Logistic Population Growth - Online Tutor, Practice Problems & Exam Prep

The logistic population growth model accounts for environmental limitations, contrasting with the exponential model, which assumes unlimited resources. The carrying capacity (k) represents the maximum sustainable population size, creating a sigmoidal growth curve. As population size (n) approaches k, growth rate decreases, eventually reaching zero. The equation for instantaneous growth rate incorporates a term, $1-\frac{n}{k}$, differentiating it from exponential growth. Understanding these dynamics is crucial for studying population ecology and resource management.

Logistic Growth in a Limited Environment

In this video, we're going to talk about the 3rd and final population growth model in our lesson, which is the logistic population growth model, and we'll talk about how logistic growth occurs in a limited environment. Now, recall from our previous lesson videos that in nature, unlimited exponential population growth is somewhat rare. And when it does occur, it does not occur for too long, and this is because of limited resources in nature. And so this logistic population growth model is pretty much exactly the same as the exponential population growth model, except for the very important fact that the logistic population growth model accounts for environmental limitations on population growth, and this includes the limitations that are caused by density-dependent factors. And so recall from our previous lesson videos that density-dependent factors are associated with a carrying capacity.

And so these density-dependent factors that are part of this logistic growth model will prevent the population size, or n, from permanently exceeding the carrying capacity, or k. And so once again, recall from our previous lesson videos that the carrying capacity is the theoretical maximum population size that an area can sustain at any given time. And so notice that in this graph down below where we have time on the x-axis and population size on the y-axis, that this horizontal green dotted line here represents the carrying capacity, which again is abbreviated with the variable k. And the carrying capacity mathematically serves as a horizontal asymptote that somewhat creates a lid or a cap on the population growth, again, preventing the population size from permanently exceeding the carrying capacity. Now notice over here on the right hand side, we're showing you the equation for the instantaneous population growth rate for this logistic growth model.

And what you'll notice is that this part of the equation highlighted here with black text is exactly the same as the instantaneous growth rate equation for the exponential growth model. And, really, the only difference here is the addition of this term, which is being multiplied as 1 minus nk. And so the addition of this term is really what differentiates the exponential growth model from the logistic growth model, and the addition of this term is what accounts for the environmental limitations on population growth. And so the addition of this term is ultimately what gives the logistic growth curve a sigmoidal curve. So it's a sigmoid curve, which means that it is an S-shaped curve, and we can see that here in this graph where the curve somewhat resembles the shape of an S.

Now what's really important to note is that when n or population size is relatively small in the logistic growth model, the logistic population growth is going to be approximately, but not perfectly equal to, the exponential population growth. And we can see that over here with this blue highlighted region showing you how early on the growth is approximately exponential in this logistic growth model. However, once the population size is equal to half of the carrying capacity or k, the population growth rate in the logistic model will start to slow down, and that's unique in this model. And so we are indicating that with this grayish background, that you can see throughout the rest of this curve. And so what's important to note is that as the population size or n increases and when n starts to approach the carrying capacity k, the logistic population growth will also approach 0.

And so notice that later growth falls to 0, and that's what creates this somewhat of a horizontal line here in the population growth, as, again, the population size is approaching the carrying capacity. Now it is important to note that in reality, it is possible for population size to surpass the carrying capacity. But, again, this is only temporary because shortly after, the population size will decrease or drop either to the carrying capacity just beneath it or it will crash, you know, significantly further in some cases. But in many cases, it is possible for the data to kind of fluctuate right around the carrying capacity as you see here and then stabilize at the carrying capacity over long enough periods of time. And so this here concludes our lesson on the logistic growth in a limited environment, and moving forward, we'll be able to apply these concepts in many problems.

So I'll see you all there.

Logistic Population Growth Example 1

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.

In some cases, a population can exceed its carrying capacity for a brief period of time. In a population of dragonflies that approaches its carrying capacity, many females use the last of their energy reserves to reproduce. As a result, the population size reaches 675 dragonflies, but the estimated carrying capacity ($K$) of the area is 550 dragonflies. The $r_{\max }$ value for this population is 0.5. Calculate the rate of population growth (dN/dt) & the per capita rate of growth, r(1 – N/k).

• A. dN/dt = –97, r(1 – N/k) = –0.14
• B. dN/dt = –77, r(1 – N/k) = –0.11
• C. dN/dt = –153, r(1 – N/k) = –0.23
• D. dN/dt = 828, r(1 – N/k) = 0.23

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 ($r$) is 0.25 hyenas/year/hyena. What is the instantaneous rate of population growth of the spotted hyena population?

• A. 22 hyenas/year.
• B. 7 hyenas/year.
• C. 15 hyenas/year.
• D. 13 hyenas/year.

Exponential Growth vs. Logistic Growth

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.

Summary of Key Features & Assumptions for Each Population Growth Model

In this video, we're going to summarize many of the key features and assumptions that we already covered in our previous lesson videos for each of the population growth models, and so there's no new information in this video. If you're already feeling really confident about these three population 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. Notice that down below in this table on the far left, we're listing out either the feature or assumption in a yes or no question format, making it really easy for us to fill out this table. We've got columns for each of the population growth models: the linear, exponential, and logistic population growth models.

The first question asks whether there is a constant population growth rate, or is there a constant delta n over delta t? It turns out that a constant population growth rate is a unique feature of the linear population growth model, so we can put yes here to answer this question. The linear population growth model has a constant population growth rate or a constant delta n over delta t, regardless of the value of n or the population size. This is not going to be the case for the exponential or logistic population growth models, whose population growth rate continuously changes as the population size changes. The next question is asking if the population growth rate or delta n over delta t is proportional to the population size, n.

It turns out that a perfectly proportional relationship between the population growth rate and the population size is a unique feature of the exponential population growth model. So we can put yes here for that question. In fact, the population growth rate is directly proportional to the population size, which means as the population size increases, the population growth rate will also continuously increase at a rate directly proportional in the exponential population growth model. In the linear population growth model, the answer to this question is no.

Again, it has a constant population growth rate regardless of whatever the population size is. For the logistic model, the answer is also no, because although it can get really close to being perfectly proportional, the logistic model is never actually perfectly proportional. The next question here asks whether the population growth rate or delta n over delta t initially increases, then decreases as the population size or n gets large. This initial increase and then decrease is a unique feature of the logistic population growth model. So we can put yes over here for the logistic population growth model to answer this question.

For the linear population growth model, the answer to this question is no. Again, the population growth rate is constant regardless of the population size in the linear model, and the answer to this question is also no in the exponential population growth model, because the population growth rate continually increases in the exponential model. So there's no decreasing happening of the population growth rate in the exponential model. This is a unique feature of the logistic population growth model. The next question is asking if there is a constant per capita population growth rate or a constant r value?

Recall that for the linear population growth model, it is not on a per capita basis. So it essentially ignores per capita growth, so this question is simply not applicable to the linear population growth model. For the exponential population growth model, the answer to this question is yes. There is a constant per capita population growth rate, and it's always assumed to be constant in the exponential population growth model. For the logistic population growth model, the answer to this question is no.

Although it does utilize a constant r value, that r value is being modified by this additional term here, 1 minus n over k. This term is essentially going to cause the value of r to continually decrease as the population size increases in the logistic model. Because of this term, the value of r is not constant. The next question is asking, are there unlimited resources? The only model that accounts for limitations in environmental resources is the logistic growth model.

So we can put, no, there are not unlimited resources in the logistic growth model because it's the only model where resources are considered limited. The other two models, the answer to this question is essentially yes. There are no limitations in resources in those two models. The next question we're asking is, is there a carrying capacity or is there a value of k? Recall that the carrying capacity is unique to the logistic growth model.

So the answer to this question is yes. There is a carrying capacity, and that carrying capacity is what causes resources to be limited. In the other two models, there is no carrying capacity, so we put no. The next question is asking if the model considers density-dependent factors. Recall from previous lesson videos that density-dependent factors are associated with a carrying capacity, so the only model that considers density-dependent factors is the logistic population growth model, so we can put yes over here for that.

The other two models don't account for density-dependent factors, so we can put no. Notice that down below, the rest of all of these that we see here all have the same exact answer, which you'll notice is that these are the assumptions that we had covered in our previous lesson videos that are consistent for all three of these models. In terms of considering a closed population where the population size is only determined by births and deaths and it essentially ignores the effects of immigration and emigration or assumes that they completely balance each other out so that they don't need to be considered, that is an assumption of all of these models. Assuming a homogeneous environment where the environment is uniform, that is also an assumption of all three of these models. Ignoring age structure, ignoring sex ratio, and ignoring external factors that affect n or population size, such as ignoring density-independent factors, these are all going to be the case for all of these models.

So for all of these models, they have the same answer to these questions, and the answer is yes. This here concludes our summary of the key features and assumptions of each model, and I'll see you all in our next video.