This video, we're going to do a side-by-side comparison of the exponential and logistic population growth models. Really, there's no new information in this video that we haven't already covered in our previous lesson videos. So, if you're already feeling confident about the exponential and logistic growth models, then you can feel free to skip this video if you'd like. But if you're struggling even just a little bit, then stick around because this video could be really helpful for you. More specifically in this video, we're going to observe how these two models differ from each other when we apply them to the same population.
We're going to set the initial population size or n as 50, the intrinsic growth rate or the per capita population growth rate or r as 1.0, and the carrying capacity or k as 3,000. When we do that and plot the data on a graph like this one down below that has time in years on the x-axis and the population size on the y-axis, we get these two curves. And, of course, this curve here in blue, which has a j-shape or the tail of an e, if you will, is going to be the exponential growth curve. And this other curve that you can see here in this brownish color, which has more of an s-shape, a sigmoidal shape, is going to be the logistic growth curve. What you'll notice here in comparing these two growth models is that the exponential growth curve is uninhibited without limitations, and the population growth continuously increases in a directly proportional manner as the population size increases.
Really, there's no sign of slowing down. It just keeps going up and up. Now with the logistic growth model, on the other hand, recall that it's pretty much the same as the exponential model, but it includes and accounts for environmental limitations on the population's growth. Notice that the carrying capacity, which is again set to 3,000, really only applies to this logistic growth model. The exponential one just zooms and soars right past it without any regard for limitations whatsoever.
What you'll notice with the logistic growth model is that early on, when the population size is small, it is very similar to the exponential growth. But as the population size continuously gets larger, these two models start to drift more and more apart from one another. When the population size reaches half of the carrying capacity, you'll notice that in the logistic growth model, it will start to slow down in terms of its growth. Eventually, once it reaches or approaches the carrying capacity, the logistic growth falls to 0, and that's why we have a flat line here. You can think of this carrying capacity as somewhat of a lid or cap to the logistic population growth.
So really, that's it when we look at this graph and compare these two models. Now, what we're going to do is look at this table form of the same exact graph just to give you another perspective on how these two models differ from each other when we compare them to the same population. Even though it looks extremely complex, it's not nearly as bad as it looks, and I just want you to understand a few basic things here. So, let's break it down. On the far left, we have the total population size or n, and this column applies to both models since again we're applying these models to the same population.
The next three columns apply only to the exponential population growth model, and the last three columns on the far right apply only to the logistic population growth model. Here in this column, we have r or the per capita population growth rate, which we set to 1.0. In the exponential growth model, r is always assumed to be constant, so it's always 1.0 for the entire exponential growth model. The next column is the equation for the instantaneous population growth for the exponential model, which from our previous lesson videos, is just dn/dt=r⋅n. All we need to do to get the data in this column is take the r value, which again is set to 1.0, and multiply it by the value of n, which is in this first column. For example, it takes 1.39 years for this population to reach a population size of 200 when it's growing exponentially.
Over here on the far right, we're showing you the r value for the logistic growth model, which is modified by this term 1 minus nk. The r value in the logistic growth model continuously decreases over time, whereas with the exponential growth model, the r value was consistent throughout. This decreasing r value ultimately accounts for the environmental changes and allows the logistic growth to compensate for limitations on population growth. The next column is the equation for the instantaneous population growth for the logistic model. Notice that this first part here in black text is exactly the same as it is in the exponential growth model, and really the only difference is the addition of this term 1 minus nk.
To get the data in this particular column, all we need to do is take this value here in this column, which is r times that term, and multiply it by n. For example, to get the figure we need, take 0.33 and multiply it by 2,000, and you will get an answer of positive 666. On the far right over here, what we have is the amount of time it takes in years for the population to reach the particular population size when it's growing via the logistic growth model. And so now that we've broken down the table, notice that the exponential growth model times are always going to be smaller than the times for the logistic growth model, so the exponential growth model is growing without limitations, uninhibited, but the logistic growth model is growing with limitations, and it is inhibited.
Even early on, before the population size reaches half the carrying capacity, what you'll notice is that the amount of time it takes is quite similar, but not exactly the same; in the logistic growth model, it's just a little bit slower, a little bit lagged. You'll notice that when the population size reaches half of the carrying capacity, which was set to 3,000, the logistic growth model reaches a peak instantaneous population growth of +750, but for the exponential growth model, there is no peak. It just keeps getting higher and higher, and there are no signs of it slowing down. However, you'll notice that as you go beyond the population size of half the carrying capacity and approach the carrying capacity, this instantaneous population growth continuously goes lower and lower until once it reaches the carrying capacity, the population growth reaches 0 for the logistic growth, and it takes an infinite amount of time, represented by the infinity symbol here, for the logistic growth model to actually reach that 3,000 mark. In reality, it is possible for populations to surpass their carrying capacity. In the logistic growth model, if it were to go up to 4,000, well above the carrying capacity, you'll notice that the value of r becomes negative, and a negative r value leads to a decrease in the population size. And so, the population growth also becomes a negative value as well. If the population does surpass the carrying capacity in the logistic growth model, it's going to quickly fall below the carrying capacity shortly after.
This concludes our video comparing the exponential to logistic growth models. Hopefully, this was helpful for you, and moving forward, we'll be able to apply these concepts in problems. I'll see you all there.