A population of spotted hyenas in a South African National Park currently has 87 individuals. The carrying capacity for spotted hyenas in the park is estimated to be 125 hyenas & the intrinsic growth rate ( r r ) is 0.25 hyenas/year/hyena. What is the instantaneous rate of population growth of the spotted hyena population?

So here we have a practice problem that says a population of spotted hyenas in a South African national park currently has 87 individuals. The carrying capacity for spotted hyenas in the park is estimated to be 125 hyenas, and the intrinsic growth rate or the per capita population growth rate or r is 0.25 hyenas per year per hyena. What is the instantaneous rate of population growth of the spotted hyena population? And we've got these 4 potential answer options down below. Now notice that this problem gives us a carrying capacity, and because that's the case, that's a big hint that these spotted hyenas are growing via the logistic population growth model. And so recall from our previous lesson videos that the equation for the instantaneous rate of population growth for the logistic growth model is this equation that you can see right here, which is d n over d t, the instantaneous rate of population growth, is equal to r times n times this factor here, which is 1 minus n over k. And so all we need to do is plug in values into this equation, type them into our calculators to get the answer. And so when we do that, what we get is dn over dt is equal to r, which is given to us as 0.25, times n, which is the population size given to us as 87 individuals, times 1 minus n, again, 87 individuals, over k, which is the carrying capacity given to us as 125 hyenas. And so when you type all of this into your calculator, what you get is an answer of about 6.6, and that's approximately equal to 7. And so notice that this answer here of 7 matches with answer option b, which says 7 hyenas per year. And so that is the instantaneous rate of population growth under these conditions. So option b is the correct answer to this practice problem. That concludes this practice, and I'll see you all in our next video.