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

The exponential population growth model describes how populations grow exponentially when the per capita growth rate (r) is positive and constant. This growth is represented by the equation nt=n0⁢ert, where n0 is the initial population size. The growth rate can be analyzed using dn/dt=rn for instantaneous rates and ∆n/∆t=r∆tn for average rates over intervals. This model highlights the limitations of resource availability in natural settings.

Exponential Population Growth

This video, we're going to talk about the exponential population growth model. And so as long as the per capita population growth rate or r is both positive and constant, any population will grow exponentially over time. And so this exponential population growth really just refers to uninhibited population growth, where the population growth is increasing proportionally with the population size. And therefore, as population size or n increases, the population growth rate or ΔnΔt will also continuously increase in a manner that is directly proportional. And so this perfectly directly proportional relationship between the population growth and the population size is unique to this exponential population growth model.

Now exponential population growth does occur in nature, but it only occurs in very specific situations and over relatively short periods of time. And so exponential population growth is practically impossible in nature for long or extended periods of time because resources in nature are actually limited. And those limited resources somewhat inhibit or prevents populations from growing exponentially forever. And so notice that in a graph like the one down below where we have time on the x-axis and population size on the y-axis, exponential population growth will create somewhat of a curve that has a J shape, if you will, or because the e in exponential growth is somewhat unique, you could think of it as the shape of a lowercase e where the tail kind of curls up, if you will. So that could be helpful for some of you.

And again, any population will grow exponentially as long as the per capita population growth rate or r is both positive and constant, and that's always assumed to be the case in the exponential population growth model. And so notice that for this particular curve, the r is indicated as 1.0. Now also recall from previous lesson videos that there is a maximum per capita population growth rate or a maximum r value that is species-specific and only occurs under ideal conditions, and we refer to this as the r_max value. And so also recall that the greater the value of r, the faster the population size will increase. And so if we assume that the r_max value here is 1.5, then we can expect the exponential growth curve to increase even faster and look something like that.

And again, if the r value is smaller like 0.5, then it can still grow exponentially, it'll just grow slower. So the exponential curve might look something like this, for example. And so this here concludes our lesson on the exponential population growth model. Moving forward, we'll be able to apply these concepts, and then we'll talk about the equations for this exponential population growth model. So I'll see you all in our next video.

Exponential Population Growth Example 1

So here we have an example problem that asks, a population will grow exponentially under which of the following conditions? And we've got these four potential answer options down below, where option a says unlimited food, b says unlimited space, c says no competition from other species, and d says no density dependent factors. And so what we need to recall from our previous lesson videos is that the availability of food, space, and the presence and amount of competition are all examples of density dependent factors, and so option d somewhat encompasses options a, b, and c. And recall from previous lesson videos that density dependent factors are factors that limit a population's growth and whose impact on that limitation depends directly on the density of the population. And, usually, it's the case where the greater the density of the population and the closer the population size is to its carrying capacity, the greater the impact that these density dependent factors have on limiting that population's growth.

And also recall from previous lesson videos that populations that grow exponentially will grow uninhibited, and, therefore, they are not limited, and they will not be limited by density dependent factors. And so what we're saying here is that answer option d is the best choice because there are no density dependent factors that are going to have a significant limitation on populations that are growing exponentially. And so d here is the best option for this example problem. That concludes this example, and I'll see you all in our next video.

You’re studying 2 different populations of the same bacteria in the lab; both in the same unlimited, ideal conditions. Population-A starts with 2,000 individuals, while Population-B starts with 20,000 individuals. Assuming both populations grow exponentially, which population will have a higher population growth rate & why?

• A. Population-A because it has fewer individuals & therefore has more space to grow.
• B. Population-A because populations in ideal environments have a higher growth rate when N is smaller.
• C. Population-B because it can utilize resources more efficiently.
• D. Population-B because with exponential growth, larger populations have a higher growth rate.

Exponential Population Growth Equations

This video, we're going to talk about the 3 exponential population growth equations, which you can see down below in these boxes, and we'll also talk about how these equations relate to this graph here, which has time on the x-axis and population size on the y-axis. Now usually, it's the case that your professor will provide these equations to you, and in those cases, you don't need to worry about memorizing these equations, and instead, you just need to worry about what these equations describe so that you can apply them appropriately in problems. And so that's what we're going to focus on in this video. There's quite a lot going on in this graph and each of these boxes, so let's go ahead and push all of this over to the left-hand side. Let's wipe this graph clean and let's wipe out each of these boxes clean as well, so that we can approach this one step at a time starting with this equation over here on the far left.

So let's go ahead and make this equation bigger and shift it over to the right so we can see it more clearly. And this equation is n ( t ) = n 0 ⁢ e r t . Now what you should know about this equation is that it describes the exponential curve, which creates this j shape, if you will. This equation here, when you compare it to the equation of an exponential curve, they are actually identical to each other, and really, this equation just uses population growth variables. So, n(t) is the final population size, n0 is the initial population size, e is a constant which is Euler's number, which is approximately equal to 2.71828, r is the per capita population growth rate or the intrinsic rate of increase which is always assumed to be constant in the exponential growth model, and t is the elapsed time.

That's it for this first equation, and we can go ahead and minimize it and check it off and move on to the next equation here. Now let's go ahead and make this equation larger and shift it over to the right so we can see it more clearly. And this equation is d n d t = r n . The d's in this equation are calculus notation, but don't worry at all because you don't need to know any calculus at all in order to understand this equation. This equation describes the slope of the line tangent to any single point on the curve.

If we choose a point, right over here, where the coordinates of this point are going to be 1, 100, where we have a time unit of 1 and population size of 100, if we take the line that's tangent to this point, the slope of this line is described by this equation. d n d t literally just the slope of that line. This represents the instantaneous population growth rate at any time instant. r is the per capita population growth rate or the intrinsic rate of increase, which again is always assumed to be constant in this exponential population growth model, and then n is the population size at that time instant.

If we were to choose another point, let's say up here, this time with coordinates of 2, 1,000, where the time unit is 2, and the population size is 1,000, again, if we take the tangent line to this point, the slope of that tangent line is again described by the same exact equation. The slope of these tangent lines continuously increases over time for this exponential population growth model. That's it for this equation, we can go ahead and minimize it and check it off our list and move on to the 3rd and final equation here, this equation here in yellow.

Let's go ahead and make this equation larger and shift it up so we can see it more clearly. And this equation is Δ n Δ t = r Δ t n , where the Δt here is actually the subscript of the r. This equation describes the slope of the line connecting any two points on the curve. If we draw a line connecting two points, as you see here, the slope of this yellow line is described by this equation. Δ n Δ t , as we know from previous lesson videos, is the population growth rate. More specifically here, it represents the average population growth rate over a discrete time interval, and this will change depending on what the time interval is. rΔt is the apparent per capita population growth rate in that time interval. rΔt is different than r in the first two equations because where r is constant in these first two equations, rΔt is not a constant. It depends on the time interval as the notation here indicates with Δt. Last but not least, n is the initial population size. In this case, it would be 100.

That's it for this equation here, so we can go ahead and check it off our list as well. Here are the 3 equations in their large form so that you can see them more clearly. And again, we have this important note that r is always assumed to be constant in the exponential growth model. r appears in the first and second equation; and in this third equation, it's rΔt, so it's not a constant in this third equation, it's actually the apparent per capita population growth rate. Let me go ahead and remove my image so you can see all of these, very clearly.

That concludes this video, and I'll see you all in our next one.

Exponential Population Growth Example 2

So here we have a 3 part example problem where we can apply the equations of the exponential population growth model. And so part number 1 says, to calculate the population size at t equals 3 years using the initial population size of 100 and a constant per capita population growth rate, or r, of 2.303 individuals per year per individual. Now, of course, recall from our previous lesson videos that any population will grow exponentially as long as the per capita population growth rate, or r, is both positive and constant. And so 2.303 is positive and we're told by the problem that it's also constant. So this population will be growing exponentially.

Let's go ahead and pull up the equations for the exponential population growth model. So how do I know which of these three equations I need to use for this first part? Well, we need to look for clues in the problem, and so notice this problem wants us to calculate the population size, more specifically, the final population size after 3 years. Recall that the final population size is abbreviated with the variable n_t, and notice that the only equation that allows us to calculate the final population size or n_t directly is this first equation on the far left where n_t is already isolated. So that's the equation that we're going to use here for this problem, so let's go ahead and get rid of these other equations.

All we need to do now is just go ahead and plug in some variables and type them into our calculators. So n_t, the final population size we're trying to calculate, is going to be equal to n₀, which is the initial population size given to us as 100, and this is going to be times ert. R is given to us as 2.303, and t is given to us as 3 years. And when we type all this into our calculators, we get an answer of about 100,125, and the units are going to be individuals. And so this is the correct answer to part number 1, the final population size under these conditions.

Let's go ahead and move on to part number 2, and in part number 2, it says given the slope of the line between two points (1,100) and (2,1000), on an exponential growth curve described by the equation, calculate the average per capita population growth rate or r_Δt between t equals 1 and t equals 2. Again, let's pull up our 3 equations of the exponential population growth model. This problem is asking us to calculate the average per capita growth rate or r_Δt, which is only found in this equation over here on the far right. And so that's the equation that we're going to use for this part of the problem, so let's go ahead and get rid of these other equations and shift this over to the left, so that we have a little bit more room. We need to isolate r_Δt, and so we can do that by rearranging this equation algebraically by dividing both sides of the equation by n.

What we get is r_Δt is equal to Δn/_Δt all divided by n. Now we just plug in values into this equation. For Δn, we take the difference in population size, which is going to be 1000 minus 100, and for Δt, we take the difference in the time, which is 2 minus 1. This is all going to be over n, which is the initial population size, which is 100 here. So, 1000 minus 100 gives 900, and divided by 100, we get 9. And the units are going to be individuals per year per individual, which we will just abbreviate. This answer of 9 individuals per year per individual is the correct answer to part number 2.

Let's move on to part number 3, which is the last part of this problem. Part number 3 says, given the slopes of the lines tangent to the exponential growth curves at the points (1,100) and (2,1000), which are described by the equation dndt = rn, calculate the instantaneous population growth rate at t equals 1 and t equals 2. This problem makes it a little bit easy for us to decide which equation we need to use since it tells us, basically to use dndt = rn, and that's exactly what we want to use since dndt is the instantaneous population growth rate.

So, at t equals 1, with r still at 2.303 and n as 100, the calculation gives 230.3 individuals per year. For t equals 2, with r still at 2.303 and n now 1,000, the calculation results in 2,303 individuals per year. These are the two answers to part number 3, concluding this three-part practice problem. Hopefully, this was helpful for you, and I'll see you all in our next video.

If a bacterial population starts with 100 cells and has a per capita population growth rate ($r$) of 0.1 cells per hour per cell, what would be the population size after 5 hours?

• A. 165 individuals.
• B. 111 individuals.
• C. 272 individuals.
• D. 150 individuals.

Imagine a regional population of Atlantic Bluefin Tuna has an annual intrinsic growth rate ($r$) of 0.45. Their population ($N$) at the beginning of year 1 is 1,500. Calculate the regional population of Atlantic Bluefin Tuna at the end of year 1.

• A. 2,175 Tuna.
• B. 4,077 Tuna.
• C. 1,835 Tuna.
• D. 2,352 Tuna.

Using the same data as the previous question, calculate the regional population of Atlantic Bluefin Tuna at the end of year 2 & at the end of year 3.

• A. 3,689 Tuna; 5,786 Tuna.
• B. 3,670 Tuna; 5,505 Tuna.
• C. 3,204 Tuna; 4,056 Tuna.
• D. 4,085 Tuna; 5,963 Tuna.

A chameleon population ($N$) in a shrubland in Cyprus has an intrinsic growth rate ($r_{\max}$) of 0.38. You estimate the total population to be 550. Assuming this population has experienced uninhibited exponential growth, what was the chameleon population 3 years ago?

• A. 376 chameleons.
• B. 257 chameleons.
• C. 176 chameleons.
• D. 202 chameleons.