So here we have an example problem that says, for a population of barn owls, the intrinsic growth rate or the per capita population growth rate, or r, is 0.15 owls per year per owl, and the carrying capacity or k is 500 owls. Calculate the instantaneous population growth rate when the population size is 50 individuals, 250 individuals, and 450 individuals. First, we need to realize that this problem gives us a carrying capacity, and that's a huge hint that this population of barn owls is probably growing via the logistic population growth model. Recall from our last lesson video that the equation for the instantaneous population growth rate for the logistic population growth model is this equation, which is dN/dt = r∙N∙(1 - N/K).
For the first population size of 50, what this will look like is dN/dt = 0.15∙50∙(1 - 50/500). When you type all this into your calculator, what you get is that the dN/dt, or the instantaneous population growth rate, is going to be 6.75, and the units will be owls per year. That's when the population size is 50 individuals.
For each of these other ones, all we need to do is basically just copy the same exact thing and substitute the values of N. So, for the population size of 250, the calculation becomes dN/dt = 0.15∙250∙(1 - 250/500) which equals 18.75 owls per year. This is the answer for when the population size is 250 individuals.
For the last, with a population size of 450, the calculation is dN/dt = 0.15∙450∙(1 - 450/500) which once again provides a result of 6.75 owls per year.
Notice that this answer is the same as the first and that's because, once the population size reaches half of the carrying capacity, which notice 250 is half of 500, which is the carrying capacity, the population growth rate begins to decrease. In the stages, when the population size is increasing towards 250, this growth rate will continuously increase. Once it hits half the carrying capacity, it reaches a peak in the instantaneous population growth rate at 18.75. From that point on, as the population size increases, the population growth rate continuously decreases. So, for 50 and 450, being equidistant from half the carrying capacity, they will share the same instantaneous population growth rate.
These are the answers to this example problem. That concludes this example, and I'll see you all in our next video.