The population of the citizens inside a village has the carrying capacity of 600 and a growth rate of four percent. If the initial population is 120 citizens, what is the population of the citizens in the village at any time?
We’re told some information about the population of citizens inside of a village. We’re told that this population has a carrying capacity of 600 and it has a growth rate of four percent. We’re also told that the initial population inside of the village is 120 citizens. We need to use this information to model the population of the citizens in the village at any time.
Since we’re asked to model the population inside of a village and we’re given the carrying capacity, growth rate, and initial population of our citizens, we can do this by using the logistic growth model. So we’ll call the population of our village capital 𝑃. And we’ll call the time 𝑡. And it’s important to remember that the units of time will need to match the units that our growth rate is given in. In this case, our growth rate is not given with a unit of time, so we don’t need to include units. However, it’s important to keep this in mind.
We recall the logistic growth model will then tell us that d𝑃 by d𝑡 is equal to 𝑘 times 𝑃 multiplied by one minus 𝑃 divided by 𝐿, where 𝑘 is the growth rate of our population and 𝐿 is the carrying capacity of our population. And we’re given the growth rate and carrying capacity of this population. So we could try and solve this differential equation by using what we know about separable differential equations. However, we can also find the solution in the general case and then use this to find it in our specific case. So it’s often easier to remember the solution and then apply it.
We get 𝑃 is equal to 𝐿 divided by one plus 𝐴 times 𝑒 to the power of negative 𝑘 times 𝑡, where 𝐴 is equal to 𝐿 minus 𝑃 of zero all divided by 𝑃 of zero. And 𝑃 of zero will be the population when 𝑡 is equal to zero, in other words, the initial population. And we’re given this information as well. 𝑃 of zero is equal to 120. So we’re almost ready to start using our logistic growth model. First, our value of 𝑘, the growth rate, is four percent. We need to write this in decimal form as 0.04.
Next, we need to remember, in the question, we’re told the carrying capacity of our population is 600, so we’ll set 𝐿 equal to 600. Now that we know the carrying capacity 𝐿 is 600 and our initial population is 120, we can find the value of 𝐴. Substituting these values into our expression, we get 𝐴 is equal to 600 minus 120 all divided by 120. And if we evaluate this expression, we get four. And now that we know the value of our carrying capacity, our growth rate, and the constant 𝐴, we can find an expression for the population at time 𝑡.
Substituting in 𝐿 is equal to 600, 𝑘 is equal to 0.04, and 𝐴 is equal to four, we get that our population at time 𝑡 is equal to 600 divided by one plus four times 𝑒 to the power of negative 0.04𝑡. And we could leave our answer like this. However, we’ll simplify this by multiplying both our numerator and our denominator by 𝑒 to the power of 0.04𝑡. Doing this, we get 600𝑒 to the power of 0.04𝑡 all divided by 𝑒 to the power of 0.04𝑡 plus four. And the last thing we’ll do is rearrange the two terms in our denominator. And this gives us our final answer of 600𝑒 to the power of 0.04𝑡 divided by four plus 𝑒 to the power of 0.04𝑡.
In this question, we were given a real-world problem involving the population of citizens inside a village. And we were able to use the information given to model the population. Our model showed the population after time 𝑡 would be approximately equal to 600 times 𝑒 to the power of 0.04𝑡 divided by four plus 𝑒 to the power of 0.04𝑡.