logarithm of 𝑒 to the power of negative 0.05𝑡 is equal to the natural logarithm of one over 13.

Now, remember, the natural logarithm function and the exponential function are inverses. In other words, the natural logarithm of 𝑒 to the power of 𝑥 is just equal to 𝑥. So, the natural logarithm of 𝑒 to the power of negative 0.05𝑡 is just equal to negative 0.05𝑡. And now, we can just solve for 𝑡 by dividing both sides of the equation through by negative 0.05. So, we get that our value of 𝑡 is equal to the natural logarithm of one divided by 13 divided by negative 0.05.

And if we calculate this, we see we get approximately 51.3 years. However, since this is a model and we want to know approximately how many years our wolf population will take to reach 60, we’ll round this the nearest number of years, which is 51. Therefore, by using a logistic growth model, we were able to show that in approximately 51 years, the wolf population in this national park will reach 60.