It is believed that some foxes have started colonizing a small town. Observations suggest that there are currently 14 foxes. Using a logistic growth model with 𝑘 equal to 0.15, using a year as a unit of time, and a carrying capacity of 68 for this town, estimate the number of foxes that will be in this town in five years. Give your answer to the nearest whole number.
The question tells us about some foxes that have started colonizing a small town. We’re told that currently there are 14 foxes in the town. The question tells us that the population can be estimated by using a logistic growth model. And in our logistic growth model for the population of the foxes, we’re told our value of 𝑘 will be 0.15, our carrying capacity will be 68, and we’re using years as the unit of time. We need to use our logistic growth model to estimate the number of foxes that will be in the town in five years. We need to give our answer to the nearest whole number.
To start, if we call our function 𝑃 the population of the foxes after 𝑡 years, then the question wants us to estimate the value of 𝑃 of five. And since we know this follows a logistic growth model, let’s start by recalling what a logistic growth model is. This model tells us d𝑃 by d𝑡 will be equal to 𝑘 times 𝑃 multiplied by one minus 𝑃 over 𝐿, where 𝑘 is a measure of the growth of our population and 𝐿 is the carrying capacity.
But remember, the question wants us to approximate the population of foxes after five years. So, we want to find an expression for 𝑃 of 𝑡. To do this, we could solve our differential equation. However, we know in a logistic growth model, 𝑃 of 𝑡 will be equal to 𝐿 divided by one plus 𝐴 times 𝑒 to the power of negative 𝑘𝑡, where 𝐴 is equal to 𝐿, the carrying capacity, minus the initial population all divided by the initial population.
Now, let’s look at our question for information we’re given about our logistic growth model. First, we’re told that initially there are 14 foxes. So, in our case, 𝑃 of zero is equal to 14. Next, we’re told that the value of 𝑘 is equal to 0.15. Finally, we’re told that our carrying capacity, 𝐿, is equal to 68. So, now, we have values for 𝐿 and 𝑃 of zero which we can use to find 𝐴. And we know the value of 𝑘. So, we can find our solution 𝑃 of 𝑡.
Let’s start by finding the value of 𝐴. 𝐴 is equal to 𝐿 minus the initial population all divided by the initial population. In our case, that’s 68 minus 14 all divided by 14. And if we calculate this, we get 27 divided by seven. We’re now ready to use our logistic model to find an approximation for 𝑃 of 𝑡. In our numerator, we have 𝐿, which is equal to 68. Then, in our denominator, we have one plus 𝐴, which is 27 over seven times 𝑒 to the power of negative 𝑘𝑡. And 𝑘, in this case, is equal to 0.15.
And in this case, the question is asking us what the logistic model will approximate the population of our foxes will be in five years. So, to do this, we need to substitute 𝑡 is equal to five into our expression for 𝑃 of 𝑡. Substituting in 𝑡 is equal to five, we get 68 divided by one plus 27 over seven times 𝑒 to the power of negative 0.15 multiplied by five. And if we just calculate this expression, we get, to the nearest whole number, 24. And this is our final answer. Therefore, by using a logistic growth model, we were able to approximate the number of foxes in a town after five years. We got that after five years, there would be 24 foxes.